On Coalitional Manipulation for Multiwinner Elections: Shortlisting

Shortlisting of candidates--selecting a group of"best"candidates--is a special case of multiwinner elections. We provide the first in-depth study of the computational complexity of strategic voting for shortlisting based on the perhaps most basic voting rule in this scenario, l-Bloc (every voter approves l candidates). In particular, we investigate the influence of several tie-breaking mechanisms (e.g., pessimistic versus optimistic) and group evaluation functions (e.g., egalitarian versus utilitarian). Among other things, conclude that in an egalitarian setting strategic voting may indeed be computationally intractable regardless of the tie-breaking rule. Altogether, we provide a fairly comprehensive picture of the computational complexity landscape so far in the literature of this neglected scenario.


Introduction
A university wants to select the two favorite compositions in classical style to be played during the next graduation ceremony.The students were asked to submit their favorite compositions.Then a jury consisting of seven members (three juniors and four seniors) from the university staff selects from the six most frequently submitted compositions as follows: Each jury member approves two compositions and the two winners are those obtaining most of the approvals.The six options provided by the students are "Beethoven: Piano Concerto No. 5 (b 1 )", "Beethoven: Symphony No. 6 (b 2 )", "Mozart: Clarinet Concerto (m 1 )", "Mozart: Requiem K626 (m 2 )", "Uematsu: Final Fantasy (o 1 )", and "Badelt: Pirates of the Caribbean (o 2 )".The three junior jury members are excited about recent audio-visual presentation arts (both interactive and passive) and approve o 1 and o 2 .Two of the senior jury members are Mozart enthusiasts, and the other two senior jury members are Beethoven enthusiasts.Hence, when voting truthfully, two of them would approve the two Mozart compositions and the other two would approve the two Beethoven compositions.The winners of the selection process would be o 1 and o 2 , both receiving three approvals whereas every other composition receives only two approvals.
The senior jury members meet every Friday evening and discuss important academic issues including the graduation ceremony music selection processes, why movie background noise recently counts as classical music,1 and the influence of video games on the ability of making important decisions.During such a meeting they agreed that a graduation ceremony should always be accompanied by a composition of a first-class composer and by another composition of a second-class composer (two compositions of a first-class composer may overstrain students and junior researchers).Thus, finally all four senior jury members approve m 1 and b 1 making these two compositions playing during the graduation ceremony.
Already this little example above (which will be the base of our running example throughout the paper) illustrates important aspects of strategic voting in multiwinner elections.In case of coalitional manipulation for single-winner elections (where a coalition of voters casts untruthful votes in order to influence the outcome of an election; a topic which has been intensively studied in the literature [Brandt et al., 2016, Rothe, 2015]) one can always assume that a coalition of manipulators agrees on trying to make a distinguished alternative win the election.In case of multiwinner elections, however, already determining concrete possible goals of a coalition seems to be a non-trivial task: There may be exponentially many different outcomes which can be reached through strategic votes of the coalition members and each member could have its individual evaluation of these outcomes.
Multiwinner voting rules come up very naturally whenever one has to select from a large set of candidates a smaller set of "the best" candidates.Surprisingly, although at least as practically relevant as single-winner voting rules, the multiwinner literature is much less developed than the single-winner literature.In recent years (see a survey of Faliszewski et al. [2017a]), however, research into multiwinner voting rules, their properties, and algorithmic complexity grew significantly [Aziz et al., 2015, 2017a,b, Barberà and Coelho, 2008, 2010, Barrot et al., 2013, Betzler et al., 2013, Elkind et al., 2017, Faliszewski et al., 2016, 2017b, Meir et al., 2008, Obraztsova et al., 2013, Skowron, 2015, Skowron et al., 2015].When selecting a group of winning candidates, various goals can be interesting; namely, proportional representation, diversity, or a short list (see Elkind et al. [2017]).We focus on the last scenario.Here the goal is to select the best (say highest-scoring) group of candidates.
Shortlisting comes very naturally in the context of selection committees; for instance, for human resources departments that need to select, for a fixed number of positions, the best qualified applicants.A standard way of candidate selection in the context of shortlisting is to use scoring-based voting rules.We focus on the two most natural ones: SNTV (single non-transferable vote-each voter gives one point to one candidate) and ℓ-Bloc (each voter gives one point to each of ℓ different candidates, so SNTV is the same as 1-Bloc).Obviously, for such voting rules it is trivial to determine the score of each individual candidate.The main goal of our work is to model and under-stand coalitional manipulation in a computational sense-that is, to introduce a formal description of how a group of manipulators can influence the election outcome by casting strategic votes and whether it is possible to find an effective strategy for the manipulators to change the outcome in some desired way.In this fashion, we complement well-known work: manipulation for singlewinner rules initiated by Bartholdi III et al. [1989], coalitional manipulation for single-winner rules initiated by Conitzer et al. [2007], and (non-coalitional) manipulation for multiwinner rules initiated by Meir et al. [2008].In coalitional manipulation scenarios, given full knowledge about other voters' preferences, one has a set of manipulative voters who want to influence the election outcome in a favorable way by casting their votes strategically.
To come up with a useful framework for coalitional manipulation for multiwinner elections, we first have to identify the exact mathematical model and questions to be asked.The straightforward extensions of coalitional manipulation for single-winner elections or (non-coalitional) manipulation for multiwinner elections do not fit.Extending the single-winner variant directly, one would probably assume that the coalition agrees on making a distinguished candidate part of the winners or that the coalition agrees on making a distinguished candidate group part of the winners.The former is unrealistic because in multiwinner settings one typically cares about more than just one candidateespecially in shortlisting it is natural that one wants rather some group of "similarly good" candidates to become part of winners instead of only one representative of such a group.Agreeing on a distinguished candidate group to be part of the winners is also problematic since there may be exponentially many "equally good" candidate groups for the coalition.Notably, this was not a problem in the single-winner case; there, one can test for a successful manipulation towards each possible candidate avoiding an exponential increase of the running time (compared to the running time of such a test for a single candidate).The single-manipulator model for multiwinner rules of Meir et al. [2008] is a helpful first step: the manipulator specifies the utility of each candidate; the utility for a candidate group is obtained by adding up the utilities of each group member.Aggregating utilities, however, becomes non-trivial for a coalition of manipulators which may have totally different utilities for single candidates but still have strong incentives to work together (e.g., as we have seen in our introductory example).Besides formalizing this either in a utilitarian or egalitarian way, our modeling also aims to distinguish between optimistic and pessimistic manipulators.In case of a tie, optimistic manipulators assume that the tie is broken in favor of them while pessimistic ones assume the opposite scenario.It turns out that this difference in manipulators' behavior leads to significant differences in terms of computational complexity.Technically, analyzing the issues discussed above requires to study tie-breaking mechanisms and (winning) group evaluation functions.
Related Work.To the best of our knowledge, there is no previous work on coalitional manipulation in the context of multiwinner elections.We refer to recent textbooks for an overview of the huge literature on single-winner (coalitional) manipulation [Brandt et al., 2016, Rothe, 2015].Most relevant to this paper, Lin [2011] showed that coalitional manipulation in single-winner elections under ℓ-Approval is solvable in linear time by a greedy algorithm.Meir et al. [2008] introduced (noncoalitional) manipulation for multiwinner elections.While identifying manipulation for several voting rules as NP-hard problems, they showed that manipulation remains polynomial-time solvable for Bloc (which can be interpreted as a multiwinner equivalent of 1-Approval).Obraztsova et al. [2013] extended the latter result for different tie-breaking strategies and identified further tractable special cases of multiwinner scoring rules but conjectured manipulation to be hard in general for (other) scoring rules.Summarizing, Bloc is simple but comparably well-studied and was, hence, selected as a showcase for our study of the presumably computationally harder coalitional manipulation.
Organization.Section 2 introduces basic notations and formal concepts.In Section 3, we develop our model for coalitional manipulation in multiwinner elections.Its variants respect different ways of evaluating candidate groups (utilitarian vs. egalitarian) and two kinds of manipulators behavior (optimistic vs. pessimistic).In Section 4, we present algorithms and complexity results for computing the output of several tie-breaking rules that allow to model optimistic and pessimistic manipulators.In Section 5, we formally define the coalitional manipulation problem and explore its computational complexity using ℓ-Bloc as a showcase.We refer to our conclusion and Table 1 (Section 6) for a detailed overview of our findings.

Preliminaries
For a positive integer n, let [n] := {1, 2, . . ., n}.We use the toolbox of parameterized complexity [Cygan et al., 2015, Downey and Fellows, 2013, Flum and Grohe, 2006, Niedermeier, 2006] to analyze the computational complexity of our problems in a fine-grained way.To this end, we always identify a parameter ρ that is typically a positive integer.We call a problem parameterized by ρ fixed-parameter tractable O(1) time, where |I| is the size of a given instance encoding, ρ is the value of the parameter, and f is an arbitrary computable (typically super-polynomial) function.To preclude fixed-parameter tractability, we use an established complexity hierarchy of classes of parameterized problems, It is widely believed that all inclusions are proper.The notions of hardness for parameterized classes are defined through parameterized reductions similar to classical polynomial-time manyone reductions-in this work, it suffices to ensure that the value of the parameter in the problem we reduce to depends only on the value of the parameter of the problem we reduce from.Occasionally, we use a combined parameter ρ ′ + ρ ′′ which is a more explicit way of expressing a parameter ρ = ρ ′ + ρ ′′ .
An election (C, V ) consists of a set C of m candidates and a multiset V of n votes.Votes are linear orders over C-for example, for C = {c 1 , c 2 , c 3 } we write c 1 ≻ v c 2 ≻ v c 3 to express that candidate c 1 is the most preferred and candidate c 3 is the least preferred according to vote v.We write ≻ if the corresponding vote is clear from the context.
A multiwinner voting rule2 is a function that, given an election (C, V ) and an integer k ∈ [|C|], outputs a family of co-winning size-k subsets of C representing the co-winning k-excellencegroups.We use k-egroup as an abbreviation for k-excellence-group.The reason we do not use the established term "committee" is that in shortlisting applications "committee" traditionally rather refers to voters and not to candidates.
We consider scoring rules which assign points to candidates based on their positions in the votes.By score(c), we denote the total number of points that candidate c obtains, and we use score V ′ (c) when restricting the election to a subset V ′ ⊂ V of voters.A (multiwinner) scoring rule selects a family X of co-winning k-egroups with the maximum total sum of scores.It holds that X ∈ X if and only if ∀c ∈ X, c ′ ∈ C \ X : score(c) ≥ score(c ′ ).We focus on the family of ℓ-Bloc multiwinner voting rules which assign, for each vote, one point to each of the top ℓ < |C| candidates. 3xample 1. Referring back to our introductory example, we have a set of candidates The voters v y 1 , v y 2 , and v y 3 represent the three junior jury members, whereas v b 1 , v b 2 and v m 1 , v m 2 represent, respectively, the Beethoven and Mozart enthusiasts among the senior jury members.In the example, we described a way of manipulating the election by the senior jury members which leads to selecting two music pieces.There are several ways to illustrate this manipulation using our model.Below we present one of the possible sets of casted votes that represents the manipulated election: Following the introductory example, we are choosing an egroup of size k = 2. Using the Bloc multiwinner voting rules (which coincides with our introductory example), the winning 2-egroup consist of candidates b 1 and m 1 .However, under the SNTV voting rule the situation would change, and the winners would be o 1 and b 1 .SNTV and Bloc alike output a single winning egroup in this example, and thus tie-breaking is negligible.
To select a single k-egroup from the set of co-winning k-egroups one has to consider tiebreaking rules.A multiwinner tie-breaking rule is a mapping that, given an election and a family of co-winning k-egroups, outputs a single k-egroup.Among them, there is a set of natural rules that is of particular interest in order to model the behavior of manipulative voters.Indeed, in case of a single manipulator both pessimistic tie-breaking as well as optimistic tie-breaking have been considered in addition to lexicographic and randomized tie-breaking [Meir et al., 2008, Obraztsova et al., 2013].To model optimistic and pessimistic tie-breaking in a meaningful manner 4 , we use the model introduced by Obraztsova et al. [2013] in which a manipulative voter v is described not only by the preference order ≻ v of the candidates but also by a utility function u : C → N. To cover this in the tie-breaking process, coalition-specific tie-breaking rules get-in addition to the original election, the manipulators' votes, and the co-winning excellence-groups-the manipulators' utility functions in the input.The formal implementations of these rules and their properties are discussed in Subsection 3.2.

Model for Coalitional Manipulation
In this section, we formally define and explain our model and the respective variants.To this end, we discuss how we evaluate a k-egroup in terms of utility for a coalition of manipulators and introduce tie-breaking rules that model optimistic or pessimistic viewpoints of the manipulators.

Evaluating k-egroups
As already discussed in the introduction, one should not extend the model of coalitional manipulation for single-winner elections to multiwinner elections in the simplest way (e.g., by assuming that the manipulators agree on some distinguished candidate or on some distinguished egroup).Instead, we follow Meir et al. [2008] and assume that we are given a utility function over the candidates for each manipulator and a utility level which, if achieved, indicates a successful manipulation.Meir et al. [2008] compute the utility of an egroup by summing up the utility values the manipulator assigns to each member of the egroup.
At first glance, summing up the utility values assigned by each manipulator to each member of an egroup seems to be the most natural extension for a coalition of manipulators.However, this utilitarian variant does not guarantee single manipulators to gain non-zero utility.In extreme cases it could even happen that some manipulator is worse off compared to voting sincerely, as demonstrated in Example 2.

Example 2. Consider the election
Additionally, consider two manipulators, u 1 and u 2 , that report utilities to the candidates as depicted in the table below.
Let us analyze the winning 2-egroup under the SNTV voting rule.Observe that if the manipulators vote sincerely, then together they give one point to b 1 and one to m 2 (one point from each manipulator).Combining the manipulators' votes with the non-manipulative ones, the winning 2-egroup consists of candidates o 1 and m 2 that both have score two; no other candidate has greater or equal score, so tie-breaking is unnecessary.The value of such a group is equal to seven according to the utilitarian evaluation variant.Manipulator u 2 's utility is seven.However, both manipulators can do better by giving their points to candidate b 1 .Then, the winners are candidates o 1 and b 1 , giving the total utility of 11 (according to the utilitarian variant).Observe that in spite of growth of the total utility, the utility value gained by u 2 , which is one, is lower than in the case of sincere voting.
In Example 2 manipulator u 2 devotes its satisfaction to the utilitarian satisfaction of the group of the manipulators; that is, u 2 is worse off voting strategically compared to voting sincerely.Despite this issue, however, the utilitarian viewpoint can be justified if the manipulators are able to compensate such losses of utility of some manipulators, for example, by paying money to each other.For cases where manipulators cannot do that, we introduce two egalitarian evaluation variants.The (egroup-wise) egalitarian variant aims at maximizing the minimum satisfaction of the manipulators with the whole k-egroup.The candidate-wise egalitarian variant aims at maximizing the manipulators' satisfaction resulting from the summation of the minimum satisfactions every single candidate contributes.We do not distinguish "candidate-wise utilitarian" variant since this variant would be equivalent to the (regular) utilitarian variant.
We formalize the described variants of k-egroup evaluation (for r manipulators) with Definition 1.
Intuitively, these functions determine the utility of a k-egroup S according to, respectively, the utilitarian and the egalitarian variants of evaluating S by a group of r manipulators (identifying manipulators with their utility functions).We omit subscript U when U is clear from the context.To illustrate Definition 1 we apply it in Example 3.
Example 3. Consider our example set of candidates C = {b 1 , b 2 , m 1 , m 2 , o 1 , o 2 } and two manipulators u 1 , u 2 whose utility functions over the candidates are depicted in the table below.
Analyzing Example 3, we observe that we can compute the utilitarian value of an egroup by summing up the overall utilities each candidate contributes to all manipulators.Analogously, we can deal with the candidate-wise egalitarian variant by taking the minimum utility associated to each candidate as the overall utility of this candidate.In both variants we obtain a single utility function.
Observation 1.Without loss of generality, one can assume that there is a single non-zero valued utility function over the candidates under the utilitarian or candidate-wise egalitarian evaluation.

Breaking Ties
Before formally defining our tie-breaking rules, we briefly discuss some necessary notation and central concepts.Consider an election (C, V ), a size k for the egroup to be chosen, and a scoringbased multiwinner voting rule R. We can partition the set of candidates C into three sets C + , P , and C − as follows: The set C + contains the confirmed candidates, that is, candidates that are in all co-winning k-egroups.The set P contains the pending candidates, that is, candidates that are only in some co-winning k-egroups.The set C − contains the rejected candidates, that is, candidates that are in no co-winning k-egroup.Observe that |C + | ≤ k, |C + ∪ P | ≥ k, and that every candidate from P ∪ C − receives fewer points than every candidate from C + .Additionally, all candidates in P receive the same number of points.
We define the following families of tie-breaking rules which are considered in this work.In order to define optimistic and pessimistic rules, we assume that in addition to C + , P , and k, we are given a family of utility functions which are used to evaluate the k-egroups as discussed in Subsection 3.1.
Lexicographic F lex .A tie-breaking F belongs to F lex if and only if ties are broken lexicographically with respect to some predefined order > F of the candidates from C. That is, F selects all candidates from C + and the top k − |C + | candidates from P with respect to > F .
Optimistic F eval opt , eval ∈ {util, egal, candegal}.A tie-breaking belongs to F eval opt if and only if it always selects some k-egroup S such that C + ⊆ S ⊆ (C + ∪ P ) and there is no other k-egroup S ′ with C + ⊆ S ′ ⊆ (C + ∪ P ) and eval(S ′ ) > eval(S).
Pessimistic F eval pess , eval ∈ {util, egal, candegal}.A tie-breaking belongs to F eval pess if and only if it always selects some k-egroup S such that C + ⊆ S ⊆ (C + ∪ P ) and there is no other k-egroup S ′ with C + ⊆ S ′ ⊆ (C + ∪ P ) and eval(S ′ ) < eval(S).

Limits of Lexicographic Tie-Breaking
From the above discussion, we can conclude that lexicographic tie-breaking is straightforward in the case of scoring-based multiwinner voting rules.Basically any subset of the desired cardinality from the set of pending candidates can be chosen.In particular, the best pending candidates with respect to the given order can be chosen.We remark that applying lexicographic tie-breaking may be more complicated for general multiwinner voting rules.
It remains to be clarified whether one can find a reasonable order of the pending candidates in order to model optimistic or pessimistic tie-breaking rules in a simple way.We show that this is possible for every F eval bhav , eval ∈ {util, candegal}, bhav ∈ {opt, pess}, using the fact that in these cases we can safely assume that there is only one non-zero valued utility function (see Observation 1).On the contrary, there is a counterexample for eval = egal and bhav ∈ {opt, pess}.On the way to prove these claims we need to formally define what it means that one family of tiebreaking rules can be used to simulate another family of tie-breaking rules.Definition 2. Let C be a fixed set of candidates.Let C + be a set of confirmed and P be a set of pending candidates.Let U be a family of utility functions and k be a size of an egroup.Consider some subset P of the set {C + , P , U, k}.For two families F and F ′ of tie-breaking rules we say that F can P-simulate F ′ if there are some rules F ∈ F and F ′ ∈ F ′ such that F and F ′ have the same output for all possible elections assuming all elements from P together with set C are fixed.We call rule F a P-simulator.
At first glance Definition 2 might seem overcomplicated.However, it is tailored to grasp different degrees of simulation possibilities.On the one hand, one can always find a lexicographic order and use it for breaking ties if all: confirmed candidates, pending candidates, utility functions, and the size of an egroup are known.Thus, one needs some flexibility in the definition of simulation for it to be non-trivial.On the other hand, it is somewhat obvious that without fixing the utility functions, one cannot simulate optimistic or pessimistic tie-breaking rules.In other words, we have: Observation 2. Let bhav ∈ {opt, pess} and eval ∈ {util, candegal, egal}.The family of lexicographic tie-breaking rules does not {C + , P, k}-simulate F eval bhav .Next, we show that for some cases it is sufficient to fix just the utility functions in order to simulate optimistic or pessimistic tie-breaking rules (see Proposition 1).For other cases, however, one has to fix all: confirmed candidates, pending candidates, utility functions, and the size of an egroup (see Proposition 2).
Proposition 1.Let C be a set of candidates, U be a family of utility functions, bhav ∈ {opt, pess}, and eval ∈ {util, candegal}.Let |C| = m and |U | = r.Then the family of lexicographic tiebreaking rules F lex can {U }-simulate F eval bhav , and a {U }-simulator Proof.Recall from Observation 1 that if eval ∈ {util, candegal}, then there is always a set of utility functions with just one non-zero valued utility function u ′ that is equivalent to U .Hence, we compute such a function u ′ in O(m • r) time as follows: In the utilitarian case, function u ′ assigns every candidate the sum of utilities the manipulators give to the candidate.Considering the candidate-wise egalitarian evaluation, function u ′ assigns every candidate the minimum utility value among utilities given to the candidate over all manipulators.We say an order > F of the candidates is consistent with some utility function u if c > F c ′ implies u(c) ≥ u(c ′ ) for optimistic tie-breaking and c > F c ′ implies u(c) ≤ u(c ′ ) for pessimistic tie-breaking.Any lexicographic tie-breaking rule defined by an order > F that is consistent with the utility function u ′ simulates F eval bhav .We compute a consistent order by sorting the candidates according to u ′ in O(m • log m) time.
Proposition 1 describes a strong feature of optimistic utilitarian and candidate-wise egalitarian tie-breaking and their pessimistic variants.Intuitively, the proposition says that for these tiebreaking mechanisms one can compute a respective linear order of candidates.Then one can forget all the details of the initial tie-breaking mechanism and use the order to determine winners.The order can be computed even without knowing the details of an election.Unfortunately, the simulation of pessimistic and optimistic egalitarian tie-breaking turns out to be more complicated.
Proposition 2. Let C be a set of candidates, U be a family of utility functions, C + be a set of confirmed candidates, P be a set of pending candidates, and k be a size of an egroup.For each P ⊆ {C + , P, U, k}, |P| < 4, the lexicographic tie-breaking family of rules does not P-simulate F egal bhav assuming bhav ∈ {opt, pess}.
Proof.From Observation 2 we already know that the family of lexicographic tie-breaking rules cannot {C + , P, k}-simulate the family of egalitarian pessimistic tie-breaking rules or the family of egalitarian optimistic tie-breaking rules.
Next, we build one counterexample for each of the remaining size-three subsets of {C + , P, U, k} to show our theorem.To this end, let us fix a set of candidates C = {b 1 , b 2 , m 1 , m 2 , o 1 , o 2 } (compatible with our running example) and a family U = {u 1 , u 2 } of utility functions as depicted in the table below.
First, we prove that the family We consider the optimistic variant of egalitarian tie-breaking, so we are searching for the best possible 1-egroup.Looking at the values of U , we see that candidate m 1 gives the best possible egalitarian evaluation value which is four.This means that a {C + , P, U }-simulator F ∈ F lex has to use an order where m 1 precedes both b 1 and m 2 .However, it turns out that if we set k = 2, then the best 2-egroup consists exactly of candidates b 1 and m 2 .This leads to a contradiction because now candidates b 1 and m 2 should precede m 1 in F 's lexicographic order.Consequently, family F lex does not {C + , P, U }-simulate F egal opt .Using the same values of utility functions and the same sequence of the values of k we get a proof for the pessimistic variant of egalitarian evaluation.
Second, we prove that the family F lex cannot {P, k, U }-simulate F egal bhav (i.e., set C + is unfixed) for bhav ∈ {opt, pess}.This time, we fix P = C \ {o 1 , o 2 }, k = 2.We construct the first case by setting C + = {o 1 }.Using the fact that in both functions candidate o 1 has utility zero, we choose exactly the same candidate as in the proof of {C + , P, U }-simulation for the case k = 1; that is, for the optimistic variant, the winning 2-egroup is m 1 and o 1 .Consequently, this leads to the fact that m 1 precedes b 1 and m 2 in the potential {P, k, U }-simulator's lexicographic order.Towards a contradiction, we set C + = ∅.The situation is exactly the same as in the proof of the {C + , P, U }simulation case.Now, the winning 2-egroup consists of b 1 and m 2 which ends the proof for the optimistic case.By almost the same argument, the result holds for the pessimistic variant.
Finally, we prove that the family F lex cannot {C + , k, U }-simulate F egal bhav (i.e., set P is unfixed) for bhav ∈ {opt, pess}.We fix C + = ∅, k = 2.For the first case we pick P = {b 2 , m 1 , m 2 }.The best egalitarian evaluation happens for the 2-egroup consisting of b 2 and m 1 .This imposes that, in the potential {C + , k, U }-simulator's order, b 2 and m 1 precede the remaining candidates (in particular, m 1 precedes m 2 ).However, for P = C the best 2-egroup changes to that consisting of b 1 and m 2 which gives a contradiction (m 2 precedes m 1 ).As in the previous cases, the same argument provides a proof for the pessimistic variant.
Proposition 2 implies that pessimistic and optimistic egalitarian tie-breaking cannot be simulated without having full knowledge about an election.In terms of computational complexity, however, pessimistic egalitarian tie-breaking remains tractable whereas optimistic egalitarian tiebreaking is intractable.In the next section, among other things, we show that the egalitarian opti-mistic tie-breaking significantly differs from the egalitarian pessimistic tie-breaking with respect of hardness of computing winners.

Complexity of Tie-Breaking
It is natural to ask whether the tie-breaking rules proposed in Subsection 3.2 are practical in terms of their computational complexity.If not, then there is little hope for coalitional manipulation because tie-breaking is a subtask to be solved by the manipulators.
Clearly, we can apply every lexicographic tie-breaking rule that is defined through some predefined order of the candidates in linear time.Hence, we focus on the rules that model optimistic or pessimistic manipulators.To this end, we analyze the following computational problem.F eval bhav -TIE-BREAKING (F eval bhav -TB), eval ∈ {util, egal, candegal}, bhav ∈ {opt, pess} Input: A set of candidates C partitioned into a set P of pending candidates and a set C + of confirmed candidates, the size k of the excellence-group such that |C + | < k < |C|, a family of manipulators' utility functions U = {u 1 , u 2 , . . ., u r } where u i : C → N, and a non-negative, integral evaluation threshold q.Question: Is there a size-k set S ⊆ C such that S is selected according to F eval bhav , C + ⊆ S, and eval(S) ≥ q?
Naturally, we may assume that the number of candidates and the number of utility functions are polynomially bounded in the size of the input.However, both the evaluation threshold and the utility function values are encoded in binary.
Note that an analogous problem has not been considered for single-winner elections since, for single-winner elections, optimistic and pessimistic tie-breaking rules can be easily simulated by straightforward lexicographic tie-breaking rules.

Utilitarian and Candidate-Wise Egalitarian: Tie-Breaking Is Easy
As a warm-up, we observe that tie-breaking can be applied and evaluated efficiently if the k-egroups are evaluated according to the utilitarian or candidate-wise egalitarian variant.The corresponding result follows almost directly from Proposition 1.
Corollary 1.Let m denote the number of candidates and r denote the number of manipulators.Then one can solve Proof.The algorithm works in two steps.First, compute a lexicographic tie-breaking rule F lex that simulates F eval bhav in O(m • (r + log m)) time as described in Proposition 1.Second, apply tiebreaking rule F lex , and evaluate the resulting k-egroup in O(k•r) time.The running time of applying a lexicographic tie-breaking rule is linear with respect to the input length (see Subsection 3.3).

Egalitarian: Being Optimistic Is Hard
In this subsection, we consider the optimistic and pessimistic tie-breaking rules when applied for searching a k-egroup evaluated according to the egalitarian variant.First, we show that applying and evaluating egalitarian tie-breaking is computationally easy for pessimistic manipulators but computationally intractable for optimistic manipulators even if the size of the egroup is small.Being pessimistic, the main idea is to "guess" the manipulator that is least satisfied and select the candidates appropriately.We show the computational hardness of the optimistic case via a reduction from the W[2]-complete SET COVER problem parameterized by solution size [Downey and Fellows, 2013].
Theorem 1.Let m denote the number of candidates, r denote the number of manipulators, q denote the evaluation threshold, and k denote the size of an egroup.Then one can solve -hard when parameterized by k even if q = 1 and every manipulator only gives either utility one or zero to each candidate.
Proof.For the pessimistic case, it is sufficient to "guess" the least satisfied manipulator x by iterating through r possibilities.Then, select the |C + | − k pending candidates with the smallest total utility for this manipulator in O(m log m) time.Finally, comparing the k-egroup with the worst minimum satisfaction over all manipulators with the lower bound q on satisfaction level given in the input solves the problem.
We prove the hardness for the optimistic case reducing from the W[2]-hard SET COVER problem which, given a collection S = {S 1 , S 2 , . . ., S m } of subsets of universe X = {x 1 , x 2 , . . ., x n } and an integer h, asks whether there exists a family S ′ ⊆ S of size at most h such that S∈S ′ S = X.Let us fix an instance I = (X, S, h) of SET COVER.To construct an F egal opt -TIE-BREAKING instance, we introduce pending candidates P = {c 1 , c 2 , . . ., c m } representing subsets in S and manipulators u 1 , u 2 , . . ., u n representing elements of the universe.Note that there are no confirmed and rejected candidates.Each manipulator u i gives utility one to candidate c j if set S j contains element x i and zero otherwise.We set the excellence-group size k := h and the threshold q to be 1.
We observe that if there is a size-k subset S ⊆ P such that min i∈[n] s∈S u i (s) ≥ 1, then there exists a family S ′ -consisting of the sets represented by candidates in S-such that each element of the universe belongs to the set S∈S ′ S. On the contrary, if we cannot pick a group of candidates of size k for which every manipulator's utility is at least one, then instance I is a 'no' instance.This follows from the fact that for each size-k subset S ⊆ P there exists at least one manipulator u * for whom s∈S u * (s) = 0.This translates to the claim that there exists no size-h subset S ′ ⊆ S such that all elements in X belong to the union of the sets in S ′ .
Since SET COVER is NP-hard and W [2]-hard with respect to parameter h, we obtain that our problem is also NP-hard and W [2]-hard when parameterized by the size k of an excellence-group.
Inspecting the W[2]-hardness proof of Theorem 1, we learn that a small egroup size (alone) does not make F egal opt -TIE-BREAKING computationally tractable even for very simple utility functions.Next, using a parameterized reduction from the W[1]-complete MULTICOLORED CLIQUE problem [Fellows et al., 2009], we show that there is still no hope for fixed-parameter tractability (under standard assumptions) even for the combined parameter "number of manipulators and egroup size"; intuitively, this parameter covers situations where few manipulators are going to influence an election for a small egroup.
Theorem 2. Let k denote the size of an egroup and r denote the number of manipulators.Then, parameterized by r + k, F egal opt -TIE-BREAKING is W[1]-hard.Proof.We describe a parameterized reduction from the W[1]-hard MULTICOLORED CLIQUE problem [Fellows et al., 2009].In this problem, given an undirected graph G = (V, E), a non-negative integer h, and a vertex coloring φ : V → {1, 2, . . ., h}, we ask whether graph G admits a colorful h-clique, that is, a size-h vertex subset Q ⊆ V such that the vertices in Q are pairwise adjacent and have pairwise distinct colors.Without loss of generality, we assume that the number of vertices of each color is the same; to be referred as y in the following.
and let E(i, j) = {e i,j 1 , e i,j 2 , . . ., e i,j |E(i,j)| }, defined for i, j ∈ [h], i < j, denote the set of edges that connect a vertex of color i to a vertex of color j.
Candidates.We create one confirmed candidate c * and |V | + |E| pending candidates.More precisely: for each ℓ ∈ [y], we create one vertex candidate a i ℓ for each vertex v i ℓ ∈ V (i), i ∈ [h] and for each i, j ∈ [h] such that i < j we create one edge candidate b i,j t for each edge e i,j t ∈ E(i, j), t ∈ [E(i, j)].We set the size k of the egroup to h+ h 2 +1 and set the evaluation threshold q := y+1.Next, we describe the manipulators and explain the high-level idea of the construction.
Manipulators and main idea.Our construction will ensure that there is a k-egroup X with c * ∈ X and egal(X) ≥ q if and only if X contains h vertex candidates and h 2 edge candidates that encode a colorful h-clique.To this end, we introduce the following manipulators.1.For each color i ∈ [h], there is a color manipulator µ i ensuring that the k-egroup contains a vertex candidate a i z i corresponding to a vertex of color i.Herein, variable z i denotes the id of the vertex candidate (resp.vertex) that is selected for color i.

Verification manipulator ν
, utility q − ℓ for each candidate corresponding to an edge that connects vertex v i ℓ to a vertex of color j, and utility zero for the remaining candidates.
4. Verification manipulator ν ′ i,j , i, j ∈ [h], i = j, has utility q − ℓ for candidate a i ℓ , ℓ ∈ [y], utility ℓ for each candidate corresponding to an edge that connects vertex v i ℓ , to a vertex of color j, and utility zero for the remaining candidates.
Correctness.We argue that the graph G admits a colorful clique of size h if and only if there is a k-egroup X with c * ∈ X and egal(X) ≥ q.
Suppose that there exists a colorful clique H of size h.Create the k-egroup X as follows.Start with {c * } and add every vertex candidate that corresponds to some vertex of H and every edge candidate that corresponds to some edge of H.Each color manipulator and color pair manipulator receives total utility y + 1, because H contains, by definition, one vertex of each color and one edge connecting two vertices for each color pair.It is easy to verify that the verification manipulator ν i,j must receive utility ℓ from a vertex candidate and utility q − ℓ from an edge candidate and that the verification manipulator ν ′ i,j must receive utility q − ℓ from a vertex candidate and utility ℓ from an edge candidate.Thus, egal(X) = q = y + 1.
Suppose that there exists a k-egroup X ⊆ C such that egal(X) ≥ q.Since each color manipulator cannot achieve utility y + 1 unless c * belongs to the winning k-egroup, it follows that c * ∈ X.Because each color manipulator µ i receives total utility at least y + 1, X must contain some vertex candidate a i z i corresponding to a vertex of color i for some z i ∈ [y].We say that X selects vertex v i z i .Since each color pair manipulator µ i,j receives total utility at least y + 1, X must contain some edge candidate b i,j z i,j corresponding to an edge connecting a vertex of color i and a vertex of color j for some z i,j .We say that X selects edge e i,j z i,j .We implicitly assumed that each color manipulator and color pair manipulator contributes exactly one selected candidate to X.This assumption is true because there are exactly k − 1 such manipulators and each needs to select at least one candidate; hence, X is exactly of the desired size.In order to show that the corresponding vertices and edges encode a colorful h-clique, it remains to show that no selected edge is incident to a vertex that is not selected.Assume towards a contradiction that X selects an edge e i,j z i,j and some vertex v i z i / ∈ e i,j z i,j .However, either verification manipulator ν i,j or verification manipulator ν ′ i,j receives the total utility at most q − 1; a contradiction.Finally, devising an ILP formulation, we show that F egal opt -TIE-BREAKING becomes fixed-parameter tractable when parameterized by the combined parameter "number of manipulators and number of different utility values".This parameter covers situations with few manipulators that have simple utility functions; in particular, when few voters have 0/1 utility functions.Together with Theorem 1 and Theorem 2, following Theorem 3 shows that neither few manipulators nor few utility functions make F egal opt -TB fixed-parameter tractable, but only combining these two parameters allows us to deal with the problem in FPT time.
Theorem 3. Let u diff denote the number of different utility values and r denote the number of manipulators.Then, parameterized by r+u diff , F egal opt -TIE-BREAKING is fixed-parameter tractable.
Proof.We define the type of any candidate c i to be the size-r vector t = (u 1 (c i ), u 2 (c i ), . . ., u r (c i )).
Let T = {t 1 , t 2 , . . ., t |T | } be the set of all possible types.Naturally, the size of T is upper-bounded by u r diff .We denote the set of candidates of type t i ∈ T by T i .Now, the ILP formulation of the problem using exactly |T | + 1 variables reads as follows.For each type t i ∈ |T |, we introduce variable x i indicating the number of candidates of type t i in an optimal k-egroup.We use variable s to represent the minimal value of the total utility achieved by manipulators.We define the following ILP with the goal to maximize s (indicating the utility gained by the least satisfied manipulator) subject to: (1) ∀ℓ ∈ [r] : Constraint set (1) ensures that the solution is achievable with given candidates.Constraint ( 2) guarantees a choice of an egroup of size k.The last set of constraints imposes that s holds at most the minimal value of the total utility gained by manipulators.By a famous result of Lenstra [1983], this ILP formulation with the number of variables bounded by u r diff +1 yields that F egal opt -TIE-BREAKING is fixed-parameter tractable when parameterized by the combined parameter r + u diff .

Complexity of Coalitional Manipulation
In the previous section, we have seen that breaking ties optimistically or pessimistically-an essential subtask to be solved by the manipulators-can be computationally challenging; in most cases, however, this problem turned out to be computationally easy.In this section, we move on to our full framework and analyze the computational difficulty of voting strategically for a coalition of manipulators.To this end, we formalize our central computational problem.Let R be a multiwinner voting rule and let F be a multiwinner tie-breaking rule.R-F-eval-COALITIONAL MANIPULATION (R-F-eval-CM), eval ∈ {util, egal, candegal} Input: An election (C, V ), a searched egroup size k < |C|, r manipulators represented by their utility functions U = {u 1 , u 2 , . . ., u r } such that ∀ i∈[r] u i : C → N, and a non-negative, integral evaluation threshold q.Question: Is there a size-r multiset W of manipulative votes over C such that k-egroup S ⊂ C wins the election (C, V ∪ W ) under R and F, and eval(S) ≥ q?
The R-F-eval-CM problem is defined very generally; namely, one can consider any multiwinner voting rule R (in particular, any single-winner voting rule is a multiwinner voting rule with k = 1).In our paper, however, we focus on ℓ-Bloc; hence, from now on, we narrow down our analysis of R-F-eval-CM to the ℓ-Bloc-F-eval-CM problem.
As a step on the way to show our results we also use a restricted version of ℓ-Bloc-F-eval-COALITIONAL MANIPULATION that we call ℓ-Bloc-F-eval-COALITIONAL MANIPULATION with consistent manipulators.In this variant, the input stays the same, but all manipulators cast exactly the same vote to achieve the objective.
To increase readability, we decided to represent manipulators by their utility functions.As a consequence, we frequently use, for example, u 1 referring to the manipulator itself, even if we do not care about the values of utility function u 1 at the moment of usage.In the paper, we also stick to the term "voters" meaning the set V of voters of an input election.We never call manipulators "voters"; however, we speak about the manipulative votes they cast.
As for the encoding of the input of R-F-eval-CM, we use a standard assumption; namely, that the number of candidates, the number of voters, and the number manipulators are polynomially upper-bounded in the size of the input.Analogously to F eval bhav -TIE-BREAKING, both the evaluation threshold and the utility function values are encoded in binary.

Utilitarian and Candidate-Wise Egalitarian: Manipulation Is Tractable
We show that ℓ-Bloc-F-eval-COALITIONAL MANIPULATION can be solved in polynomial time for any constant ℓ ∈ N, any eval ∈ {util, candegal}, and any F ∈ {F lex , F eval opt , F eval pess }.For Bloc (i.e., ℓ = k), we give a quadratic-time algorithm with respect to the size of the input.We start with an algorithm for ℓ-Bloc-F-eval-CM with consistent manipulators and show that this algorithm solves also Bloc-F-eval-CM.The algorithm "guesses" the minimum score among all members of the winning egroup and then carefully (with respect to the tie-breaking method) selects the best candidates that can reach this score.
In several proofs in Subsection 5.1 we use the value of a candidate for manipulators (coalition) and say that a candidate is more valuable or less valuable than another candidate.Although we cannot directly measure the value of a candidate for the whole manipulators' coalition in general, thanks to Observation 1, we can assume a single non-zero utility function when discussing the utilitarian and candidate-wise egalitarian variants.Thus, assigning a single value to each candidate is justified.
Proposition 3. Let m denote the number of candidates, n denote the number of voters, and r denote the number of manipulators.Then one can solve ℓ-Bloc-F-eval-COALITIONAL MANIPULATION with consistent manipulators in O(m(m + r + n)) time for any eval ∈ {util, candegal} and F ∈ {F lex , F eval opt , F eval pess }.
Proof.Consider an instance of ℓ-Bloc-F lex -eval-CM with consistent manipulators with an election E = (C, V ) where C is a candidate set and V is a multiset of non-manipulative votes, r manipulators, an egroup size k, and a lexicographic order > F used by F lex to break ties.In essence, we introduce a constrained solution form called a canonical solution and argue that it is sufficient to analyze only this type of solutions.Then we provide an algorithm that efficiently seeks for an optimal canonical solution.
At the beginning, we observe that when manipulators vote consistently, then we can arrange the top ℓ candidates of a manipulative vote in any order.Hence, the solution to our problem is a size-ℓ subset (instead of an order) of candidates which we call a set of supported candidates; we call each member of this set a supported candidate.
Strength order of the candidates.Additionally, we introduce a new order > S of the candidates.It sorts them descendingly with respect to the score they receive from voters and, as a second criterion, according to the position in > F .Intuitively, the easier it is for some candidate to be a part of a winning k-egroup, the higher is the candidate's position in > S .As a consequence, we state Claim 1.
Claim 1.Let us fix an instance of ℓ-Bloc-F lex -eval-CM with consistent manipulators and a solution X which leads to a winning k-egroup S. For every supported (resp.unsupported) candidate c, the following holds: • If c is part of the winning k-egroup, then every supported (resp.unsupported) predecessor of c, according to > S , belongs to S.
• If c is not part of the winning k-egroup, then every supported (resp.unsupported) successor of c, according to > S , does not belong to S.
Claim 1 justifies thinking about > S as a "strength order"; hence, in the proof we use the terms stronger and weaker candidate.Using Claim 1, we can fix some candidate c as the weakest in the winning k-egroup and then infer candidates that have to be and that cannot be part of this k-egroup.
To formalize this idea, we introduce the concept of a canonical solution.
Canonical solutions.Assuming the case where k ≤ ℓ, we call a solution X leading to a winning k-egroup S canonical if all candidates of the winning egroup are supported; that is, S ⊆ X.In the opposite case, k > ℓ, solution X is canonical if X ⊂ S and X is a set of the ℓ weakest candidates in S. For the latter case, the formulation describes the solution which favors supporting weaker candidates first and ensures that no approval is given to a candidate outside the winning k-egroup.
Canonical solutions are achievable from every solution without changing the outcome.Observe that one cannot prevent a candidate from winning by supporting the candidate more because this only increases the candidate's score.Consequently, we can always transfer approvals to all candidates from the winning k-egroup.For the case of k > ℓ, we then have to rearrange the approvals in such a way that only the weakest members of the k-egroup are supported.However, such a rearrangement cannot change the outcome because, according to Claim 1, we can transfer an approval from some stronger candidate c to weaker c ′ keeping both of them in the winning k-egroup.
Dropped and kept candidates.Observe that for every solution (including canonical solutions), we can always find the strongest candidate who is not part of the winning egroup.We call this candidate the dropped candidate.Note that we use the strength order in the definition of the dropped candidate; this order does not take manipulative votes into account.Moreover, without loss of generality, we can assume that the dropped candidate is not a supported candidate.This is because if the dropped candidate is not in the winning k-egroup even if supported, then we can support any other candidate outside of the winning k-egroup without changing the winning k-egroup (see Claim 1).There always exists some candidate to whom we can transfer our support because ℓ < m.Naturally, by definition of the dropped candidate, all candidates stronger than the dropped candidate are members of the winning k-egroup.We call these candidates kept candidates.
High-level description of the algorithm.The algorithm solving ℓ-Bloc-F lex -eval-CM with consistent manipulators iteratively looks for an optimal canonical solution for every possible (nonnegative) number t of kept candidates (alternatively the algorithm checks all feasible possibilities of choosing the dropped candidate).Observe that k − ℓ ≤ t ≤ k.The upper bound k is the consequence of the fact that each kept candidate is (by definition) in the winning k-egroup.Since all candidates except for kept candidates have to be supported to be part of the winning egroup, we need at least k − ℓ kept candidates, in order to be able to complete the k-egroup.
Running time.To analyze the running time of the algorithm described in the previous paragraph, several steps need to be considered.At the beginning we have to compute values of candidates and then sort the candidates with respect to their value.This step runs in O(rm • m log m) time.Similarly, computing > S takes O(ℓn • m log m) time.Having both orders, Procedure 1 (described in detail later in this proof) needs O(m) to find an optimal canonical solution for some fixed number t of kept candidates.Finally, we have at most ℓ + 1 possible values of t.Summing the times up, together with the fact that ℓ < m, we obtain a running time O(m(m + r + n)).
What remains to be done.Procedure 1 describes how to look for an optimal canonical solution for a fixed number t of kept candidates.First, partition the candidate set in the following way.By C * we denote the kept candidates (which are the top t candidates according to > S ).Consequently, the (t + 1)-st strongest candidate is the dropped candidate; say c * .For every value of t, the corresponding dropped candidate, by definition, is not allowed to be part of the winning egroup.Let be the set of distinguished candidates.Each distinguished candidate, if supported, is preferred over c * to be selected into the winning k-egroup.Consequently, the distinguished candidates are all candidates who can potentially be part of the winning k-egroup.We remark that to fulfil our assumption that the dropped candidate is not part of a winning egroup, it is obligatory to support at least k − t distinguished candidates.Note that C * ∪ {c * } ∪ D is not necessarily equal to C. The remaining candidates cannot be a part of the winning k-egroup under any circumstances assuming t kept candidates.Making use of the described division into c * , D, and C * , Procedure 1 incrementally builds set X of supported candidates associated with an optimal solution until all possible approvals are used.
Detailed description of the algorithm.Before studying Procedure 1 in detail, consider Figure 1 illustrating the procedure on example data.In line 1, the procedure builds set X of supported candidates using the k − t best valued distinguished candidates.Since only the distinguished candidates might be a part of the winning k-egroup besides the kept candidates, there is no better outcome achievable.Then, in line 2, the remaining approvals, if they exist, are used to support secured candidates.This operation does not change the resulting k-egroup.Then Procedure 1 checks whether all ℓ approvals were used; that is, whether ℓ = |X|.If not, then there are exactly ℓ − |X| remaining approvals to use.Note that at this stage set X contains k supported candidates who correspond to the best possible k-egroup, however, without spending all approvals.Let us call this k-egroup S. It is possible that there is no way to spend the remaining ℓ − |X| approvals without changing the Procedure 1: A procedure of finding an optimal set of supported candidates.
Input: Election E = (C, V ); number ℓ of approvals in ℓ-Bloc rule; size k of the winning k-egroup; a partition of C into kept candidates C * (such that | C * | = t and k − ℓ ≤ t ≤ k), a dropped candidate c * , and distinguished candidates D.
winning k-egroup S. Then substitutions of candidates occur.The new candidates in the k-egroup can be only those that are distinguished and so far unsupported whereas the exchanged ones can be only so far supported distinguished candidates.This means that each substitution lowers the overall value of the winning k-egroup.So, the best what can be achieved is to find the minimal number of substitutions and then pick the most valuable remaining candidates from D to be substituted.The minimal number of substitutions can be found by analyzing how many candidates would be exchanged in the winning k-egroup if the weakest ℓ − |X| previously unsupported candidates were supported.The procedure makes such a simulation and computes the number p of necessary substitutions, in lines 4-6.Supporting the ℓ − |X| − p weakest unsupported candidates and then the p most valuable so far unsupported distinguished candidates gives the optimal k-egroup for t kept candidates (when all approvals are spent).Note that the number ℓ of approvals is strictly lower than the number of candidates, so one always avoids supporting c * .
The algorithm we presented can be applied also for pessimistic and optimistic evaluation because of the possibility of simulating these evaluations by a lexicographic order in time O(m(r + log(m))) (see Proposition 1).
For Bloc, we will show that manipulators can always vote identically to achieve an optimal kegroup.In a nutshell, for every egroup the manipulators can only increase the scores of its members by voting exactly for them.This fact leads to the next corollary.
Corollary 2. Let m denote the number of candidates, n denote the number of voters, and r denote the number of manipulators.One can solve Bloc-F-eval-COALITIONAL MANIPULATION in O(m(m + r + n)) time for any eval ∈ {util, candegal} and F ∈ {F lex , F eval opt , F eval pess }.
Proof.We show that for Bloc-F-eval-COALITIONAL MANIPULATION the manipulators have no incentive to deviate from one optimal profile (i.e., they vote in the same manner).Let us fix an has to be substituted; naturally, it is optimal to pick the most valuable possible candidate as a replacement for the substituted one.Supported candidates are double-edged and the winning k-egroup is starred.
optimal k-egroup S. If there exists a candidate c ∈ S which is not by some manipulator u * , then there exists also some candidate c ′ / ∈ S which is approved by u * (u * approves at most k − 1 candidates from S). Observe that in the Bloc voting rule by shifting a candidate up in a preference order we only increase the candidate's score; as a result, we cannot prevent the candidate from winning by doing such a shift.Using this observation, we can exchange candidate c with candidate c ′ in the preference order of u * without preventing c from winning.We repeat exchanging candidates until all manipulators approve only candidates from S. Then we obtain an optimal vote by fixing a preference order over those candidates arbitrarily (there might be more than one optimal vote but all of them place only candidates from set S at the first k places).Concluding, we can use the algorithm from Proposition 3 which works in the given time.
To complete our analysis of coalitional manipulation for ℓ-Bloc under utilitarian and candidatewise egalitarian evaluation, we provide a polynomial-time algorithm that solves a general case of ℓ-Bloc-F-eval-COALITIONAL MANIPULATION, eval ∈ {util, candegal}, F ∈ {F lex , F eval opt , F eval pess }.In the general case, the number of approvals may differ from the size of the excellence-group and the manipulators can vote differently from each other.Proof.We prove the theorem for the lexicographic tie-breaking rule F lex .This is sufficient since, using Claim 1, one can generalize the proof for the cases of utilitarian and candidate-wise egalitarian variants.The basic idea of our algorithm is to fix certain parameters of a solution and then to reduce the resulting subproblem to a variant of the KNAPSACK problem with polynomial-sized weights.The algorithm iterates through all possible value combinations of the following two parameters: • the lowest final score z < |V ∪ W | of any member of the k-egroup and • the candidate ĉ that is the least preferred member of the k-egroup with final score z with respect to tie-breaking rule F lex .
Having fixed z and ĉ, let C + denote the set of candidates who get at least z + 1 approvals from the non-manipulative voters or who are preferred to ĉ according to F lex and get exactly z approvals from the non-manipulative voters.Assuming that the combination of parameter values is correct, all candidates from C + ∪ {ĉ} must belong to the k-egroup.Let k + := |C + |.For sanity, we check whether k + < k, that is, whether candidate ĉ can belong to the k-egroup if the candidate obtains final score z.We discard the corresponding combination of solution parameter values if the check fails.Next, we ensure that ĉ obtains the final score exactly z.If ĉ receives less than z − r or more than z approvals from non-manipulative voters, then we discard this combination of solution parameter values.Otherwise, let ŝ := z −score V (ĉ) denote number of additional approvals candidate ĉ needs in order to get final score z.Let k * := k −k + −1 be the number of remaining (not yet fixed) members of the k-egroup.Let s * := r • ℓ − ŝ be the number of approvals to be distributed to candidates in C \ {ĉ}.Now, the manipulators have to influence further k candidates to join the k-egroup (so far only consisting of C + ∪ {ĉ}) and distribute exactly s * approvals in total to candidates in C \ {ĉ} but at most r approvals per candidate.To this end, let C * denote the set of candidates which can possibly join the k-egroup.For each candidate c ∈ C \ (C + ∪ {ĉ}) it holds that c ∈ C * if and only if A straightforward idea is to select the k * elements from C * which have the highest values (that is, utility) for the coalition.However, there can be two issues: First, s * might be too small; that is, there are too few approvals to ensure that each of the k * best-valued candidates gets the final score at least z (resp.at least z + 1).Second, s * might be too large; that is, there are too many approvals to be distributed so that there is no way to do this without causing unwanted candidates to get a final score of at least z (resp.at least z + 1).
Fortunately, we can easily detect these cases and deal with them efficiently.In the former scenario we reduce the remaining problem to an instance of EXACT k-ITEM KNAPSACK-the problem in which, for a given set of items, their values and weights, and a knapsack capacity, we search for k items that maximize the overall value and do not exceed the knapsack capacity.In the latter case, we show that we can discard the corresponding combination of solution parameters.
First, if s * ≤ r • k * , then one can certainly distribute all s * approvals (e.g., to the k * candidates that will finally join the k-egroup).Of course, it could still be the case that there are too few approvals available to push the desired candidates into the k-egroup in a greedy manner.To solve this problem, we build an EXACT k-ITEM KNAPSACK instance where each candidate in C * is mapped to an item.We set the weight of each c * ∈ C * to z − score V (c * ) if c * is preferred to ĉ with respect to F lex and otherwise to (z + 1) − score V (c * ).We set the value of each c * ∈ C * to be equal to the utility that candidate c * contributes to the manipulators.Now, an optimal solution (given the combinations of parameter values is correct) must select exactly k * elements from C * such that the total weight is at most s * .This corresponds to EXACT k-ITEM KNAPSACK if we set our knapsack capacity to s * .Furthermore, finding any such set with maximum total value leads to an optimal solution.Even if the final total weight s ′ of the chosen elements is smaller than s * , we can transfer the EXACT k-ITEM KNAPSACK solution to the correct solution of our problem.The total weight corresponds to the number of approvals used.Thus, with the EXACT k-ITEM KNAPSACK solution we spend s ′ approvals and, because of the monotonicity of Bloc together with the assumption that s * ≤ r • k * , we use s * − s ′ approvals to approve the chosen candidates even more.
Second, if s * > r • k * , then one can certainly ensure for any set of k * candidates from C * the final score at least z (resp.at least z + 1).In many cases, it will not be a problem to distribute the approvals; for example, one can safely spend up to r approvals for each candidate from C \ C * , that is, to candidates that have no chance to get enough points to join the k-egroup or to candidates which are already fixed to be in the k-egroup.Furthermore, each candidate from C * can be safely approved z − score V (c * )− 1 times (resp.z − score V (c * ) times) without reaching final score z (resp.z + 1).We denote by s + the total number of approvals which can be safely distributed to candidates in C \{ĉ} without causing one of the candidates from C * to reach score at least z (resp.at least z+1).
If s * ≤ s + + r • k * (note that we assume s * > r • k * ), then we can greedily push the k * mostvalued candidates from C * into the k-egroup (spending r • k * approvals) and then safely distribute the remaining approvals within C \{ĉ} as discussed.If s * > s + +r•k * , then there is no possibility of distributing approvals in a way that ĉ is part of the k-egroup.Towards a contradiction let us assume that ĉ is part of the k-egroup obtained after distributing s + + r • k * + 1 approvals.This means that we spend all possible s + approvals so that ĉ is not beaten and r • k * approvals to push k * candidates to the winning k-egroup.Giving one more approval to some candidate c ′ from C * that is not yet in the k-egroup, by definition of C * and s + , means that the score of c ′ is enough to push ĉ out of the final k-egroup; a contradiction.Consequently, for the case of s * > s + + r • k * , we discard the corresponding combination of solution parameters.
As for the running time, the first step is sorting the candidates according to their values in O(m(r + log(m))) time.Then let us consider the running time of two cases s * ≤ r • k * and s * > r • k * separately.In the former case, we solve EXACT k-ITEM KNAPSACK in O(k 2 mr) time by using dynamic programming based on analyzing all possible total weights of the selected items until the final value is reached [Kellerer et al., 2004, Chapter 9.7.3]5(note that the maximum possible total weight is upper-bounded by kr).If s * > r • k * , then we approve at most m candidates which gives the running time O(m).Thus, we can conclude that the running time of the discussed cases is O(k 2 mr).Additionally, there are at most n + r values of z and at most m values of ĉ.Summarizing we get running time O(k 2 m 2 (n + r)).

Egalitarian: Hard Even for Simple Tie-Breaking
In Subsection 4.2, we showed that already breaking ties might be computationally intractable.These intractability results only hold with respect to the egalitarian evaluation and optimistic manipulators.We now show that this intractability transfers to coalitional manipulation for any tie-breaking rule and the egalitarian evaluation.This includes the pessimistic egalitarian case which we consider to be highly relevant as it naturally models searching for a "safe" voting strategy.
Proposition 4.There is a polynomial-time many-one reduction from F egal opt -TIE-BREAKING to ℓ-Bloc-F-egal-COALITIONAL MANIPULATION for any tie-breaking rule F.
Proof.We reduce an instance of F egal opt -TIE-BREAKING to ℓ-Bloc-F-egal-COALITIONAL MA-NIPULATION; however, before we describe the actual reduction, we present a useful observation concerning F egal opt -TIE-BREAKING in the next paragraph.Let us fix an instance I of F egal opt -TIE-BREAKING with a confirmed set C + , a pending set P , a size k of an egroup, a threshold q, and a set of manipulators represented by a family U of utility functions.We can construct a new equivalent instance I ′ of F egal opt -TIE-BREAKING with a larger set of manipulators' utility functions U ′ ⊇ U .The construction is a polynomial-time many-one reduction which proves that we can "pump" the number of manipulators arbitrarily for instance I. To add a manipulator, it is enough to set to q the utility that the manipulator gives to every candidate.
Naturally, such a manipulator cannot have the total utility than q, so the correct solution for I is also correct for I ′ .Contrarily, when there is no solution for I, it means that for every possible k-egroup S ′ there is some manipulator ū such that egal ū(S ′ ) < q.Consequently, one cannot find a solution for I ′ as well, because the set of possible k-egroups and their values of egalitarian utility do not change.
Now we can phrase our reduction from F egal opt -TIE-BREAKING to ℓ-Bloc-F-egal-COALITIO-NAL MANIPULATION.Let us fix an instance I of F egal opt -TIE-BREAKING with a confirmed set C + , a pending set P , a size k of an egroup, a threshold q, and a set U of r utility functions.Because of the observation about "pumping" instances of F egal opt -TIE-BREAKING, we can assume, without loss of generality, that ℓ • r ≥ k − |C + | holds.In the constructed instance of ℓ-Bloc-F-egal-CM equivalent to I, we build an election that yields sets P and C + .However, it is likely that we need to add a set of dummy candidates that we denote by D. It is important to ensure that the dummy candidates cannot be the winners of the constructed election.To do so, we keep the score of each dummy candidate to be at most 1, the score of each pending candidate to be r + 2, and the score of each confirmed candidate to be at least 2r + 3. The construction starts from ensuring the scores of the confirmed candidates.Observe, that in this step we add at most (2r + 3) • |C + | voters (in case ℓ = 1).If ℓ > |C + |, then we have to add some dummy candidates in this step.We can upper-bound the number of the added dummy candidates by ((2r + 3) Analogously, we add new voters such that each pending candidate has score exactly r + 2. At this step we have the election where we are able to spend ℓ • r ≥ k − |C + | approvals.We can select every possible subset of pending candidates to form the winning k-egroup by approving candidates in this subset exactly once.However, to be sure that we are able to distribute all approvals such that there is no tie, we ensure that the remaining (ℓ • r) − (k − |C + |) approvals can be distributed to some candidates without changing the outcome.To achieve this goal we add exactly (ℓ • r) − (k − |C + |) dummy candidates with score 0. We set the evaluation threshold of the newly constructed instance to q.
By our construction, we are always able to approve enough pending candidates to form a kegroup without considering ties, and we cannot make a dummy candidate a winner under any circumstances.Thus, if F egal opt -TIE-BREAKING has a solution S, then we approve every candidate c ∈ S such that c was in the pending set P before, and we obtain a solution to the reduced instance.In the opposite case, if there is no such a k-egroup whose egalitarian utility value is at least q, then the corresponding instance of ℓ-Bloc-F-egal-COALITIONAL MANIPULATION also has no solution since the possible k-egroups are exactly the same.The reduction runs in polynomial time.
The reduction keeps the egroup size and the number of manipulators for the case ℓ•r ≥ k−|C + |.For the opposite case, pumping the instance of F egal opt -TIE-BREAKING increases the parameter r , but the increase is polynomially bounded in k.Indeed, we increase r by at most k−|C + | ℓ < k.Thus, together with Theorem 1 and Theorem 2, Proposition 4 leads to the following corollary.
Corollary 3. ℓ-Bloc-F-egal-COALITIONAL MANIPULATION is NP-hard.Let r denote the number of manipulators, q denote the evaluation threshold and k denote the size of an egroup.Parameterized by r + k, ℓ-Bloc-F-egal-CM is W[1]-hard.Parameterized by k, ℓ-Bloc-F-egal-CM is W [2]-hard even if q = 1 and every manipulator only gives utility one or zero to each candidate.
Finally, by using ideas from Theorem 4 and an adaptation of the ILP from Theorem 3 as a subroutine, we show that, for the combined parameter "the number of manipulators and the number of different utility values", fixed-parameter tractability of F egal opt -TIE-BREAKING transfers to coalitional manipulation for both optimistic and pessimistic tie-breaking.
Theorem 5. Let r denote the number of manipulators and u diff denote the number of different utility values.Parameterized by r + u diff , ℓ-Bloc-F-egal-COALITIONAL MANIPULATION with F ∈ {F egal pess , F egal opt } is fixed-parameter tractable.
Proof.In a nutshell, we divide ℓ-Bloc-F egal pess -egal-CM and ℓ-Bloc-F egal opt -egal-CM into subproblems solvable in FPT time with respect to the combined parameter "number of manipulators and number of different utility values".We show that solving polynomially many subproblems is enough to solve the problems.
The main idea.We split the proof into two parts.In the first part, we define subproblems and show how to find a solution assuming that the subproblems are solvable in FPT time with respect to the parameter.In the second part, we show that, indeed, the subproblems are fixed-parameter tractable using their ILP formulations.The inputs for ℓ-Bloc-F egal pess -egal-CM and ℓ-Bloc-F egal optegal-CM are the same, so let us consider an arbitrary input with an election E = (C, V ) where |V | = n, |C| = m, a size k of an excellence-group, and r manipulators represented by a set U = {u 1 , u 2 , . . ., u r } of their utility functions.Let u diff be the number of different utility values.
An election resulting from a manipulation and a corresponding k-egroup emerging from the manipulation can be described by three non-negative integer parameters: 1. the lowest final score z of any member of the k-egroup; 2. the number p of promoted candidates from the k-egroup with a score higher than z that, at the same time, have score at most z without taking manipulative votes into consideration; 3. the number b of border candidates with score z.
Observe that if as a result of a manipulation the lowest final score of members in a final k-egroup is z, then the promoted candidates are part of the k-egroup regardless of the tie-breaking method used.
For border candidates, however, it might be necessary to run the tie-breaking rule to determine the k-egroup.In other words, border candidates become pending candidates unless all of them are part of the k-egroup.By definition, no candidate scoring lower than the border candidates is a member of the k-egroup; thus, the term border candidates.From now on, we refer to the election situation characterized by parameters z, p, b as a (input) state.Additionally, we call a set of manipulators' votes a manipulation.
Part 1: High-level description of the algorithm.For now, we assume that there is a procedure P which runs in FPT time with respect to the combined parameter "number of manipulators and number of different utility values".Procedure P, takes values z, p, b and an instance of the problem, and finds a manipulation which leads to a k-egroup maximizing egalitarian utility under either egalitarian optimistic or egalitarian pessimistic tie-breaking with respect to the input state.If such a manipulation does not exist, then procedure P returns "no".The algorithm solving ℓ-Bloc-F egal pessegal-CM and ℓ-Bloc-F egal opt -egal-CM runs P for all possible combinations of values z, p, and b.Eventually, it chooses the best manipulation returned by P or returns "no" if P always returned so.Since the value of z is at most |V + W | and b together with p are both upper-bounded by the number of candidates, we run P at most (n + r)m 2 times.Because the input size grows polynomially with respect to the growth of values r, m, and n, the overall algorithm runs in FPT time with respect to the combined parameter "number of manipulators and number of different utility values".
Part 2: Basics and preprocessing for the ILP.To complete the proof we describe procedure P used by the above algorithm.In short, the procedure builds and solves an ILP program that finds a manipulation leading to the state described by the input values.Before we describe the procedure in details, we start with some notation.Fix some values of z, b, p and some election E = (C, V ) that altogether form the input of P. For each candidate c ∈ C, let a size-r vector t = (u 1 (c), u 2 (c), . . ., u r (c)), referred to as a type vector, define the type of c.We denote the set of all possible type vectors by T = {t 1 , t 2 , . . ., t |T | }.Observe that |T | ≤ u r diff .With each type vector t i , i ∈ [|T |], we associate a set T i containing only candidates of type t i .We also distinguish the candidates with respect to their initial score compared to z.A candidate of type t i ∈ T , i ∈ |T |, with score z − j, j ∈ [r] ∪ {0}, belongs to group G j i .We denote all candidates with a score higher than z by C + , whereas by C − we denote the candidates with the score strictly lower than z − r.For each type t i ∈ T of a candidate, we define function obl(t i ) = |C + ∩T i |, which holds the number of candidates of type t i that are obligatory a part of the winning k-egroup.
At the beginning, procedure P tests whether the input values z, b, and p represent a correct state.From the fact that there has to be at least one candidate with score z, we get the upper bound k − |C + | − 1 for value p.To have enough candidates to complete the k-egroup, we need at least k − |C + | − p candidates with score z after the manipulation which gives b ≥ k − |C + | − p. Finally, the state is incorrect if the corresponding set C + contains k or more candidates.If the input values are incorrect, then P returns "no".Otherwise, P continues with building a corresponding ILP program.We give two separate ILP programs-one for the optimistic egalitarian tie-breaking and the other one for the pessimistic egalitarian tie-breaking.Both programs consist of two parts.The first part models all possible manipulations leading to the state described by values z, p, and b.The second one is responsible for selecting the best k-egroup assuming the particular tie-breaking and considering all possible manipulations according to the first part.Although the whole programs are different from each other, the first parts stay the same.Thus, we postpone distinguishing between programs until we describe the second parts.For the sake of readability, we present the ILP programs step by step.
∪ {0}, we introduce variables x j i and x j+ i indicating the numbers of, respectively, border and promoted candidates from group G j i .Additionally, we introduce variables o and ō.The former represents the number of approvals used to get the obligatory numbers of border and promoted candidates.The latter indicates the number of approvals which are to be spent without changing the final k-egroup (thus, in some sense a complement of the obligatory approvals) resulting from the manipulation (e.g., candidates in C + , who are part of the winning k-egroup anyway, cannot change the outcome).We begin our ILP program with ensuring that the values of x j i and x j+ i are feasible: x j+ i = p, (5) x j i = b, (6) ∀t i ∈ T : The expressions ensure that exactly p candidates are selected to be promoted (5), exactly b candidates are selected to be border ones (6), and that, for every group, the sum of border and promoted candidates is not greater that the cardinality of the group ( 4).The last two formulae ensure that candidates who have score z have to be either promoted or border candidates ( 7) and that candidates with initial score z − r cannot be promoted (i.e., get a score higher than z) (8).Next, we add the constraints concerning the number of approvals we need to use to perform the manipulation described by all variables x j i and x j+ i .We start with ensuring that the manipulation does not exceed the number of possible approvals.As mentioned earlier, we store the number of required approvals using variable o.
Then, we model spending the ō remaining votes (if any) to use all approvals.
The upper bound on the number of votes one can spend without changing the outcome presented in equation ( 11) consists of three summands.The first one indicates the number of approvals which can be spent for candidates whose initial score was either too high or too low to make a difference in the outcome of the election resulting from the manipulation.The second summand counts the approvals we can spend for potential promoted and border candidates that eventually are not part of the winning k-egroup; we can give them less approvals than are needed to make them border candidates.The last summand represents the number of additional approvals that we can spend on the promoted candidates to reach the maximum of r approvals candidate.This completes the first part of the ILP program in which we modeled the possible variants of promoted and border candidates for the fixed state (z, b, p).
ILP extension for optimistic egalitarian tie-breaking.In the second part, we find the final k-egroup by completing it with the border candidates according to the particular tie-breaking mechanism.Let us first focus on the case of the optimistic egalitarian tie-breaking.We introduce constraints allowing us to maximize the total egalitarian utility value of the final egroup; namely, for each group , we add a non-negative, integral variable x j• i indicating the number of border candidates of the given group chosen to be in the final k-egroup.The following constraints ensure that we select exactly k − |C + | − p border candidates to complete the winning egroup and that, for each group G j i , we do not select more candidates than available.
To complete the description of the ILP, we add the final expression defining the egalitarian utility s of the final k-excellence-group.The goal of the ILP program is to maximize s.
∀q ∈ [r] : Since the goal is to maximize s, our program simulates the egalitarian optimistic tie-breaking.
ILP extension for pessimistic egalitarian tie-breaking.To solve a subproblem for the case of pessimistic egalitarian tie-breaking, we need a different approach.We start with an additional notation.For each type of candidate t i ∈ T , let b i = j∈[r]∪{0} x j i denote the number of border candidates of this type.For each type t i ∈ T and manipulator u q , q ∈ [r], we introduce a new integer variable d q i .Its value corresponds to the number of border candidates of type t i who are part of the worst possible winning k-egroup according to manipulator's u q preferences; we call these candidates the designated candidates of type t i of manipulator u q .For each variable d q i , we define a binary variable used[d q i ] which has value one if at least one candidate of type t i is a designated candidate of manipulator u q .Similarly, we define fullyused[d q i ] to indicate that all candidates of type t i are designated by manipulator u q .To give a program which solves the case of pessimistic egalitarian tie-breaking, we copy the first part of the previous ILP program (expressions from ( 4) to ( 12)) and add new constraints.First of all, we ensure that each manipulator designates not more than the number of available border candidates from each type and that every manipulator designates exactly k − p − |C + | candidates.
∀q ∈ [r] : The following forces the semantics of the variables used; that is, a used r], has value one if and only if variable d q i is at least one.
Similarly, for the variables fullyused, we ensure that fullyused , is one if and only if manipulator u q designates all available candidates of type t i .
Since our task is to perform pessimistic tie-breaking, we have to ensure that the designated candidates for each manipulator are the candidates whom the manipulator gives the least utility.We impose it by forcing that the more valuable candidates (for a particular manipulator) are used only when all candidates of all less valuable types (for the manipulator) are used (i.e., they are fully used).To achieve this we make use of the used and fullyused variables in the following constraint.
∀q ∈ Finally, we give the last expression where s represents the pessimistic egalitarian k-egroup's utility which our ILP program wants to maximize: ∀q ∈ [r] : The ILP programs, for both tie-breaking variants, use at most O(rt) variables so, according to Lenstra [1983], are in FPT with respect to the combined parameter r + u diff .Consequently, procedure P is in FPT with respect to the same parameter.
After presenting the FPT result for egalitarian coalitional manipulation with optimistic or pessimistic egalitarian tie-breaking in Theorem 5, we proceed with an analogous result for egalitarian coalitional manipulation with one of the four remaining tie-breaking rules (that is, {optimistic, pes-simistic} × {utilitiarian, candidate-wise utilitarian}) in Theorem 6. Theorem 6.Let r denote the number of manipulators and u diff denote the number of different utility values.Parameterized by r + u diff , ℓ-Bloc-F-egal-COALITIONAL MANIPULATION is fixedparameter tractable for F ∈ {F lex , F eval opt , F eval pess } where eval ∈ {util, candegal}.
Proof.The general proof idea is to show an algorithm which solves problem ℓ-Bloc-F lex -egal-COALITIONAL MANIPULATION.Then, by Proposition 1, we extend this result for the remaining tie-breaking rules.
To solve ℓ-Bloc-F lex -egal-COALITIONAL MANIPULATION we create an ILP program for all possible value combinations of the following parameters: • the lowest final score z < |V ∪ W of any member of the k-egroup and • the candidate ĉ which is the least preferred member of the k-egroup with final score z with respect to the tie-breaking rule F lex .
Having z fixed, let C + denote the set of candidates which get at least z + 1 approvals from the non-manipulative voters or which are preferred to ĉ with respect to F and get exactly z approvals from the non-manipulative voters.Assuming that the combination of parameter values is correct, all candidates from C + ∪ {ĉ} must belong to the k-egroup.We check whether |C + | < k, that is, whether there is space for candidate ĉ in the k-egroup.If the check fails, then we skip the corresponding combination of solution parameter values.Next, we ensure that ĉ obtains final score exactly z.If ĉ receives less than z − r or more than z approvals from non-manipulative voters, then we discard this combination of solution parameter values.Otherwise, let ŝ := z − score V (ĉ) denote number of additional approvals candidate ĉ needs in order to get final score z.
We define the type of some candidate c i to be the size-r vector t j = (u 1 (c i ), u 2 (c i ), . . ., u r (c i )).We denote by T = {t 1 , t 2 , . . ., t |T | } the set of all possible types.Observe that |T | ≤ u r diff .With each type vector t i , i ∈ [|T |], we associate a set T i containing only the candidates of type t i .Having ĉ (and z) fixed, we distinguish candidates according to types further.For j ∈ [r] ∪ {0}, all candidates with score z − j that are preferred (resp.not preferred) to candidate ĉ according to F, fall into group G j+ i (resp.G j− i ).For each type t i ∈ T of a candidate, we define function obl(t i ) = |C + ∩ T i | which holds the number of candidates of type t i who are obligatory part of the winning k-egroup.We denote by C r candidates which do not fall to any of such groups.
Intuitively, Mful ĉ z is the number of approvals used to make potential winners the winners.Also, we define Fbid ĉ z represents the number of approvals which cannot be used if one wants to avoid pushing candidates outside of the solution (given by values of the variables x) to the winning k-egroup; for example, if some candidate c needs j approvals to be part of the winning committee, then we subtract r − j + 1 approvals from the whole pool of r approvals for this candidate because we can use only j − 1 approvals not to push c into the k-egroup.We define the following constraints to construct our program the goal of which is to maximize s: ∀t i ∈ T : ∀q ∈ [r] : Constraint ( 24) ensures that the candidates picked into a solution are available and can be part of the solution.Observe that candidates in G 0+ i have to be part of the solution and candidates in G z− i cannot be part of the solution.These two facts are ensured by Constraints ( 25) and ( 26).Constraint ( 27) forbids spending more votes than possible to push some candidates to the k-egroup.The same role for "wasted" approvals plays Constraint (28).The upper bound of wasted approvals is counted in the following way: From the number of all "places" of putting approvals (we subtract one from the number of candidates because we cannot put any approvals except for ŝ to candidate ĉ), we first subtract the approvals already given to candidates in the k-egroup (i.e., Mful ĉ z ).Next, we subtract all "places" of approvals that will cause the unchosen potential candidates to be chosen (i.e., Fbid ĉ z ).Constraint (29) ensures that, altogether, we spend exactly as many approvals as required, and Constraint (30) holds only when a proper number of candidates are pushed to be part of kegroup.The last equation forces maximization of the egalitarian utility of the winning k-egroup when s is maximized.
Using our technique we can obtain a solution by making O(nm) ILPs with at most 2ru r diff + 2 variables.According to Lenstra's famous result [Lenstra, 1983], the constructed ILPs yield fixedparameter tractability with respect to the combined parameter r + u diff .Because of Proposition 1, we can add a preprocessing phase in which we transfer any instance of the problem for tie-breaking F ∈ {F eval opt , F eval pess } with eval ∈ {util, candegal} to an equivalent problem with lexicographic tie-breaking.The running time of the preprocessing (see Proposition 1) is O(m • (r + log m)).

Conclusion
We developed a new model for and started the first systematic study of coalitional manipulation for multiwinner elections.Our analysis revealed that multiwinner coalitional manipulation requires models which are significantly more complex than those for single-winner coalitional manipulation or multiwinner non-coalitional manipulation.On the one hand, we generalized tractability results Table 1: Computational complexity of tie-breaking and coalitional manipulation.Our results for ℓ-Bloc hold for any ℓ ≥ 1, and thus cover SNTV.The parameters are the size k of the excellencegroup, the number r of manipulators, and the number u diff of different utility values.Furthermore, m is the number of candidates and n is the number of voters.The result marked with † holds for all possible combinations of the respective evaluation and behavior variants.
for coalitional manipulation of ℓ-Approval by Conitzer et al. [2007] and Lin and for noncoalitional manipulation of Bloc by Meir et al. [2008] and Obraztsova et al. [2013] to tractability of coalitional manipulation of ℓ-Bloc in case of utilitarian or candidate-wise egalitarian evaluation of egroups.On the other hand, we showed that coalitional manipulation becomes computationally intractable in case of egalitarian evaluation of egroups.
Let us discuss a few findings in more detail (Table 1 surveys all our results).We studied lexicographic, optimistic, and pessimistic tie-breaking and showed that, with the exception of egalitarian group evaluation, winner groups can be determined very efficiently.The intractability (NPhardness, parameterized hardness in form of W[1]and W [2]-hardness) for the egalitarian case, however, turns out to hold even for quite restricted scenarios.We also demonstrated that numerous tie-breaking rules can be "simulated" by (carefully chosen) lexicographic tie-breaking, again except for the egalitarian case.Interestingly, the hardness of egalitarian tie-breaking holds only for the optimistic case while for the pessimistic case it is efficiently solvable.Hardness for the egalitarian optimistic scenario, however, translates into hardness results for coalitional manipulation regardless of the specific tie-breaking rule.On the contrary, coalitional manipulation becomes tractable for the other two evaluation strategies-"candidate-wise" egalitarian and utilitarian.
In our study, we entirely focused on shortlisting as one of the simplest tasks for multiwinner elections to analyze our evaluation functions.It is interesting and non-trivial to develop models for multiwinner rules that aim for proportional representation or diversity.For shortlisting, extending our studies to non-approval-like scoring-based voting correspondences would be a natural next step.In this context, already seeing what happens if one extends the set of individual scores from being only 0 or 1 to more (but few) numbers is of interest.Moreover, we focused on deterministic tiebreaking mechanisms, ignoring randomized tie-breaking-another issue for future research.
Going beyond the above, further research on manipulators' behavior directing towards gametheoretic aspects seems promising as well.Intuitively, the utility for every voter that is a part of the manipulating coalition should not be below the utility the voter receives when voting sincerely.This is of course only a necessary condition to ensure the stability of a coalition.A more sophisticated analysis of stability needs to consider game-theoretic aspects such as Nash or core stability [Nisan et al., 2007].

Figure 1 :
Figure 1: An illustrative example of a run of Procedure 1 for t = 2, nine candidates, 7-Bloc, and 4-egroup.The horizontal position indicates the strength of a candidate and the vertical position indicates the value of a candidate.Since the number r of manipulators determines only the set of distinguished candidates, we do not specify r explicitly.We indicate the set of distinguished candidates instead.Subfigures 1a to 1d step by step present the execution of Procedure 1 on the way to find an optimal 4-egroup.

Theorem 4 .
Let m denote the number of candidates, n the number of voters, k the size of a searched egroup, and r the number of manipulators.One can solve ℓ-Bloc-F-eval-COALITIONAL MANIPU-LATION in O(k 2 m 2 (n + r)) time for any eval ∈ {util, candegal} and F ∈ {F lex , F eval opt , F eval pess }.

i
and G j− i , i ∈ |T |, j ∈ [r] ∪ {0}, we introduce variables x j+ i and x j− i respectively.The variables indicate, respectively, the number of candidates from groups G j+ i and G j− i whom we push to the winning k-egroup.Also, we introduce two additional variables s and u.The former one represents the minimal value of the total utility achieved by manipulators.The latter one indicates the number of votes which were spent without changing the outcome.To shorten the ILP we define Mful ĉ z := t i ∈T ,j∈[r] t i ∈T ,j∈[r]∪{0} (r − j + 1)(|G j+ i | − x j+ i ) + (r − j)(|G j− i | − x j− i ) .