The Complexity of Computing Minimal Unidirectional Covering Sets
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DOI: 10.1007/s00224-012-9437-9
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- Baumeister, D., Brandt, F., Fischer, F. et al. Theory Comput Syst (2013) 53: 467. doi:10.1007/s00224-012-9437-9
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Abstract
A common thread in the social sciences is to identify sets of alternatives that satisfy certain notions of stability according to some binary dominance relation. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer (in Math. Soc. Sci. 56(2):254–268, 2008) proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal unidirectional (i.e., either upward or downward) covering set. For both problems, we raise this lower bound to the \(\varTheta_{2}^{p}\) level of the polynomial hierarchy and provide a \(\varSigma_{2}^{p}\) upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size unidirectional covering sets are hard or complete for either of NP, coNP, and \(\varTheta_{2}^{p}\). An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer’s result that minimal bidirectional covering sets are polynomial-time computable.
Keywords
Computational social choice Computational complexity Minimal upward covering sets Minimal downward covering sets1 Introduction
A common thread in the social sciences is to identify sets of alternatives that satisfy certain notions of stability according to some binary dominance relation. Applications range from cooperative to non-cooperative game theory, from social choice theory to argumentation theory, and from multi-criteria decision analysis to sports tournaments (see, e.g., [7, 33] and the references therein). To give an example from cooperative game theory, von Neumann and Morgenstern [40] introduced the notion of stable set as the set of (“efficient” and “individually rational”^{1}) payoff vectors in a cooperative game that satisfies both internal stability (no vector in this set is dominated by another vector in the set) and external stability (every vector outside this set is dominated by some vector inside the set). The underlying dominance relation is defined as follows: A payoff vector \(\vec{x}=(x_{1},x_{2},\ldots,x_{n})\)dominates a payoff vector \(\vec{y}=(y_{1},y_{2},\ldots,y_{n})\) if there is a nonempty coalition C of players such that x_{i}>y_{i} for all i∈C and ∑_{i∈C}x_{i} is bounded above by the profit the players in C can make on their own. Stable sets exist for some, but not for all cooperative games [34], and if they exist, they need not be unique [35]. Brandt and Fischer [7] proved that every stable set is a “minimal upward covering set” and thus contained in the “upward uncovered set” (these notions, which are central to the present paper, will be defined formally in Sect. 2).
In settings of social choice, the most common dominance relation is the pairwise majority relation, where an alternative x is said to dominate another alternative y (written x≻y) if the number of individuals preferring x to y exceeds the number of individuals preferring y to x. McGarvey [36] proved that every asymmetric dominance relation can be realized via a particular preference profile, even if the individual preferences are linear.
2 | 1 | 1 | 1 | 1 |
---|---|---|---|---|
a | d | c | b | d |
b | a | d | c | a |
c | b | b | d | c |
d | c | a | a | b |
A well-known paradox due to the Marquis de Condorcet [13] says that the majority relation may contain cycles and thus does not always admit maximal elements, even if all of the underlying individual preferences do. Consider, for example, the three individual preference relations a>_{1}b>_{1}c, b>_{2}c>_{2}a, and c>_{3}a>_{3}b. Here, a majority prefers a to b and b to c, but also c to a. This means that although the individual preferences >_{i} are each transitive, the resulting dominance relation (a≻b≻c≻a) is not, so the concept of maximality is rendered useless in such cases. For this reason, various alternative solution concepts that can be used in place of maximality for nontransitive relations (see, e.g., [33]) have been proposed. In particular, concepts based on covering relations—transitive subrelations of the dominance relation at hand—have turned out to be very attractive [16, 20, 39].
In this paper, we study the computational complexity of problems related to the notions of upward and downward covering sets in dominance graphs. An alternative x is said to upward cover another alternative y if x dominates y and every alternative dominating x also dominates y. The intuition is that x “strongly” dominates y in the sense that there is no alternative that dominates x but not y. Looking for example at the dominance graph (A,≻) in Fig. 1, although alternative a dominates alternative b, a does not upward cover b, since alternative d dominates a but not b. On the other hand, alternative b does upward cover alternative c, since b dominates c, and the only alternative dominating b, namely a, also dominates c.
Similarly, an alternative x is said to downward cover another alternative y if x dominates y and every alternative dominated by y is also dominated by x. The intuition here is that x “strongly” dominates y in the sense that there is no alternative dominated by y but not by x. Again looking at the dominance graph (A,≻) from Fig. 1, a downward covers b, since a dominates both b and c, the only alternative dominated by b. However, although b dominates c, b does not downward cover c, since b does not dominate d, which is dominated by c.
A minimal upward or minimal downward covering set is defined as an inclusion-minimal set of alternatives that satisfies certain notions of internal and external stability with respect to the upward or downward covering relation [7, 16] (cf. the von Neumann and Morgenstern stable sets in cooperative game theory mentioned in the first paragraph of the introduction), as will be formally stated in Definition 3 in Sect. 2.
Overview of complexity results for the various types of upward covering set problems. As indicated, previously known results are due to Brandt and Fischer [7]; all other results are new to this paper
Problem type | \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\) | \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\) |
---|---|---|
Size | NP-complete, see Thm. 11 | NP-complete, see Thm. 11 |
Member | \(\varTheta_{2}^{p}\)-hard and in \(\varSigma_{2}^{p}\), see Thm. 12 | \(\varTheta_{2}^{p}\)-complete, see Thm. 13 |
Member-All | coNP-complete, see [7] | \(\varTheta_{2}^{p}\)-complete, see Thm. 13 |
Unique | coNP-hard and in \(\varSigma_{2}^{p}\), see Thm. 14 | coNP-hard and in \(\varTheta_{2}^{p}\), see Thm. 16 |
Test | coNP-complete, see Thm. 14 | coNP-complete, see Thm. 15 |
Find | not in polynomial time unless P=NP, see Thm. 17 | not in polynomial time unless P=NP, see Thm. 17 |
Overview of complexity results for the various types of downward covering set problems. As indicated, previously known results are due to Brandt and Fischer [7]; all other results are new to this paper
Problem type | \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\) | \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\) |
---|---|---|
Size | NP-complete, see Thm. 27 | NP-complete, see Thm. 27 |
Member | \(\varTheta_{2}^{p}\)-hard and in \(\varSigma_{2}^{p}\), see Thm. 30 | coNP-hard and in \(\varTheta_{2}^{p}\), see Thm. 28 |
Member-All | coNP-complete, see [7] | coNP-hard and in \(\varTheta_{2}^{p}\), see Thm. 28 |
Unique | coNP-hard and in \(\varSigma_{2}^{p}\), see Thm. 31 | coNP-hard and in \(\varTheta_{2}^{p}\), see Thm. 28 |
Test | coNP-complete, see Thm. 31 | coNP-complete, see Thm. 29 |
Find | not in polynomial time unless P=NP (follows from [7], see Thm. 32) | not in polynomial time unless P=NP, see Thm. 32 |
Put into a wider perspective, this work adds to a growing body of complexity and hardness results for the lower levels of the polynomial hierarchy, for problems arising in various areas, such as optimization problems in logic (see, e.g., [29, 49] and also the surveys by Schaefer and Umans [46, 47]), logic programming and reasoning (see, e.g., [17, 18] and also the survey by Eiter and Gottlob [19]), graph theory (see, e.g., [26, 27, 29, 51]), voting problems in social choice theory (see, e.g., [23, 28, 45] and also the survey by Hemaspaandra et al. [24]), and fair division problems in multiagent resource allocation (see, e.g., [6]).
This paper is organized as follows. Section 2 provides the needed definitions and notation, and Sect. 3 states all results and a discussion of the results. After presenting the constructions for minimal and minimum-size upward covering sets in Sect. 4.1, the proofs of the results on minimal and minimum-size upward covering sets are given in Sect. 4.2. Section 5.1 presents the constructions for minimal and minimum-size downward covering sets and Sect. 5.2 gives the proofs of the results on minimal and minimum-size downward covering sets. Finally Sect. 6 concludes this paper.
2 Definitions and Notation
In this section, we define the necessary concepts from social choice theory and complexity theory.
Definition 1
(Covering Relations)
Let A be a finite set of alternatives, let B⊆A, and let ≻ ⊆A×A be a dominance relation on A, i.e., ≻ is asymmetric and irreflexive.^{2} A dominance relation ≻ on a set A of alternatives can be conveniently represented as a dominance graph, denoted by (A,≻), whose vertices are the alternatives from A, and for each x,y∈A there is a directed edge from x to y if and only if x≻y.
xupward coversyinB, denoted by \(x\, C_{u}^{B}\, y\), if x≻y and for all z∈B, z≻x implies z≻y, and
xdownward coversyinB, denoted by \(x\, C_{d}^{B}\, y\), if x≻y and for all z∈B, y≻z implies x≻z.
Definition 2
(Uncovered Set)
Example 1
(Upward and Downward Uncovered Set)
b upward covers c in A (i.e., \(b\, C_{u}^{A} \, c\)), but no element in A except c is upward covered, and
a downward covers b in A (i.e., \(a\, C_{d}^{A} \,b\)), but no element in A except b is downward covered,
For both the upward and the downward covering relation (henceforth both will be called unidirectional covering relations), transitivity of the relation implies nonemptiness of the corresponding uncovered set for each nonempty set of alternatives. The intuition underlying covering sets is that there should be no reason to restrict the selection by excluding some alternative from it (internal stability) and there should be an argument against each proposal to include an outside alternative into the selection (external stability).
Definition 3
(Minimal Covering Set)
Internal stability: UC_{C}(B)=B.
External stability: For all y∈A−B, \(y \not\in \mathrm{UC}_{C}(B \cup\{y\})\).
A covering set M for A under C is said to be (inclusion-)minimal if no M′⊂M is a covering set for A under C.
Example 2
(Minimal Upward and Downward Covering Set)
internal stability, i.e., UC_{u}({b,d})={b,d}, and
external stability, i.e., neither a∈UC_{u}({a,b,d})={b,d} nor c∈UC_{u}({b,c,d})={b,d}, the latter equality holding due to b (which is undominated in {b,c,d}) upward covering c.
If the dominance relation a≻c were missing in (A,≻), then the resulting dominance graph would have two minimal upward covering sets for A, {a,c} and {b,d}. That is, minimal upward covering sets are not guaranteed to be unique.
internal stability, i.e., UC_{d}({a,c,d})={a,c,d}, and
external stability, i.e., \(b \not\in\mathrm{UC}_{d}(A) = \{a,c,d\}\), as we have seen above,
Every upward uncovered set contains one or more minimal upward covering sets, whereas minimal downward covering sets may not always exist,^{3} and if they exist, they need not be unique [7]. Dutta [16] proposed minimal covering sets in the context of tournaments, i.e., complete dominance relations. In tournaments, both notions of covering coincide because the set of alternatives dominating a given alternative x consists precisely of those alternatives not dominated by x. Minimal unidirectional covering sets are one of several possible generalizations to incomplete dominance relations (for more details, see [7]). Occasionally, it might be helpful to specify the dominance relation explicitly to avoid ambiguity. In such cases we refer to the dominance graph used and write, e.g., “M is an upward covering set for (A,≻)”.
- 1.
\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\): Given a set A of alternatives, a dominance relation ≻ on A, and a positive integer k, does there exist some minimal upward covering set for A containing at most k alternatives?
- 2.
\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\): Given a set A of alternatives, a dominance relation ≻ on A, and a distinguished element d∈A, is d contained in some minimal upward covering set for A?
- 3.
\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\): Given a set A of alternatives, a dominance relation ≻ on A, and a distinguished element d∈A, is d contained in all minimal upward covering sets for A?
- 4.
\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\): Given a set A of alternatives and a dominance relation ≻ on A, does there exist a unique minimal upward covering set for A?
- 5.
\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\): Given a set A of alternatives, a dominance relation ≻ on A, and a subset M⊆A, is M a minimal upward covering set for A?
- 6.
\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\): Given a set A of alternatives and a dominance relation ≻ on A, find a minimal upward covering set for A.
If we replace “upward” by “downward” above, we obtain the six corresponding “downward covering” versions, denoted by \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\), \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\), \(\mbox{\textsc{MC}$_{\mathrm{d}}$}\hbox{-}\allowbreak\mbox{\textsc{Member}}\hbox{-}\allowbreak\mbox{\textsc{All}}\), \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\), \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\), and \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\). And if we replace “minimal” by “minimum-size” in the twelve problems just defined, we obtain the corresponding “minimum-size” versions: \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\), \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\), \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\), \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\), \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\), \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\), \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\), \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\), \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\), \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\), \({\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}}\), and \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\).
Note that the four problems \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\), \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\), \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\), and \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) are search problems, whereas the other twenty problems are decision problems.
We assume that the reader is familiar with the basic notions of complexity theory, such as polynomial-time many-one reducibility and the related notions of hardness and completeness, and also with standard complexity classes such as P, NP, coNP, and the polynomial hierarchy [38, 48] (see also, e.g., the textbooks [41, 44]). In particular, coNP is the class of sets whose complements are in NP. \(\varSigma_{2}^{p} = \mathrm {NP}^{ \mathrm {NP}}\), the second level of the polynomial hierarchy, consists of all sets that can be solved by an NP oracle machine that has access (in the sense of a Turing reduction) to an NP oracle set such as SAT. SAT denotes the satisfiability problem of propositional logic, which is one of the standard NP-complete problems (see, e.g., Garey and Johnson [21]) and is defined as follows: Given a boolean formula in conjunctive normal form, does there exist a truth assignment to its variables that satisfies the formula?
Papadimitriou and Zachos [43] introduced the class of problems solvable in polynomial time via asking \(\mathcal{O}(\log n)\) sequential Turing queries to NP. This class is also known as the \(\varTheta_{2}^{p}\) level of the polynomial hierarchy (see Wagner [52]), and has been shown to coincide with the class of problems that can be decided by a P machine that accesses its NP oracle in a parallel manner (see [22, 31]). Equivalently, \(\varTheta_{2}^{p}\) is the closure of NP under polynomial-time truth-table reductions. It follows immediately from the definitions that \(\mathrm {P}\subseteq \mathrm {NP}\cap \mathrm {coNP}\subseteq \mathrm {NP}\cup \mathrm {coNP}\subseteq \varTheta_{2}^{p} \subseteq\varSigma_{2}^{p}\).
\(\varTheta_{2}^{p}\) captures the complexity of various optimization problems. For example, the problem of testing whether the size of a maximum clique in a given graph is an odd number, the problem of deciding whether two given graphs have minimum vertex covers of the same size, and the problem of recognizing those graphs for which certain heuristics yield good approximations for the size of a maximum independent set or for the size of a minimum vertex cover each are known to be complete for \(\varTheta_{2}^{p}\) (see [26, 27, 51]). Hemaspaandra and Wechsung [29] proved that the minimization problem for boolean formulas is \(\varTheta_{2}^{p}\)-hard. In the field of computational social choice, the winner problems for Dodgson [15], Young [54], and Kemeny [30] elections have been shown to be \(\varTheta_{2}^{p}\)-complete in the nonunique-winner model [23, 28, 45], and also in the unique-winner model [25].
3 Results and Discussion
Results
- 1.
to raise Brandt and Fischer’s NP-hardness lower bounds for \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) and \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) to \(\varTheta_{2}^{p}\)-hardness and to provide (simple) \(\varSigma_{2}^{p}\) upper bounds for these problems, and
- 2.
to extend the techniques we developed to apply also to the 22 other covering set problems defined in Sect. 2, in particular to the search problems.
Theorem 1
The complexity of the covering set problems defined in Sect. 2is as shown in Table 1for upward covering set problems and as shown in Table 2for downward covering set problems.
The detailed proofs of the single results collected in Theorem 1 will be presented in Sects. 4.2 for minimal and minimum-size upward covering sets and in Sect. 5.2 for minimal and minimum-size downward covering sets, and the technical constructions establishing the properties that are needed for these proofs are given in Sects. 4.1 for minimal and minimum-size upward covering sets and in Sect. 5.1 for minimal and minimum-size downward covering sets.
Discussion
We consider the problems of finding minimal and minimum-size upward and downward covering sets (\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\), \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\), \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\), and \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\)) to be particularly important and natural.
Regarding upward covering sets, we stress that our result (see Theorem 17) that, assuming P≠NP, \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) and \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) are hard to compute does not seem to follow directly from the NP-hardness of \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) in any obvious way. The decision version of \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) is: Given a dominance graph, does it contain a minimal upward covering set? However, this question has always an affirmative answer, so the decision version of \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) is trivially in P. Note also that \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) can be reduced in a “disjunctive truth-table” fashion to the search version of \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) (“Given a dominance graph (A,≻) and an alternative d∈A, find some minimal upward covering set for A that contains d”) by asking this oracle set about all alternatives in parallel.^{5} So \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) is no harder (with respect to disjunctive truth-table reductions) than that problem. The converse, however, is not at all obvious. Brandt and Fischer’s results only imply the hardness of finding an alternative that is contained in all minimal upward covering sets [7]. Our reduction that raises the lower bound of \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) from NP-hardness to \(\varTheta_{2}^{p}\)-hardness, however, also allows us to prove that \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) and \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) cannot be solved in polynomial time unless P=NP.
Regarding downward covering sets, the result that \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) cannot be computed in polynomial time unless P=NP is an immediate consequence of Brandt and Fischer’s result that it is NP-complete to decide whether there exists a minimal downward covering set [7, Thm. 9]. We provide an alternative proof based on our reduction showing that \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) is \(\varTheta_{2}^{p}\)-hard (see the proof of Theorem 32). In contrast to Brandt and Fischer’s proof, our proof shows that \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) is hard to compute even when the existence of a (minimal) downward covering set is guaranteed. As indicated in Tables 1 and 2, coNP-completeness of \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\) and \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\) was also shown previously by Brandt and Fischer [7].
As mentioned above, the two problems \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) and \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) were already known to be NP-hard [7] and are here shown to be even \(\varTheta_{2}^{p}\)-hard. One may naturally wonder whether raising their (or any problem’s) lower bound from NP-hardness to \(\varTheta_{2}^{p}\)-hardness gives us any more insight into the problem’s inherent computational complexity. After all, P=NP if and only if \(\mathrm {P}= \varTheta_{2}^{p}\). However, this question is a bit more subtle than that and has been discussed carefully by Hemaspaandra et al. [24]. They make the case that the answer to this question crucially depends on what one considers to be the most natural computational model. In particular, they argue that raising NP-hardness to \(\varTheta_{2}^{p}\)-hardness potentially (i.e., unless longstanding open problems regarding the separation of the corresponding complexity classes could be solved) is an improvement in terms of randomized polynomial time (i.e., for the class RP introduced by Adleman [1]) and in terms of unambiguous polynomial time (i.e., for the class UP introduced by Valiant [50]): Since it is neither known whether NP=RP implies \(\varTheta_{2}^{p} = \mathrm {RP}\) nor whether NP=UP implies \(\varTheta_{2}^{p} = \mathrm {UP}\), proving \(\varTheta_{2}^{p}\)-hardness for the problems considered in this paper potentially gives a higher level of evidence (than merely NP-hardness) that these problems are neither in RP nor in UP [24].
4 Minimal and Minimum-Size Upward Covering Sets
In this section, we consider minimal and minimum-size upward covering sets.
4.1 Constructions
We start by giving the constructions that will be used for establishing results on the minimal and minimum-size upward covering set problems. Brandt and Fischer [7] proved the following result. Since we need their reduction in Construction 7 and Sect. 4.2, we give a proof sketch for Theorem 2.
Theorem 2
(Brandt and Fischer [7])
Deciding whether a designated alternative is contained in some minimal upward covering set for a given dominance graph is NP-hard. That is, \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\)is NP-hard.
Proof Sketch
For each i, 1≤i≤n, there is a cycle \(x_{i} \succ \overline{x}_{i} \succ x_{i}' \succ\overline{x}_{i}' \succ x_{i}\);
if variable v_{i} occurs in clause c_{j} as a positive literal, then x_{i}≻y_{j};
if variable v_{i} occurs in clause c_{j} as a negative literal, then \(\overline{x}_{i} \succ y_{j}\); and
for each j, 1≤j≤r, we have y_{j}≻d.
As we will use this reduction to prove results for both \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) and some of the other problems stated in Sect. 2, we now analyze the minimal and minimum-size upward covering sets of the dominance graph constructed in the proof sketch of Theorem 2. Brandt and Fischer [7] showed that each minimal upward covering set for A contains exactly two of the four alternatives corresponding to any of the variables, i.e., either x_{i} and \(x_{i}'\), or \(\overline{x_{i}}\) and \(\overline{x_{i}}'\), 1≤i≤n. We now assume that if φ is not satisfiable then for each truth assignment to the variables of φ, at least two clauses are unsatisfied (which can be ensured, if needed, by adding two dummy variables). It is easy to see that every minimal upward covering set for A not containing alternative d must consist of at least 2n+2 alternatives, where 2n alternatives are from the variables and at least two are from the unsatisfied clauses. Also, every minimal upward covering set for A containing d consists of exactly 2n+1 alternatives, where again 2n alternatives are from the variables, none from the clauses and alternative d. Thus, φ is satisfiable if and only if every minimum-size upward covering set consists of 2n+1 alternatives. These minimum-size upward covering sets always include alternative d. We summarize these observations in the following proposition for later use.
Proposition 1
- 1.
Every minimal upward covering set forAcontaining alternativedconsists of exactly 2n+1 alternatives.
- 2.
Every minimal upward covering set forAnot containing alternativedmust consist of at least 2n+2 alternatives.
- 3.
φis satisfiable if and only if every minimum-size upward covering set consists of 2n+1 alternatives (including d).
We now provide another construction that transforms a given boolean formula into a dominance graph with quite different properties.
Construction 3
(for coNP-hardness of upward covering set problems)
Given a boolean formula in conjunctive normal form, φ(w_{1},w_{2},…,w_{k})=f_{1}∧f_{2}∧…∧f_{ℓ}, over the setW={w_{1},w_{2},…,w_{k}} of variables, we construct a set of alternativesAand a dominance relation ≻ on A. Without loss of generality, we may assume that ifφis satisfiable then it has at least two satisfying assignments. This can be ensured, if needed, by adding dummy variables.
For each i, 1≤i≤k, there is a cycle\(u_{i} \succ \overline{u}_{i} \succ u_{i}' \succ\overline{u}_{i}' \succ u_{i}\);
if variablew_{i}occurs in clausef_{j}as a positive literal, thenu_{i}≻e_{j}, \(u_{i} \succ e'_{j}\), \(e_{j} \succ\overline{u}_{i}\), and\(e'_{j} \succ\overline{u}_{i}\);
if variablew_{i}occurs in clausef_{j}as a negative literal, then\(\overline{u}_{i} \succ e_{j}\), \(\overline{u}_{i} \succ e'_{j}\), e_{j}≻u_{i}, and\(e'_{j} \succ u_{i}\);
if variablew_{i}does not occur in clause f_{j}, then\(e_{j} \succ u_{i}'\)and\(e_{j}' \succ\overline{u}_{i}'\);
for eachj, 1≤j≤ℓ, we havea_{1}≻e_{j}and\(a_{1} \succ e'_{j}\); and
there is a cyclea_{1}≻a_{2}≻a_{3}≻a_{1}.
We now show some properties of the dominance graph created by Construction 3 in general. We will need these properties for the proofs in Sect. 4.2. The first property, stated in Claim 4, has already been seen in the example above.
Claim 4
Consider the dominance graph (A,≻) created by Construction 3, and fix any j, 1≤j≤ℓ. For each minimal upward covering set M for A, if M contains the alternative e_{j} then all other alternatives are contained in M as well (i.e., A=M).
Proof
To simplify notation, we prove the claim only for the case of j=1. However, since there is nothing special about e_{1} in our argument, the same property can be shown by an analogous argument for each j, 1≤j≤ℓ.
Let M be any minimal upward covering set for A, and suppose that e_{1}∈M. First note that the alternatives dominating e_{1} and \(e_{1}'\) are always the same (albeit e_{1} and \(e_{1}'\) may dominate different alternatives). Thus, for each minimal upward covering set, either both e_{1} and \(e_{1}'\) are contained in it, or they both are not. Thus, since e_{1}∈M, we have \(e_{1}' \in M\) as well.
Since the alternatives a_{1}, a_{2}, and a_{3} form an undominated three-cycle, they each are contained in every minimal upward covering set for A. In particular, {a_{1},a_{2},a_{3}}⊆M. Furthermore, no alternative e_{j} or \(e_{j}'\), 1≤j≤ℓ, can upward cover any other alternative in M, because a_{1}∈M and a_{1} dominates e_{j} and \(e_{j}'\) but none of the alternatives that are dominated by either e_{j} or \(e_{j}'\). In particular, no alternative in any of the k four-cycles \(u_{i} \succ\overline{u}_{i} \succ u'_{i} \succ\overline{u}'_{i} \succ u_{i}\) can be upward covered by any alternative e_{j} or \(e_{j}'\), and so they each must be upward covered within their cycle. For each of these cycles, every minimal upward covering set for A must contain at least one of the sets \(\{u_{i}, u'_{i}\}\) and \(\{\overline{u}_{i}, \overline{u}'_{i}\}\), since at least one is needed to upward cover the other one.^{6}
Since e_{1}∈M and by internal stability, we have that no alternative from M upward covers e_{1}. In addition to a_{1}, the alternatives dominating e_{1} are u_{i} (for each i such that w_{i} occurs as a positive literal in f_{1}) and \(\overline{u}_{i}\) (for each i such that w_{i} occurs as a negative literal in f_{1}).
First assume that, for some i, w_{i} occurs as a positive literal in f_{1}. Suppose that \(\{u_{i}, u'_{i}\} \subseteq M\). If \(\overline{u}'_{i} \not\in M\) then e_{1} would be upward covered by u_{i}, which is impossible. Thus \(\overline{u}'_{i} \in M\). But then \(\overline{u}_{i} \in M\) as well, since u_{i}, the only alternative that could upward cover \(\overline{u}_{i}\), is itself dominated by \(\overline{u}'_{i}\). For the latter argument, recall that \(\overline{u}_{i}\) cannot be upward covered by any e_{j} or \(e_{j}'\). Thus, we have shown that \(\{u_{i}, u'_{i}\} \subseteq M\) implies \(\{\overline{u}_{i}, \overline{u}'_{i}\} \subseteq M\). Conversely, suppose that \(\{\overline{u}_{i}, \overline{u}'_{i}\} \subseteq M\). Then \(u_{i}'\) is no longer upward covered by \(\overline{u}_{i}\) and hence must be in M as well. The same holds for the alternative u_{i}, so \(\{\overline{u}_{i}, \overline{u}'_{i}\} \subseteq M\) implies \(\{u_{i}, u'_{i}\} \subseteq M\). Summing up, if e_{1}∈M then \(\{u_{i}, u'_{i}, \overline{u}_{i}, \overline{u}'_{i}\} \subseteq M\) for each i such that w_{i} occurs as a positive literal in f_{1}.
By symmetry of the construction, an analogous argument shows that if e_{1}∈M then \(\{u_{i}, u'_{i}, \overline{u}_{i}, \overline{u}'_{i}\} \subseteq M\) for each i such that w_{i} occurs as a negative literal in f_{1}.
Now, consider any i such that w_{i} does not occur in f_{1}. We have \(e_{1} \succ u'_{i}\) and \(e'_{1} \succ\overline{u}'_{i}\). Again, none of the sets \(\{u_{i}, u'_{i}\}\) and \(\{\overline{u}_{i},\overline{u}'_{i}\}\) alone can be contained in M, since otherwise either u_{i} or \(\overline{u}'_{i}\) would remain upward uncovered. Thus, e_{1}∈M again implies that \(\{u_{i}, u'_{i}, \overline{u}_{i}, \overline{u}'_{i}\} \subseteq M\).
Now it is easy to see that, since \(\bigcup_{1\leq i \leq k} \{u_{i}, u'_{i}, \overline{u}_{i}, \overline{u}'_{i}\} \subseteq M\) and since a_{1} cannot upward cover any of the e_{j} and \(e_{j}'\), 1≤j≤ℓ, external stability of M enforces that \(\bigcup_{1 < j \leq\ell} \{e_{j}, e_{j}'\} \subseteq M\). Summing up, we have shown that if e_{1} is contained in any minimal upward covering set M for A, then M=A. □
Claim 5
Consider Construction 3. The boolean formula φ is satisfiable if and only if there is no minimal upward covering set for A that contains any of the e_{j}, 1≤j≤ℓ.
Proof
It is enough to prove the claim for the case j=1, since the other cases can be proven analogously.
Since every upward covering set for A must contain {a_{1},a_{2},a_{3}} and at least one of the sets \(\{u_{i}, u'_{i}\}\) and \(\{\overline{u}_{i}, \overline{u}'_{i}\}\) for each i, 1≤i≤k, B_{α} is a (minimal) upward covering set for A. Let M be an arbitrary minimal upward covering set for A. By Claim 4, if e_{1} were contained in M, we would have M=A. But since B_{α}⊂A=M, this contradicts the minimality of M. Thus \(e_{1} \not\in M\).
From right to left, let M be an arbitrary minimal upward covering set for A and suppose \(e_{1} \not\in M\). By Claim 4, if any of the e_{j}, 1<j≤ℓ, were contained in M, it would follow that e_{1}∈M, a contradiction. Thus, {e_{j} ∣ 1≤j≤ℓ}∩M=∅. It follows that each e_{j} must be upward covered by some alternative in M. It is easy to see that for each j, 1≤j≤ℓ, and for each i, 1≤i≤k, e_{j} is upward covered in \(M\cup\{e_{j}\}\supseteq\{u_{i},u'_{i}\}\) if w_{i} occurs in f_{j} as a positive literal, and e_{j} is upward covered in \(M\cup\{e_{j}\}\supseteq\{\overline{u}_{i}, \overline{u}'_{i}\}\) if w_{i} occurs in f_{j} as a negative literal. It can never be the case that all four alternatives, \(\{u_{i}, u'_{i}, \overline{u}_{i}, \overline{u}'_{i}\}\), are contained in M, because then either e_{j} would no longer be upward covered in M or the resulting set M was not minimal. Now, M induces a satisfying assignment for φ by setting, for each i, 1≤i≤k, α(w_{i})=1 if u_{i}∈M, and α(w_{i})=0 if \(\overline{u}_{i} \in M\). □
Note that in Construction 3 every minimal upward covering set for A obtained from any satisfying assignment for φ contains exactly 2k+3 alternatives, and there is no minimal upward covering set of smaller size for A when φ is unsatisfiable.
Claim 6
Consider Construction 3. The boolean formula φ is not satisfiable if and only if there is a unique minimal upward covering set for A.
Proof
Recall that we assumed in Construction 3 that if φ is satisfiable then it has at least two satisfying assignments.
From left to right, suppose there is no satisfying assignment for φ. By Claim 5, there must be a minimal upward covering set for A containing one of the e_{j}, 1≤j≤ℓ, and by Claim 4 this minimal upward covering set for A must contain all alternatives. By reason of minimality, there cannot be another minimal upward covering set for A.
From right to left, suppose there is a unique minimal upward covering set for A. Due to our assumption that if φ is satisfiable then there are at least two satisfying assignments, φ cannot be satisfiable, since if it were, there would be two distinct minimal upward covering sets corresponding to these assignments (as argued in the proof of Claim 5). □
Wagner provided a sufficient condition for proving \(\varTheta_{2}^{p}\)-hardness that was useful in various other contexts (see, e.g., [23, 26, 27, 29, 51]) and is stated here as Lemma 1.
Lemma 1
(Wagner [51])
We will apply Lemma 1 as well. In contrast with those previous results, however, one subtlety in our construction is due to the fact that we consider not only minimum-size but also (inclusion-)minimal covering sets. To the best of our knowledge, our Construction 7 and Construction 24, which will be presented later, for the first time apply Wagner’s technique [51] to problems defined in terms of minimality/maximality rather than minimum/maximum size of a solution:^{7} In Construction 7 below, we define a dominance graph based on Construction 3 and the construction presented in the proof sketch of Theorem 2 such that Lemma 1 can be applied to prove \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\)\(\varTheta_{2}^{p}\)-hard (see Theorem 12), making use of the properties established in Claims 4, 5, and 6.
Construction 7
(for applying Lemma 1 to upward covering set problems)
We apply Wagner’s lemma with the NP-complete problemS=SAT and construct a dominance graph. Fix an arbitrarym≥1 and letφ_{1},φ_{2},…,φ_{2m}be 2mboolean formulas in conjunctive normal form such that ifφ_{j}is satisfiable then so isφ_{j−1}, for each j, 1<j≤2m. Without loss of generality, we assume that for each j, 1≤j≤2m, the first variable ofφ_{j}does not occur in all clauses ofφ_{j}. Furthermore, we requireφ_{j}to have at least two unsatisfied clauses ifφ_{j}is not satisfiable, and to have at least two satisfying assignments ifφ_{j}is satisfiable. It is easy to see that ifφ_{j}does not have this property, it can be transformed into a formula that does have it, without affecting the satisfiability of the formula.
- (1)
For each i, 1≤i≤m, let (A_{2i−1},≻_{2i−1}) be the dominance graph that results from the formulaφ_{2i−1}according to Brandt and Fischer’s construction [7] given in the proof sketch of Theorem 2. We use the same names for the alternatives inA_{2i−1}as in that proof sketch, except that we attach the subscript 2i−1. For example, alternativedfrom the proof sketch of Theorem 2 now becomesd_{2i−1}, x_{1}becomesx_{1,2i−1}, y_{1}becomesy_{1,2i−1}, and so on.
- (2)
For each i, 1≤i≤m, let (A_{2i},≻_{2i}) be the dominance graph that results from the formulaφ_{2i}according to Construction 3. We use the same names for the alternatives inA_{2i}as in that construction, except that we attach the subscript 2i. For example, alternativea_{1}from Construction 3 now becomesa_{1,2i}, e_{1}becomese_{1,2i}, u_{1}becomesu_{1,2i}, and so on.
- (3)
For eachi, 1≤i≤m, connect the dominance graphs (A_{2i−1},≻_{2i−1}) and (A_{2i},≻_{2i}) as follows. Let\(u_{1,2i}, \overline{u}_{1,2i}, u_{1,2i}', \overline{u}_{1,2i}' \in A_{2i}\)be the four alternatives in the cycle corresponding to the first variable ofφ_{2i}. Then both\(u_{1,2i}'\)and\(\overline{u}_{1,2i}'\)dominated_{2i−1}. The resulting dominance graph is denoted by\((B_{i}, \succ_{i}^{B})\).
- (4)
Connect themdominance graphs\((B_{i}, \succ_{i}^{B})\), 1≤i≤m, as follows: For eachi, 2≤i≤m, d_{2i−1}dominates all alternatives inA_{2i−2}.
Before we use this construction to obtain \(\varTheta_{2}^{p}\)-hardness results for some of our upward covering set problems in Sect. 4.2, we again show some useful properties of the dominance graph constructed, and we first consider the dominance graph \((B_{i}, \succ_{i}^{B})\) (see Step (3) in Construction 7) separately,^{8} for any fixed i with 1≤i≤m. Doing so will simplify our argument for the whole dominance graph (A,≻). Recall that \((B_{i},\succ_{i}^{B})\) results from the formulas φ_{2i−1} and φ_{2i}.
Claim 8
Consider Construction 7. Alternative d_{2i−1} is contained in some minimal upward covering set for \((B_{i},\succ_{i}^{B})\) if and only if φ_{2i−1} is satisfiable and φ_{2i} is not satisfiable.
Proof
- Case 1:
φ_{2i−1}∈SAT and φ_{2i}∈SAT. Since φ_{2i} is satisfiable, it follows from the proof of Claim 5 that for each minimal upward covering set M for \((B_{i},\succ_{i}^{B})\), either \(\{u_{1,2i}, u'_{1,2i}\} \subseteq M\) or \(\{\overline{u}_{1,2i}, \overline{u}'_{1,2i}\} \subseteq M\), but not both, and that none of the e_{j,2i} and \(e'_{j,2i}\) is in M. If \(\overline{u}'_{1,2i} \in M\) but \(u'_{1,2i} \not\in M\), then \(d_{2i-1} \not\in\mathrm{UC}_{u}(M)\), since \(\overline{u}'_{1,2i}\) upward covers d_{2i−1} within M. If \(u'_{1,2i} \in M\) but \(\overline{u}_{1,2i} \not\in M\), then \(d_{2i-1} \not\in\mathrm{UC}_{u}(M)\), since \(u'_{1,2i}\) upward covers d_{2i−1} within M. Hence, by internal stability, d_{2i−1} is not contained in M.
- Case 2:
\(\varphi_{2i-1} \not\in \mathrm {SAT}\) and \(\varphi_{2i} \not\in \mathrm {SAT}\). Since \(\varphi_{2i-1} \not\in \mathrm {SAT}\), it follows from the construction used in the proof of Theorem 2 that each minimal upward covering set M for \((B_{i},\succ_{i}^{B})\) contains at least one alternative y_{j,2i−1} (corresponding to some clause of φ_{2i−1}) that upward covers d_{2i−1}. Thus d_{2i−1} cannot be in M, again by internal stability.
- Case 3:
φ_{2i−1}∈SAT and \(\varphi_{2i} \not\in \mathrm {SAT}\). Since φ_{2i−1}∈SAT, it follows from the proof of Theorem 2 (see also Proposition 1) that there exists a minimal upward covering set M′ for (A_{2i−1},≻_{2i−1}) that corresponds to a satisfying truth assignment for φ_{2i−1}. In particular, none of the y_{j,2i−1} is in M′. On the other hand, since \(\varphi_{2i} \not\in \mathrm {SAT}\), it follows from Claim 6 that A_{2i} is the only minimal upward covering set for (A_{2i},≻_{2i}). Define M=M′∪A_{2i}. It is easy to see that M is a minimal upward covering set for \((B_{i},\succ_{i}^{B})\), since the only edges between A_{2i−1} and A_{2i} are those from \(\overline{u}'_{1,2i}\) and \(u'_{1,2i}\) to d_{2i−1}, and both \(\overline{u}'_{1,2i}\) and \(u'_{1,2i}\) are dominated by elements in M not dominating d_{2i−1}.
We now show that d_{2i−1}∈M. Note that \(\overline{u}'_{1,2i}\), \(u'_{1,2i}\), and the y_{j,2i−1} are the only alternatives in B_{i} that dominate d_{2i−1}. Since none of the y_{j,2i−1} is in M, they do not upward cover d_{2i−1}. Also, \(u'_{1,2i}\) doesn’t upward cover d_{2i−1}, since \(\overline{u}_{1,2i} \in M\) and \(\overline{u}_{1,2i}\) dominates \(u'_{1,2i}\) but not d_{2i−1}. On the other hand, by our assumption that the first variable of φ_{2i} does not occur in all clauses, there exist alternatives e_{j,2i} and \(e'_{j,2i}\) in M that dominate \(\overline{u}'_{1,2i}\) but not d_{2i−1}, so \(\overline{u}'_{1,2i}\) doesn’t upward cover d_{2i−1} either. Thus d_{2i−1}∈M.
Note that, by our assumption on how the formulas are ordered, the fourth case (i.e., \(\varphi_{2i-1} \not\in \mathrm {SAT}\) and φ_{2i}∈SAT) cannot occur. Thus, the proof is complete. □
Claim 9
Consider Construction 7. For each i, 1≤i≤m, let M_{i} be a minimal upward covering set for \((B_{i},\succ_{i}^{B})\) according to the cases in the proof of Claim 8. Then each of the sets M_{i} must be contained in every minimal upward covering set for (A,≻).
Proof
The minimal upward covering set M_{m} for \((B_{m},\succ_{m}^{B})\) must be contained in every minimal upward covering set for (A,≻), since no alternative in A−B_{m} dominates any alternative in B_{m}. On the other hand, for each i, 1≤i<m, no alternative in B_{i} can be upward covered by d_{2i+1} (which is the only element in A−B_{i} that dominates any of the elements of B_{i}), since d_{2i+1} is dominated within every minimal upward covering set for B_{i+1} (and, in particular, within M_{i+1}). Thus, each of the sets M_{i}, 1≤i≤m, must be contained in every minimal upward covering set for (A,≻). □
Claim 10
Proof
To show (4.2) from left to right, suppose ∥{i ∣ φ_{i}∈SAT}∥ is odd. Recall that for each j, 1<j≤2m, if φ_{j} is satisfiable then so is φ_{j−1}. Thus, there exists some i, 1≤i≤m, such that φ_{1},…,φ_{2i−1}∈SAT and \(\varphi_{2i}, \dots, \varphi_{2m} \not\in \mathrm {SAT}\). In Case 3 in the proof of Claim 8 we have seen that there is some minimal upward covering set for \((B_{i},\succ_{i}^{B})\)—call it M_{i}—that corresponds to a satisfying assignment of φ_{2i−1} and that contains all alternatives of A_{2i}. Note that, M_{i} contains d_{2i−1}. For each j≠i, 1≤j≤m, let M_{j} be some minimal upward covering set for \((B_{j},\succ_{j}^{B})\) according to Case 1 (if j<i) and Case 2 (if j>i) in the proof of Claim 8.
To show (4.2) from right to left, suppose that ∥{i ∣ φ_{i}∈SAT}∥ is even. For a contradiction, suppose that there exists some minimal upward covering set M for (A,≻) that contains d_{1}. If \(\varphi_{1} \not\in \mathrm {SAT}\) then we immediately obtain a contradiction by the argument in the proof of Theorem 2. On the other hand, if φ_{1}∈SAT then our assumption that ∥{i ∣ φ_{i}∈SAT}∥ is even implies that φ_{2}∈SAT. It follows from the proof of Claim 4, and from Claim 9, that every minimal upward covering set for (A,≻) (thus, in particular, M) contains either \(\{u_{1,2i}, u'_{1,2i}\}\) or \(\{\overline{u}_{1,2i}, \overline{u}'_{1,2i}\}\), but not both, and that none of the e_{j,2i} and \(e'_{j,2i}\) is in M. By the argument presented in Case 3 in the proof of Claim 8, the only way to prevent d_{1} from being upward covered by an element of M, either \(u_{1,2}'\) or \(\overline{u}_{1,2}'\), is to include d_{3} in M as well.^{9} By applying the same argument m−1 times, we will eventually reach a contradiction, since d_{2m−1}∈M can no longer be prevented from being upward covered by an element of M, either \(u_{1,2m}'\) or \(\overline{u}_{1,2m}'\). Thus, no minimal upward covering set M for (A,≻) contains d_{1}, which completes the proof of (4.2). □
Furthermore, it holds that ∥{i ∣ φ_{i}∈SAT}∥ is odd if and only if d_{1} is contained in all minimum-size upward covering sets for A. This is true since the minimal upward covering sets for A that contain d_{1} are those that correspond to some satisfying assignment for all satisfiable formulas φ_{i}, and as we have seen in the analysis of Construction 3 and the proof sketch of Theorem 2 (see also Proposition 1), these are the minimum-size upward covering sets for A.
4.2 Proofs
In this section, we prove the parts of Theorem 1 that consider minimal and minimum-size upward covering sets by applying the constructions and the properties of the resulting dominance graphs presented in Sect. 4.1.
Theorem 11
It is NP-complete to decide, given a dominance graph (A,≻) and a positive integer k, whether there exists a minimal/minimum-size upward covering set forAof size at most k. That is, both\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\)and\(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\)are NP-complete.
Proof
This result can be proven by using the construction of Theorem 2. Let φ be a given boolean formula in conjunctive normal form, and let n be the number of variables occurring in φ. Setting the bound k for the size of a minimal/minimum-size upward covering set to 2n+1 proves that both problems are hard for NP. Indeed, as we have seen in the paragraph after the proof sketch of Theorem 2 (see also Proposition 1), there is a size 2n+1 minimal upward covering set (and hence a minimum-size upward covering set) for A if and only if φ is satisfiable. Both problems are NP-complete, since they can obviously be decided in nondeterministic polynomial time. □
Theorem 12
Deciding whether a designated alternative is contained in some minimal upward covering set for a given dominance graph is hard for\(\varTheta_{2}^{p}\)and in \(\varSigma_{2}^{p}\). That is, \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\)is hard for\(\varTheta_{2}^{p}\)and in \(\varSigma_{2}^{p}\).
Proof
\(\varTheta_{2}^{p}\)-hardness follows directly from Claim 10, which applies Wagner’s lemma to upward covering set problems. Specifically, this claim shows that in Construction 7 the alternative d_{1} is contained in some minimal upward covering set for A if and only if the number of underlying boolean formulas that are satisfiable is odd. For the upper bound, let (A,≻) be a dominance graph and d a designated alternative in A. First, observe that we can verify in polynomial time whether a subset of A is an upward covering set for A, simply by checking whether it satisfies internal and external stability. Now, we can guess an upward covering set B⊆A with d∈B in nondeterministic polynomial time and verify its minimality by checking that none of its subsets is an upward covering set for A. This places the problem in NP^{coNP} and consequently in \(\varSigma_{2}^{p}\). □
Theorem 13
- 1.
It is\(\varTheta_{2}^{p}\)-complete to decide whether a designated alternative is contained in some minimum-size upward covering set for a given dominance graph. That is, \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\)is\(\varTheta_{2}^{p}\)-complete.
- 2.
It is\(\varTheta_{2}^{p}\)-complete to decide whether a designated alternative is contained in all minimum-size upward covering sets for a given dominance graph. That is, \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\)is\(\varTheta_{2}^{p}\)-complete.
Proof
Wagner’s lemma can be used to show \(\varTheta_{2}^{p}\)-hardness for both problems. The remark made after Claim 10 says that in Construction 7 the alternative d_{1} is contained in all minimum-size upward covering sets for A if and only if the number of underlying boolean formulas that are satisfiable is odd. Hence \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) and \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\) are both \(\varTheta_{2}^{p}\)-hard.
To see that \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) is contained in \(\varTheta_{2}^{p}\), let (A,≻) be a dominance graph and d a designated alternative in A. Obviously, in nondeterministic polynomial time we can decide, given (A,≻), x∈A, and some positive integer ℓ≤∥A∥, whether there exists some upward covering set B for A such that ∥B∥≤ℓ and x∈B. Using this problem as an NP oracle, in \(\varTheta_{2}^{p}\) we can decide, given (A,≻) and d∈A, whether there exists a minimum-size upward covering set for A containing d as follows. The oracle is asked whether for each pair (x,ℓ), where x∈A and 1≤ℓ≤∥A∥, there exists an upward covering set for A of size bounded by ℓ that contains the alternative x. The number of queries is polynomial (more specifically in \(\mathcal{O}(\|A\|^{2})\)), and all queries can be asked in parallel. Having all the answers, determine the size k of a minimum-size upward covering set for A, and accept if the oracle answer to (d,k) was yes, otherwise reject.
To show that \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\) is in \(\varTheta_{2}^{p}\), let (A,≻) be a dominance graph and d a designated alternative in A. We now use as our oracle the set of all (x,ℓ), where x∈A is an alternative, and ℓ≤∥A∥ a positive integer, such that there exists some upward covering set B for A with ∥B∥≤ℓ and \(x \not\in B\). Clearly, this problem is also in NP, and the size k of a minimum-size upward covering set for A can again be determined by asking \(\mathcal{O}(\|A\|^{2})\) queries in parallel (if all oracle answers are no, it holds that k=∥A∥). Now, the \(\varTheta_{2}^{p}\) machine accepts its input ((A,≻),d) if the oracle answer for the pair (d,k) is no, and otherwise it rejects. □
Theorem 14
- 1.
(Brandt and Fischer [7]) It is coNP-complete to decide whether a designated alternative is contained in all minimal upward covering sets for a given dominance graph. That is, \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\)is coNP-complete.
- 2.
It is coNP-complete to decide whether a given subset of the alternatives is a minimal upward covering set for a given dominance graph. That is, \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\)is coNP-complete.
- 3.
It is coNP-hard and in\(\varSigma_{2}^{p}\)to decide whether there is a unique minimal upward covering set for a given dominance graph. That is, \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\)is coNP-hard and in\(\varSigma_{2}^{p}\).
Proof
It follows from Claim 6 that in Construction 3 the boolean formula φ is not satisfiable if and only if the entire set of alternatives A is a (unique) minimal upward covering set for A. Furthermore, if φ is satisfiable, there exists more than one minimal upward covering set for A and none of them contains e_{1} (provided that φ has more than one satisfying assignment, which can be ensured, if needed, by adding a dummy variable such that the satisfiability of the formula is not affected). This proves coNP-hardness for all three problems. \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\) and \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\) are also contained in coNP, as they can be decided in the positive by checking whether there does not exist an upward covering set that satisfies certain properties related to the problem at hand, so they both are coNP-complete. \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\) can be decided in the positive by checking whether there exists an upward covering set M such that all sets that are not strict supersets of M are not upward covering sets for the set of all alternatives. Thus, \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\) is in \(\varSigma_{2}^{p}\). □
The first statement of Theorem 14 was already shown by Brandt and Fischer [7]. However, their proof—which uses essentially the reduction from the proof of Theorem 2, except that they start from the coNP-complete problem Validity (which asks whether a given formula is valid, i.e., true under every assignment [41])—does not yield any of the other coNP-hardness results in Theorem 14.
Theorem 15
It is coNP-complete to decide whether a given subset of the alternatives is a minimum-size upward covering set for a given dominance graph. That is, \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\)is coNP-complete.
Proof
This problem is in coNP, since it can be decided in the positive by checking whether the given subset M of alternatives is an upward covering set for the set A of all alternatives (which is easy) and all sets of smaller size than M are not upward covering sets for A (which is a coNP predicate). Now, coNP-hardness follows directly from Claim 6, which shows that in Construction 3 the boolean formula φ is not satisfiable if and only if there is a unique minimal upward covering set for A and hence also a unique minimum-size upward covering set for A. □
Theorem 16
Deciding whether there exists a unique minimum-size upward covering set for a given dominance graph is hard for coNP and in \(\varTheta_{2}^{p}\). That is, \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\)is coNP-hard and in \(\varTheta_{2}^{p}\).
Proof
It is easy to see that coNP-hardness follows directly from the coNP-hardness of \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\) (see Theorem 14). Membership in \(\varTheta_{2}^{p}\) can be proven by using the same oracle as in the proof of the first part of Theorem 13. We ask for all pairs (x,ℓ), where x∈A and 1≤ℓ≤∥A∥, whether there is an upward covering set B for A such that ∥B∥≤ℓ and x∈B. Having all the answers, determine the minimum size k of a minimum-size upward covering set for A. Accept if there are exactly k distinct alternatives x_{1},…,x_{k} for which the answer for (x_{i},k), 1≤i≤k, was yes, otherwise reject. □
An important consequence of the proofs of Theorems 14 and 16 (and of Construction 3 that underpins these proofs) regards the hardness of the search problems \(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) and \(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\).
Theorem 17
Assuming P≠NP, neither minimal upward covering sets nor minimum-size upward covering sets can be found in polynomial time. That is, neither\(\mbox {\textsc {MC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\)nor\(\mbox {\textsc {MSC}$_{\mathrm {u}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\)are polynomial-time computable unless P=NP.
Proof
Consider the problem of deciding whether there exists a nontrivial minimal/minimum-size upward covering set, i.e., one that does not contain all alternatives. By Construction 3 that is applied in proving Theorems 14 and 16, there exists a trivial minimal/minimum-size upward covering set for A (i.e., one containing all alternatives in A) if and only if this set is the only minimal/minimum-size upward covering set for A. Thus, the coNP-hardness proof for the problem of deciding whether there is a unique minimal/minimum-size upward covering set for A (see the proofs of Theorems 14 and 16) immediately implies that the problem of deciding whether there is a nontrivial minimal/minimum-size upward covering set for A is NP-hard. However, since the latter problem can easily be reduced to the search problem (because the search problem, when used as a function oracle, yields the set of all alternatives if and only if this set is the only minimal/minimum-size upward covering set for A), it follows that the search problem cannot be solved in polynomial time unless P=NP. □
5 Minimal and Minimum-Size Downward Covering Sets
Now we consider minimal and minimum-size downward covering sets.
5.1 Constructions
Again we first give the constructions that will be used in Sect. 5.2 to show complexity results about minimal/minimum-size downward covering sets. we again start by giving a proof sketch of a result due to Brandt and Fischer [7], since the following constructions and proofs are based on their construction and proof.
Theorem 18
(Brandt and Fischer [7])
Deciding whether a designated alternative is contained in some minimal downward covering set for a given dominance graph is NP-hard (i.e., \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\)is NP-hard), even if a downward covering set is guaranteed to exist.
Proof Sketch
For each i, 1≤i≤n, there is a cycle \(x_{i} \succ\overline{x}_{i} \succ x_{i}' \succ\overline{x}_{i}' \succ x_{i}'' \succ\overline{x}_{i}'' \succ x_{i}\) with two nested three-cycles, \(x_{i} \succ x_{i}' \succ x_{i}'' \succ x_{i}\) and \(\overline{x}_{i} \succ \overline{x}_{i}' \succ\overline{x}_{i}'' \succ\overline{x}_{i}\);
if variable v_{i} occurs in clause c_{j} as a positive literal, then y_{j}≻x_{i};
if variable v_{i} occurs in clause c_{j} as a negative literal, then \(y_{j} \succ\overline{x}_{i}\);
for each j, 1≤j≤r, we have d≻y_{j} and z_{j}≻d; and
for each i and j with 1≤i,j≤r and i≠j, we have z_{i}≻y_{j}.
Regarding their construction sketched above, Brandt and Fischer [7] showed that every minimal downward covering set for A must contain exactly three alternatives for every variable v_{i} (either x_{i}, \(x_{i}'\), and \(x_{i}''\), or \(\overline{x}_{i}\), \(\overline{x}_{i}'\), and \(\overline{x}_{i}''\)), and the undominated alternatives z_{1},…,z_{r}. Thus, each minimal downward covering set for A consists of at least 3n+r alternatives and induces a truth assignment α for φ. The number of alternatives contained in any minimal downward covering set for A corresponding to an assignment α is 3n+r+k, where k is the number of clauses that are satisfied if α is an assignment not satisfying φ, and where k=r+1 if α is a satisfying assignment for φ. As a consequence, minimum-size downward covering sets for A correspond to those assignments for φ that satisfy the least possible number of clauses of φ.^{10}
Next, we provide a different construction to transform a given boolean formula into a dominance graph. This construction will later be merged with the construction from the proof sketch of Theorem 18 so as to apply Lemma 1 to show \(\varTheta_{2}^{p}\)-hardness for downward covering set problems.
Construction 19
(for NP- and coNP-hardness of downward covering set problems)
For eachi, 1≤i≤k, there is, similarly to the construction in the proof of Theorem 18, a cycle\(x_{i} \succ\overline{x}_{i} \succ x_{i}' \succ\overline{x}_{i}' \succ x_{i}'' \succ\overline{x}_{i}'' \succ x_{i}\)with two nested three-cycles, \(x_{i} \succ x_{i}' \succ x_{i}'' \succ x_{i}\)and\(\overline{x}_{i} \succ \overline{x}_{i}' \succ\overline{x}_{i}'' \succ\overline{x}_{i}\), and additionally we have\(z_{i}' \succ z_{i} \succ x_{i}\), \(z_{i}'' \succ z_{i} \succ \overline{x_{i}}\), \(z_{i}' \succ x_{i}\), \(z_{i}'' \succ\overline{x}_{i}\), andd≻z_{i};
if variablew_{i}occurs in clausef_{j}as a positive literal, thenx_{i}≻y_{j};
if variablew_{i}occurs in clausef_{j}as a negative literal, then\(\overline{x}_{i} \succ y_{j}\);
for eacha∈A_{1}∪A_{2}, we have\(b \succ \widehat {a}\), \(a \succ \widehat {a}\), and\(\widehat {a}\succ d\);
for eachj, 1≤j≤ℓ, we haved≻y_{j}; and
c≻d.
We now show some properties of Construction 19 in general.
Claim 20
Minimal downward covering sets are guaranteed to exist for the dominance graph defined in Construction 19.
Proof
The set A of all alternatives is a downward covering set for itself. Hence, there always exists a minimal downward covering set for the dominance graph defined in Construction 19. □
Claim 21
Consider the dominance graph (A,≻) created by Construction 19. For each minimal downward covering set M for A, if M contains the alternative d then all other alternatives are contained in M as well (i.e., A=M).
Proof
If d is contained in some minimal downward covering set M for A, then \(\{a,\widehat {a}\} \subseteq M\) for every a∈A_{1}∪A_{2}. To see this, observe that for an arbitrary a∈A_{1}∪A_{2} there is no a′∈A with \(a' \succ \widehat {a}\) and a′≻d or with a′≻a and \(a' \succ \widehat {a}\). Since the alternatives c and b are undominated, they are also in M, so M=A. □
Claim 22
Consider Construction 19. The boolean formula φ is satisfiable if and only if there is no minimal downward covering set for A that contains d.
Proof
For the direction from right to left, assume that no minimal downward covering set for A contains d. Since by Claim 20 minimal downward covering sets are guaranteed to exist for the dominance graph defined in Construction 19, there exists a minimal downward covering set B for A that does not contain d, so B≠A. It holds that {z_{i} ∣ w_{i} is a variable in φ}∩B=∅ and {y_{j} ∣ f_{j} is a clause in φ}∩B=∅, for otherwise a contradiction would follow by observing that there is no a∈A with a≻d and a≻z_{i}, 1≤i≤k, or with a≻d and a≻y_{j}, 1≤j≤ℓ. Furthermore, we have \(x_{i} \not\in B\) or \(\overline{x}_{i} \not\in B\), for each variable w_{i}∈W. By external stability, for each clause f_{j} there must exist an alternative a∈B with a≻y_{j}. By construction and since \(d \not\in B\), we must have either a=x_{i} for some variable w_{i} that occurs in f_{j} as a positive literal, or \(a=\overline{x}_{i}\) for some variable w_{i} that occurs in f_{j} as a negative literal. Now define α:W→{0,1} such that α(w_{i})=1 if x_{i}∈B, and α(w_{i})=0 otherwise. It is readily appreciated that α is a satisfying assignment for φ. □
Claim 23
Consider Construction 19. The boolean formula φ is not satisfiable if and only if there is a unique minimal downward covering set for A.
Proof
We again assume that if φ is satisfiable, it has at least two satisfying assignments. If φ is not satisfiable, there must be a minimal downward covering set for A that contains d by Claim 22, and by Claim 21 there must be a minimal downward covering set for A containing all alternatives. Hence, there is a unique minimal downward covering set for A. Conversely, if there is a unique minimal downward covering set for A, φ cannot be satisfiable, since otherwise there would be at least two distinct minimal downward covering sets for A, corresponding to the distinct truth assignments for φ, which would yield a contradiction. □
In the dominance graph created by Construction 19, the minimal downward covering sets for A coincide with the minimum-size downward covering sets for A. If φ is not satisfiable, there is only one minimal downward covering set for A, so this is also the only minimum-size downward covering set for A, and if φ is satisfiable, the minimal downward covering sets for A correspond to the satisfying assignments of φ. As we have seen in the proof of Claim 22, these minimal downward covering sets for A always consist of 5k+2 alternatives. Thus, they each are also minimum-size downward covering sets for A.
Merging the construction from the proof sketch of Theorem 18 with Construction 19, we again provide a reduction applying Lemma 1, this time to downward covering set problems.
Construction 24
(for applying Lemma 1 to downward covering set problems)
We again apply Wagner’s lemma with the NP-complete problemS=SAT and construct a dominance graph. Fix an arbitrarym≥1 and letφ_{1},φ_{2},…,φ_{2m}be 2mboolean formulas in conjunctive normal form such that the satisfiability ofφ_{j}implies the satisfiability ofφ_{j−1}, for eachj∈{2,…,2m}. Without loss of generality, we assume that for each j, 1≤j≤2m, φ_{j}has at least two satisfying assignments, ifφ_{j}is satisfiable.
- (1)
For eachi, 1≤i≤m, let (A_{2i−1},≻_{2i−1}) be the dominance graph that results from the formulaφ_{2i−1}according to Brandt and Fischer’s construction given in the proof sketch of Theorem 18. We again use the same names for the alternatives inA_{2i−1}as in that proof sketch, except that we attach the subscript 2i−1.
- (2)
For eachi, 1≤i≤m, let (A_{2i},≻_{2i}) be the dominance graph that results from the formulaφ_{2i}according to Construction 19. We again use the same names for the alternatives inA_{2i}as in that construction, except that we attach the subscript 2i.
- (3)
For eachi, 1≤i≤m, the dominance graphs (A_{2i−1},≻_{2i−1}) and (A_{2i},≻_{2i}) are connected by the alternativess_{i}, t_{i}, andr_{i} (which play a similar role as the alternativesz_{i}, \(z_{i}'\), and\(z_{i}''\)for each variable in Construction 19). The resulting dominance graph is denoted by\((B_{i},\succ_{i}^{B})\).
- (4)
Connect themdominance graphs\((B_{i},\succ_{i}^{B})\), 1≤i≤m (again similarly as in Construction 19). The alternativec^{∗}dominatesd^{∗}, andd^{∗}dominates themalternativesr_{i}, 1≤i≤m.
Claim 25
Consider Construction 24. For each i, 1≤i≤2m, let M_{i} be a minimal downward covering set for (A_{i},≻_{i}). Then each of the sets M_{i} must be contained in every minimal downward covering set for (A,≻).
Proof
For each i, 1≤i≤2m, the only alternative in A_{i} dominated from outside A_{i} is d_{i}. Since d_{i} is also dominated by the undominated alternative z_{1,i}∈A_{i} for odd i, and by the undominated alternative c_{i}∈A_{i} for even i, it is readily appreciated that internal and external stability with respect to elements of A_{i} only depends on the restriction of the dominance graph to A_{i}. □
Claim 26
Proof
- 1.
i is odd and φ_{i} is satisfiable, or
- 2.
i is even and φ_{i} is not satisfiable.
For the direction from right to left in (5.3), assume that there exists a minimal downward covering set M for A with d^{∗}∈M. By internal stability, there must exist some j, 1≤j≤k, such that r_{j}∈M. Thus, d_{2j−1} and d_{2j} must be in M, too. It then follows from the proof sketch of Theorem 18 and Claim 22 that φ_{2j−1} is satisfiable and φ_{2j} is not. Hence, ∥{i ∣ φ_{i}∈SAT}∥ is odd. □
By the remark made after Theorem 18, Construction 24 cannot be used straightforwardly to obtain complexity results for minimum-size downward covering sets.
5.2 Proofs
Now we prove the remaining parts of Theorem 1 concerning minimal and minimum-size downward covering sets by applying the constructions and the properties of the resulting dominance graphs presented in Sect. 5.1.
Theorem 27
It is NP-complete to decide, given a dominance graph (A,≻) and a positive integer k, whether there exists a minimal/minimum-size downward covering set forAof size at most k. That is, \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\)and\(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\)are both NP-complete.
Proof
Membership in NP is obvious, since we can nondeterministically guess a subset M⊆A of the alternatives with ∥M∥≤k and can then check in polynomial time whether M is a downward covering set for A. NP-hardness of \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\) and \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Size}}\) follows from Construction 19, the proof of Claim 22, and the comments made after Claim 23: If φ is a given formula with n variables, then there exists a minimal/minimum-size downward covering set of size 5n+2 if and only if φ is satisfiable. □
Theorem 28
\(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\), \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\), and\(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\)are coNP-hard and in \(\varTheta_{2}^{p}\).
Proof
It follows from Claim 23 that in Construction 19 the boolean formula φ is not satisfiable if and only if the entire set A of all alternatives is the unique minimum-size downward covering set for itself. Moreover, assuming that φ has at least two satisfying assignments, if φ is satisfiable, there are at least two distinct minimum-size downward covering sets for A. This shows that each of \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\), \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\), and \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\) is coNP-hard. For all three problems, membership in \(\varTheta_{2}^{p}\) is shown similarly to the proofs of the corresponding minimum-size upward covering set problems. However, since downward covering sets may fail to exist, the proofs must be slightly adapted. For \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\) and \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\), the machine rejects the input if the size k of a minimum-size downward covering set cannot be computed (simply because there doesn’t exist any such set). For \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\), if all oracle answers are no, it must be checked whether the set of all alternatives is a downward covering set for itself. If so, the machine accepts the input, otherwise it rejects. □
Theorem 29
It is coNP-complete to decide whether a given subset is a minimum-size downward covering set for a given dominance graph. That is, \({\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}}\)is coNP-complete.
Proof
This problem is in coNP, since its complement (i.e., the problem of deciding whether a given subset of the set A of alternatives is not a minimum-size downward covering set for A) can be decided in nondeterministic polynomial time. Hardness for coNP follows directly from Claim 23, which shows that in Construction 19 the boolean formula φ is not satisfiable if and only if there is a unique minimal downward covering set for A and hence also a unique minimum-size downward covering set for A. □
Theorem 30
Deciding whether a designated alternative is contained in some minimal downward covering set for a given dominance graph is hard for\(\varTheta_{2}^{p}\)and in\(\varSigma_{2}^{p}\). That is, \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\)is hard for\(\varTheta_{2}^{p}\)and in \(\varSigma_{2}^{p}\).
Proof
Membership in \(\varSigma_{2}^{p}\) can be shown analogously to the proof of Theorem 12, and \(\varTheta_{2}^{p}\)-hardness follows directly from Claim 26, which applies Wagner’s lemma to downward covering sets. Specifically, this claim shows that in Construction 24 the alternative d^{∗} is contained in some minimal downward covering set for A if and only if the number of underlying boolean formulas is odd. □
Theorem 31
- 1.
(Brandt and Fischer [7]) It is coNP-complete to decide whether a designated alternative is contained in all minimal downward covering sets for a given dominance graph. That is, \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\)is coNP-complete.
- 2.
It is coNP-complete to decide whether a given subset of the alternatives is a minimal downward covering set for a given dominance graph. That is, \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\)is coNP-complete.
- 3.
It is coNP-hard and in\(\varSigma_{2}^{p}\)to decide whether there is a unique minimal downward covering set for a given dominance graph. That is, \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\)is coNP-hard and in\(\varSigma_{2}^{p}\).
Proof
It follows from Claim 23 that in Construction 19 the boolean formula φ is not satisfiable if and only if the entire set of alternatives A is a unique minimal downward covering set for A. Furthermore, if φ is satisfiable, there exists more than one minimal downward covering set for A and none of them contains d (provided that φ has more than one satisfying assignment, which can be ensured, if needed, by adding a dummy variable such that the satisfiability of the formula is not affected). This proves coNP-hardness for all three problems. \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Member}}\hbox {-}\allowbreak \mbox {\textsc {All}}\) and \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Test}}\) are also contained in coNP, because they can be decided in the positive by checking whether there does not exist a downward covering set that satisfies certain properties related to the problem at hand. Thus, they are both coNP-complete. \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\) can be decided in the positive by checking whether there exists a downward covering set M such that all sets that are not strict supersets of M are not downward covering sets for the set of all alternatives. This shows that \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Unique}}\) is in \(\varSigma_{2}^{p}\). □
The first statement of Theorem 31 was already shown by Brandt and Fischer [7]. However, their proof—which uses essentially the reduction from the proof of Theorem 18, except that they start from the coNP-complete problem Validity—does not yield any of the other coNP-hardness results in Theorem 31.
An important consequence of the proofs of Theorems 28 and 31 regards the hardness of the search problems \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) and \(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\). (Note that the hardness of \(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\) also follows from a result by Brandt and Fischer [7, Theorem 9], see the discussion in Sect. 3.)
Theorem 32
Assuming P≠NP, neither minimal downward covering sets nor minimum-size downward covering sets can be found in polynomial time (i.e., neither\(\mbox {\textsc {MC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\)nor\(\mbox {\textsc {MSC}$_{\mathrm {d}}$}\hbox {-}\allowbreak \mbox {\textsc {Find}}\)are polynomial-time computable unless P=NP), even when the existence of a downward covering set is guaranteed.
Proof
Consider the problem of deciding whether there exists a nontrivial minimal/minimum-size downward covering set, i.e., one that does not contain all alternatives. By Construction 19 that is applied in proving Theorems 28 and 31, there exists a trivial minimal/minimum-size downward covering set for A (i.e., one containing all alternatives in A) if and only if this set is the only minimal/minimum-size downward covering set for A. Thus, the coNP-hardness proof for the problem of deciding whether there is a unique minimal/minimum-size downward covering set for A (see the proofs of Theorems 28 and 31) immediately implies that the problem of deciding whether there is a nontrivial minimal/minimum-size downward covering set for A is NP-hard. However, since the latter problem can easily be reduced to the search problem (because the search problem, when used as a function oracle, yields the set of all alternatives if and only if this set is the only minimal/minimum-size downward covering set for A), it follows that the search problem cannot be solved in polynomial time unless P=NP. □
6 Conclusions and Open Questions
In this paper we have systematically studied the complexity of various problems related to inclusion-minimal and minimum-size unidirectional (i.e., either upward or downward) covering sets. We have established hardness or completeness results for either of NP, coNP, and \(\varTheta_{2}^{p}\) (see Tables 1 and 2 in Sect. 3). An important consequence is that if P≠NP then neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time. This sharply contrasts with Brandt and Fischer’s result that minimal bidirectional covering sets in fact are polynomial-time computable [7].
Tables 1 and 2 also list the best upper bounds we could establish for these problems. In some cases, these upper bounds do not coincide with the lower bounds established, for example, when \(\varTheta_{2}^{p}\)-hardness but only membership in \(\varSigma_{2}^{p}\) could be proven. As an interesting task for future research, we propose to close these complexity gaps. As suggested by an anonymous reviewer, a good candidate problem for finding a reduction to prove \(\varSigma_{2}^{p}\)-completeness for problems related to minimal unidirectional covering sets is the problem of deciding whether a positive literal belongs to a minimal model of a propositional formula (see [18]).
Such payoff vectors are called imputations; see, e.g., [12, 40] for the game-theoretic notions not defined here.
In general, ≻ need not be transitive or complete. For alternatives x and y, x≻y (equivalently, (x,y)∈ ≻) is interpreted as x being strictly preferred to y (and we say “x dominates y”), e.g., due to a strict majority of voters preferring x to y (recall Fig. 1 for an example).
Consider the set A={a,b,c} of three alternatives with the dominance relation defined by a≻b≻c. Note that A is not a downward covering set for itself, since it violates internal stability (UC_{d}(A)={a,b}≠A, due to c being downward covered by b in A); both {a,b} and {b,c} violate internal stability as well (e.g., UC_{d}({a,b})={a}≠{a,b}); and external stability is violated by {a,c} (due to b∈UC_{d}({a,c}∪{b})=UC_{d}(A)={a,b}), each singleton (c∈UC_{d}({a}∪{c})={a,c} shows this for {a}, a∈UC_{d}({b}∪{a})={a} works for {b}, and b∈UC_{d}({c}∪{b})={b} works for {c}), and the empty set (due to, e.g., a∈UC_{d}(∅∪{a})={a}). Thus A has no downward covering set at all.
Consider, for example, the set A={a,b,c,d,e} of five alternatives with the dominance relation defined by a≻b≻c≻d≻a and b≻e. It is easy to see that both {a,c,e} and {b,d} are minimal upward covering sets for A, but only {b,d} is an upward covering set of minimum size for A. That is, {a,c,e} is a minimal, but not minimum-size upward covering set for A.
This type of reduction was introduced by Ladner et al. [32]. Informally stated, a disjunctive truth-table reduction between two decision problems X and Y computes, given an instance x, in polynomial time k queries y_{1},y_{2},…,y_{k} such that x∈X if and only if y_{i}∈Y for at least one i, 1≤i≤k. This reduction can be adapted straightforwardly to function problems F and G: Fdisjunctively truth-table reduces toG if, given an instance x, in polynomial time we can compute k queries y_{1},y_{2},…,y_{k} such that F(x) can be computed from G(y_{i}) for at least one i, 1≤i≤k.
The argument is analogous to that used in the construction of Brandt and Fischer [7] in their proof of Theorem 2. However, in contrast with their construction, which implies that either\(\{x_{i}, x'_{i}\}\)or\(\{\overline{x}_{i}, \overline{x}'_{i}\}\), 1≤i≤n, but not both, must be contained in any minimal upward covering set for A (see Fig. 2), our construction also allows for both \(\{u_{i}, u'_{i}\}\) and \(\{\overline{u}_{i}, \overline{u}'_{i}\}\) being contained in some minimal upward covering set for A. Informally stated, the reason is that, unlike the four-cycles in Fig. 2, our four-cycles \(u_{i} \succ\overline{u}_{i} \succ u'_{i} \succ\overline{u}'_{i} \succ u_{i}\) also have incoming edges.
For example, recall Wagner’s \(\varTheta_{2}^{p}\)-completeness result for testing whether the size of a maximum clique in a given graph is an odd number [51]. One key ingredient in his proof is to define an associative operation on graphs, ⋈, such that for any two graphs G and H, the size of a maximum clique in G⋈H equals the sum of the sizes of a maximum clique in G and one in H. This operation is quite simple: Just connect every vertex of G with every vertex of H. In contrast, since minimality for minimal upward covering sets is defined in terms of set inclusion, it is not at all obvious how to define a similarly simple operation on dominance graphs such that the minimal upward covering sets in the given graphs are related to the minimal upward covering sets in the connected graph in a similarly useful way.
Our argument about \((B_{i}, \succ_{i}^{B})\) can be used to show, in effect, DP-hardness of upward covering set problems, where DP is the class of differences of any two NP sets [42]. Note that DP is the second level of the boolean hierarchy over NP (see Cai et al. [10, 11]), and it holds that \(\mathrm {NP}\cup \mathrm {coNP}\subseteq \mathrm {DP}\subseteq \varTheta_{2}^{p}\). Wagner [51] proved appropriate analogs of Lemma 1 for each level of the boolean hierarchy. In particular, the analogous criterion for DP-hardness is obtained by using the wording of Lemma 1 except with the value of m=1 being fixed.
This implies that d_{1} is not upward covered by either \(u_{1,2}'\) or \(\overline{u}_{1,2}'\), since d_{3} dominates them both but not d_{1}.
This is different from the case of minimum-size upward covering sets for the dominance graph constructed in the proof sketch of Theorem 2. The construction in the proof sketch of Theorem 18 cannot be used to obtain complexity results for minimum-size downward covering sets in the same way as the construction in the proof sketch of Theorem 2 was used to obtain complexity results for minimum-size upward covering sets.
Acknowledgements
This work was supported in part by DFG grants BR-2312/6-1, RO-1202/12-1 (within the European Science Foundation’s EUROCORES program LogICCC), BR 2312/3-2, RO-1202/11-1, and RO-1202/15-1, and by the Alexander von Humboldt Foundation’s TransCoop program. This work was done in part while the fifth author was visiting the University of Rochester.