Sub-exponential Time Parameterized Algorithms for Graph Layout Problems on Digraphs with Bounded Independence Number

Fradkin and Seymour (J Comb Theory Ser B 110:19–46, 2015) defined the class of digraphs of bounded independence number as a generalization of the class of tournaments. They argued that the class of digraphs of bounded independence number is structured enough to be exploited algorithmically. In this paper, we further strengthen this belief by showing that several cut problems that admit sub-exponential time parameterized algorithms (a trait uncommon to parameterized algorithms) on tournaments, including Directed Feedback Arc Set, Directed Cutwidth and Optimal Linear Arrangement, also admit such algorithms on digraphs of bounded independence number. Towards this, we rely on the generic approach of Fomin and Pilipczuk (in: Proceedings of the Algorithms—ESA 2013—21st Annual European Symposium, Sophia Antipolis, France, September 2–4, 2013, pp. 505–516, 2013), where to get the desired algorithms, it is enough to bound the number of k-cuts in digraphs of bounded independence number by a sub-exponential FPT function (Fomin and Pilipczuk bounded the number of k-cuts in transitive tournaments). Specifically, our main technical contribution is a combinatorial result that proves that the yes-instances of the problems (defined above) have a sub-exponential number of k-cuts. We prove this bound by using a combination of chromatic coding, inductive reasoning and exploiting the structural properties of these digraphs.


Introduction
Tournaments form one of the most well studied families of digraphs, both algorithmically and structurally. In particular, whenever we try to generalize results that hold for undirected graphs to digraphs, arguably, one of the first families to consider is that of tournaments. Indeed, this has been the case when designing parameterized algorithms or approximation algorithms. Two problems that have been extensively studied on tournaments are Directed Feedback Vertex Set (DFVS) and Directed Feedback Arc Set (DFAS).
In the realm of approximation, we know that DFVS admits a 7/2-approximation algorithm on tournaments [20], and DFAS admits a PTAS on tournaments [18]. Here, it is worth to point out that whether or not DFAS is NP-complete on tournaments was a well known open problem in the area [4]. First, Ailon et al. [1] proved that unless NP⊆BPP, DFAS on tournaments admits no polynomial-time algorithm. Shortly afterwards, the proof that DFAS is NP-complete was attained simultaneously and independently by Alon [2] and Charbit et al. [6].
For DFVS on tournaments, the best known parameterized algorithm runs in time 1.618 k · n O( 1) [19]. Prior to this the fastest known parameterized algorithm for DFVS ran in time 2 k · n O( 1) [10], based on iterative compression. As in the case of approximation, from the viewpoint of Parameterized Complexity, DFAS on tournaments is "easier" than DFVS on tournaments. Here, we mean that for DFAS on tournaments, sub-exponential time parameterized algorithms are known. The quest for sub-exponential time parameterized algorithms for DFAS has a rich history. For a long time (even after the 2 k · n O(1) -time algorithm for DFVS was discovered), the question of the existence of an algorithm for DFAS that runs in time 2 k · n O( 1) was still being posed as an open problem.
Based on a generic method called chromatic coding (also used in our paper), Alon et al. [3] gave the first sub-exponential time parameterized algorithm for DFAS, which runs in time 2 O( √ k log 2 k) · n O( 1) . This was the first problem not confined to planar graphs (or generalizations such as apex-minor-free graphs) that was shown to admit a sub-exponential time parameterized algorithm. Later, simultaneously and independently, Feige [11] and, Karpinski and Schudy [17] gave faster algorithms that run in 1) . Fomin and Pilipczuk [13] presented a general approach, based on a bound on the number of k-cuts (defined below) in transitive tournaments, that achieved the same running time for DFAS. Using this framework they also designed the first sub-exponential time algorithms for Directed Cutwidth and Optimal Linear Arrangement (OLA) (defined later) on tournaments. Barbero et al. [5] studied Directed Cutwidth and OLA on semi-complete digraphs (that is, digraphs where for any two vertices u and v, at least one of the arcs (u, v) and (v, u) is present) and showed that these problems are NP-complete on semi-complete digraphs. Furthermore, they showed that Directed Cutwidth does not a admit polynomial kernel on semi-complete digraphs but admits a polynomial Turing kernel. Finally, they obtained a linear vertex kernel for OLA on semi-complete digraphs.
The measure of directed cutwidth plays a key role in the work of Chudnovsky and Seymour [8] where it is shown that tournaments are well-quasi-ordered under immersion. This measure was considered by Chudnovsky et al. [7] also in their algorithmic study of Immersion on tournaments. Later, Fradkin and Seymour [14] showed that the Directed Pathwidth and Topological Containment problems on tournaments are fixed parameter tractable (FPT). Fomin and Pilipczuk [12,13], and Pilipczuk [21] revisited these problems and gave the best known algorithms for them on tournaments. Fradkin and Seymour [15], to generalize their results from tournaments to broader families of graphs, introduced the idea of digraphs that have bounded independence number. In particular, tournaments have independence number 1. They showed that Edge disjoint Paths admits an XP algorithm (that is, an algorithm with running time of the form of n f (k) , where n is the number of vertices in the input graph and k is the number of pairs between which one is asked to find edge-disjoint paths) on this family of graphs.
In this paper, we study well-known cut problems (DFAS, Directed Cutwidth and OLA) on digraphs of bounded independence number. Our main contribution is proving a sub-exponential FPT bound on the number of k-cuts (defined below) in the Yes instances of these problems, which shows that the sub-exponential behavior of these problems on tournaments generalizes to digraphs of bounded independence number. Observe that for α = 1, D α is a family of tournaments. For simplicity, we assume to work with simple digraphs. However, all our results hold also when the digraph is not simple. That is, for any pair of vertices u, v, both the arcs (u, v) and (v, u) can be present in the digraph. A digraph is a DAG (Directed Acyclic Graph) if it has no directed cycles.

Problem Statements and Our Algorithms
We first study the following problem.

Directed Feedback Arc Set (DFAS)
Parameter: k Input: A digraph D and an integer k. Question: Does there exist S ⊆ E(D) of size at most k such that D − S is a DAG?
Our first theorem gives a sub-exponential time algorithm for DFAS on D α .
Towards the definition of the second problem, let D be a digraph.
The cutwidth of D, denoted by ctw(D), is the smallest possible width of an ordering of V (D). Now, the second problem is defined as follows.

Directed Cutwidth
Parameter: k Input: A digraph D and an integer k. Question: Is ctw(D) ≤ k?
We present a sub-exponential time algorithm for Directed Cutwidth on D α .
Towards the definition of the third problem, let D be a digraph. For two integers i, j, let [i > j] evaluate to 1 if i > j, and to 0 otherwise. The cost of an ordering In other words, every arc (v i , v j ) directed backward in σ costs a value equal to its length, where the length of (v i , v j ) is the distance between v i and v j in σ . Our last problem seeks an ordering of cost at most k.
Optimal Linear Arrangement (OLA) Parameter: k Input: A digraph D and an integer k. Question: Is there an ordering of V (D) of cost at most k?
Our third theorem gives a sub-exponential time algorithm for OLA on D α .

Main Contribution and Methods
Our algorithms are based on the general framework of Fomin and Pilipczuk [13] to design parameterized sub-exponential time algorithms. The main ingredient to prove in order to employ this framework is a combinatorial upper bound on the number of "k-cuts" in graphs that are Yes-instances of the problem at hand. The proof for the combinatorial bound in our case is completely different from the proof given by Fomin and Pilipczuk [13] for transitive tournaments. The bound of Fomin and Pilipczuk [13] is achieved by mapping the set of k-cuts in a transitive tournament to the set of partitions of the integer k. Then, an asymptotic bound on the partition number of an integer yields a bound on the number of k-cuts in a transitive tournament. In the case of digraphs with bounded independence number, we do not know how to attain the desired bound by utilizing such partitions of integers.
Before we go further, we define the notion of k-cuts.
The k-cut is denoted by the ordered pair (L, R). The set L is called the left part of the cut, and the set R is called the right part of the cut. The arcs in E(R, L) are the cut-arcs of (L, R).
Our first technical contribution is an upper bound on the number of k-cuts in D α .

Lemma 1.4 If D ∈ D α , then for any positive integer k, the number of k-cuts in D is
. That is, it shows that the number of k-cuts in digraphs in D α is upper bounded by a sub-exponential function in n. Clearly, such a bound is not sufficient to design sub-exponential time parameterized algorithms. If any of the problems DFAS, Directed Cutwidth or OLA on D α admits a polynomial kernel, then Lemma 1.4 can readily yield a sub-exponential time parameterized algorithm for the corresponding problem. However, we do not know whether these problems admit such kernels, and the resolution of these questions remains an interesting open problem.
Our second main technical contribution is an upper bound on the number of k-cuts in a subfamily of D α . This bound suffices to prove Theorems 1.1, 1.2 and 1.3 by embedding it in the framework of Fomin and Pilipczuk [13]. Let us first define this subfamily. Given a vertex v ∈ V (D), denote the set of out-neighbors of v in D by Observe that a digraph is d-out degenerate if and only if it has a d-out-degeneracy sequence, that is there is an ordering of the vertex set of the digraph such that each vertex has at most d edges to the vertices before it. Also observe that DAGs are 0-outdegenerate. Next, we define a class of digraphs having small independence number and bounded out-degeneracy. Formally, D α,d = {D : D ∈ D α and D is d-out-degenerate}. Note that if D ∈ D α,d , then every induced subgraph D of D belongs to D α,d . Our second main technical contribution is formally stated as follows.

Lemma 1.5 If D ∈ D α,d , then for any positive integer k, the number of k-cuts in D is
where c is a fixed absolute constant.
One can easily see that if (D, k) is a Yes-instance of DFAS, Directed Cutwidth or OLA, then D is k-out-degenerate. Thus, Lemma 1.5 implies a sub-exponential (in k) upper bound on the number of k-cuts for Yes-instances of these problems. In fact, for OLA one can show that D is 2k 2/3 -out-degenerate, and thus obtain an improved upper bound on the number of k-cuts for Yes-instances. Since the k-cuts of any digraph can be enumerated with polynomial delay [13], hence the upper bounds in Lemmas 1.4 and 1.5 are constructive.
In what follows, we present our proof strategies for the results stated above. Proof Strategy of Lemma 1. 4 We first make a very simple observation, which serves as the starting point of our proof. Let V (D) = V 1 · · · V be some partition of V (D). Then, the number of k-cuts in D is upper bounded by the product of the number of k-cuts in the digraph induced by each V i . Thus, we aim to partition V (D) into parts that induce "sufficiently structured" subdigraphs-we want the number of k-cuts in D[V i ], for any i ∈ [ ], to be "easier" to upper bound than the number of k-cuts in D directly. Moreover, since our aim is to achieve a bound of n o(k) for the total number of k-cuts in D, we want a partition V (D) = V 1 · · · V where = o(k). To this end, we utilize Gallai-Milgram's Theorem (explained next) under the canvas of chromatic coding.
On the one hand, Gallai-Milgram's Theorem states that if the size of a maximum independent set in a digraph is α, then its vertex set can be partitioned into at most α parts such that the digraph induced by each of these parts has a directed Hamiltonian path. On the other hand, chromatic coding (in its derandomized form) provides a family F of partitions of V (D) such that (i) |F| = 2 o(k) log n, (ii) for each k-cut (L, R) in D, there exists a partition P ∈ F such that all the cut arcs of (L, R) go across the parts of P, and (iii) the number of parts of each partition in F is upper bounded by O( √ k). If the cut-arcs of (L, R) go across the parts of a partition P, we say that (L, R) respects P. To see how to combine these two tools, let F be a family provided by chromatic coding. Since the number of partitions in F is 2 o(k) log n, and for each k-cut (L, R) there exists a partition in F that it respects, it suffices to bound the number of k-cuts that respect a particular (arbitrary) partition in F. Then, the total number of k-cuts in the digraph will be the product of the number of k-cuts that respect a partition in F, over all partitions in F.
Consider an arbitrary partition P ∈ F (of V (D)). Let P = P 1 · · · P . Recall that = O( √ k), and the number of k-cuts in D is at most the product of the number of k-cuts in D[P i ], over all i ∈ [ ]. Here, a crucial insight is that the number of k-cuts in D that respect P is at most the product of the number of 0-cuts in D[P i ], over all i ∈ [ ]. Thus, we have reduced our problem to upper bounding the number of 0-cuts in a digraph. Now, to upper bound the number of 0-cuts in D[P i ] by n o(k) , we utilize Gallai-Milgram's Theorem. Since D[P i ] ∈ D α , Gallai-Milgram's Theorem implies that P i can be partitioned into at most α parts, say P i = P i1 . . . P iq , q ≤ α, such that for each j ∈ [q], D[P i j ] has a directed Hamiltonian path. Thus, we have finally reduced our problem to finding 0-cuts in digraphs that have a directed Hamiltonian path. As we will see later, the number of 0-cuts in such digraphs is linear in its number of vertices. Combining everything together, we are able to bound the number of k-cuts in D by n O(α √ k) . Proof Strategy of Lemma 1.5: Each vertex in a digraph D has two choices of how to participate in a cut-it can belong either to its left side or to its right side. Thus, if |V (D)| = n, a trivial upper bound on the total number of k-cuts in D is 2 n . Suppose that we have (somehow) reached a "situation" where most of the vertices must belong to only one of the sides of a k-cut. Then, the arguments to attain the 2 n bound imply that the number of k-cuts is at most 2 q , where q is the number of vertices which possibly have both choices. By the bound in Lemma 1.4, we can further conclude that the number of k-cuts is, in fact, at most On a different note, suppose that we can identify a set of vertices in D, Our algorithm combines the two ideas above to obtain the desired bound. For any vertex v ∈ V (D), we aim to bound the number of k-cuts in D where v is "forced" to belong to the left part. We exploit the position of v in a fixed d-out-degeneracy sequence of D to conclude that a large number of vertices are forced to belong to one side of these cuts. Then, building on the second idea, we inductively find a set of vertices such that the digraph induced on it has independence number strictly smaller than the independence number of D. For such a set of vertices, we can inductively assume that the number of k-cuts in the digraph induced by them is 2 o(k) . Having this bound at hand, we are able to conclude the proof. Proof Strategy of Theorems 1.1, 1.2 and 1.3 To obtain sub-exponential FPT algorithms for DFAS, Directed Cutwidth and OLA on D α , we first use Lemma 1.5 to bound the number of k-cuts in the digraphs of the Yes-instances of these problems by 2 o(k) . Here, we rely on the observation that these digraphs must be k-out-degenerate. Though we do not explicitly state this, the procedures to bound the number of k-cuts in both Lemmas 1.4 and 1.5 are constructive. However, constructiveness is not necessary since a standard branching procedure can also enumerate all k-cuts in a digraph with polynomial delay [13]. To actually solve any of the three problem, we rely on dynamic programming procedures over the k-cuts in the input digraph.
The last two steps of this proof (namely, the enumeration and the dynamic programming procedures) are quite standard, based on the work by Fomin and Pilipczuk [13] to obtain sub-exponential FPT algorithms for DFAS, Directed Cutwidth and OLA on tournaments. For the sake of completeness, we give the full proofs in this article too.

Preliminaries
For a partition P = P 1 · · · P , each P i is referred to as a part of P. For a digraph D, V (D) denotes its vertex set and E(D) its arc set. We write In other words, X is an independent set in the underlying undirected graph of D. The independence number of a digraph is equal to the size of the maximum independent set it contains. A directed Hamiltonian path in D is a directed simple path on all vertices in D. For a set of vertices , v i is the ith vertex of the ordering.

Bounding the Number of k-cuts for Digraphs in DĮ
n this section, we prove that the number of k-cuts in any digraph on n vertices with bounded independence number is n o(k) . In particular we prove Lemma 1.4. Let us recall that a k-cut in a directed graph D is a partition of the vertex set of D into two parts, V (D) = L R, such that |E(R, L)| ≤ k. Let us note that a 0-cut in a digraph D is a partition (L, R) of the vertex set V (D) such that there are no arcs from R to L in D.
At the heart of the proof of Lemma 1.4 is a simple observation that helps us focus on parts of the digraph for which bounding the number of k-cuts is easier. This simple observation is then exploited to its fullest using two main tools -(1) the Gallai-Milgram's Theorem and (2) chromatic coding. Let us state them formally. We begin by stating this key observation, followed by formally defining both these ideas.

Lemma 2.1 Let D be a digraph and k
Proof To prove the lemma, observe that, it is enough to prove that for any k-cut (L, R) in D, there exists k-cuts Thus, if we can partition the vertex set of D into o(k) parts such that it is "easier" to bound the number of k-cuts in each of these parts, then we are done. At a high level, we will first partition the vertex set of D using chromatic coding, and then further partition each part of this partition using Gallai-Milgram's Theorem. We will then conclude by proving that the number of k-cuts in each of the sub-parts is linear in the number of vertices. We now state the Gallai-Milgram's Theorem formally.
Next, we state the technique of chromatic coding in its derandomized form. To this end, we first define universal (n, k, r )-coloring family and then state the known results about the existence of such a families of bounded size. This result is called the chromatic coding lemma. For any graph G, a proper vertex coloring of G is a function Observe that the above mentioned definition holds for digraphs too, where the notion of proper vertex coloring is defined on its underlying undirected graph.
A formulation of the Chromatic Coding lemma, in the way that is useful to us, can be seen in the following corollary.

Corollary 1 For any digraph D on n vertices, and an integer k, there exists a family F of partitions of V (D) into at most 2
√ k parts, such that, 1. for any k-cut (L, R) in D, there exists a partition P = {P 1 , . . . , P q } in the family F, such that for any cut-arc (u, v) of (L, R), there exists i, j ∈ [q], i = j, such that u ∈ P i and v ∈ P j , and Here, if for a certain i, P i = ∅, then we discard this part from the partition P h .
We will now show that F is indeed the family with the required properties. Since R) , on the vertex set of D with only the cut-arcs of (L, R). Note that |E(D (L,R) )| ≤ k. Thus, from the definition of (n, k, 2 √ k )universal coloring family, there exists a function h : such that h is a proper vertex coloring of D (L, R) . Consider the partition P h ∈ F. Let P h = P 1 · · · P 2 √ k . Since h is a proper coloring of D (L, R) and all the cut-arcs of (L, R) are in D (L, R) , for any cut-arc (u, v) of (L, R) For the rest of this section, let F denote the family described in Corollary 1 for the digraph D and integer k. For any arc (u, v) of a digraph and a partition P = P 1 · · · P q of the vertex set of the digraph, we say that the arc (u, v) goes across the parts of this partition P, if u ∈ P i , v ∈ P j and i = j. For any partition P of the vertex set of the digraph D, we say that a k-cut (L, R) in D respects P if all the cut-arcs of (L, R) go across the parts of P. The next lemma states that, the number of k-cuts in D is at most the sum of the number of k-cuts that respect a partition P, over all partitions P ∈ F. Since |F| = 2 o(k) , it is enough to bound the number of k-cuts that respect an arbitrary partition in F by n o(k) . For the digraph D, an integer k and P ∈ F, let N P be the number of k-cuts in D that respect P.

Lemma 2.2 The total number of k-cuts in D is at most
Proof To prove the lemma, we need to prove that for any k-cut (L, R) in D, there exists P ∈ F such that (L, R) respects P. This follows from Corollary 1.
Henceforth, let us fix P = P 1 · · · P q , q ≤ 2 √ k , where P is an arbitrary partition in F. We are now only interested in bounding the number of k-cuts in D that respect P. It follows from Lemma 2.1, that to bound the total number of k-cuts in D, it is sufficient to bound the number of k-cuts in D[P i ], for each i ∈ [q]. We now have the following lemma, that says something much stronger. To bound the number of k-cuts in D that respect P, it is sufficient to bound the number of just the 0-cuts in D[P i ], for all i ∈ [q]. Proof Observe that to prove the lemma it is enough to prove that for any k-cut (L, R) of D that respects P, there exists 0-cuts and v ∈ L. This contradicts that (L, R) respects P.
Thus, we have further narrowed down the class of k-cuts that we want to bound. More precisely, we are now interested in bounding the number of 0-cuts in D[P i ], for any part P i of P. Since D ∈ D α , for any P i ∈ P, D[P i ] ∈ D α . Thus, from Gallai-Milgram Theorem, the vertex set of P i can be partitioned into at most α parts, say P i = P i1 · · · P i , ≤ α, such that for each j ∈ [ ], D[P i j ] has a directed Hamiltonian path. We will now prove that for any digraph that has a directed Hamiltonian path, the number of 0-cuts in it are linear in the number of its vertices. Fig. 1 The vertex partition for the sub-exponential XP bound. P = {P 1 · · · P q } is the vertex partition obtained using chromatic coding and P i = P i1 · · · P i is the partition obtained using Gallai-Milgram's Theorem. Each P i j contains a Directed Hamiltonian Path. The cut arcs of all the cuts that respect P are marked in blue (Color figure online)

Lemma 2.4 Let D be a digraph on n vertices that has a directed Hamiltonian path.
Then the number of 0-cuts in D is at most n + 1.
Proof Since D has a directed Hamiltonian path, let {v 1 , . . . , v n } be the vertex set of D such that for each We now claim that, for all j > i, v j ∈ R. Suppose not. Then there exist a j > i, such that v j ∈ L. Since j > i, and v i appears before v j in the Hamiltonian path ordering. Thus, there is a directed path from v i to v j in D. Since v i ∈ R and v j ∈ L, an arc of this directed path is a cut-arc for (L, R), which contradicts that (L, R) is a 0-cut.
Thus, for any i ∈ [n], the number of 0-cuts in D where v i is the first vertex in the ordering (v 1 , . . . , v n ) that belongs to the right part of these cuts is exactly 1. Since any cut in D, either does not contain any vertex in its right part (there is only one such cut) or contains some vertex, the total number of 0-cuts in D is at most n + 1.
We are now ready to prove Lemma 1.4. An illustration depicting the partitioning used in the proof of Lemma 1.4 is given in Fig. 1.

Proof of Lemma 1.4
Let N be the total number of k-cuts in D. Consider the family F of Corollary 1 for the digraph D and integer k. From Corollary 1, For each partition P ∈ F, let N P be the number of k-cuts in D that respect P. From Lemma 2.2, N ≤ P∈F N P . Consider any arbitrary partition P ∈ F. Let P = P 1 . . . P q , and from Corollary 1 we have q ≤ 2 √ k . For any i ∈ [q], let N P i be the number of 0-cuts in D[P i ]. From Lemma 2.3, N P ≤ i∈[q] N P i . Since D ∈ D α , for any P i , D[P i ] ∈ D α . Thus, from Gallai-Milgram Theorem, the vertex set of P i can be partitioned into at most α parts, say P i = P i1 . . . P i , ≤ α, such that such that for each j ∈ [ ], D[P i j ] has a directed Hamiltonian path. From Lemma 2.4, the number of 0-cuts in D[P i j ] is at most n + 1. From Lemma 2.1, N P i ≤ j∈[ ] (n i, j + 1) ≤ (n + 1) ≤ (n + 1) α , where n i, j = |P i, j |. Combining everything stated above, we get that,

Improved Bounds for Digraphs in D˛with Bounded Out-Degeneracy
In this section we give the proof of Lemma Throughout this section, D is a digraph on n vertices and D ∈ D α,d . Let (v 1 , . . . , v n ) be a d-out-degeneracy sequence of D. For any i ∈ [n], we say that a k-cut (L, R) Note that the collection of the sets of type-i cuts for all i ∈ [n] 0 , forms a partition of the set of all the k-cuts. Observe that there is exactly 1 type-0 cut in any digraph.

For any k-cut (L, R) in D of type
Lemma 3.1 states that the vertex set of D can be partitioned into 4 parts with the following properties. The digraph induced on the first part is either empty or belongs to D α ,d , for α < α. To bound the number of k-cuts in such a digraph we will use an induction on α. For the second part of this partition, we prove that for any k-cut (L, R) of type-i, all the vertices of this part belong to L. Similarly, for the third part of this partition, we prove that for any k-cut (L, R) of type-i, all the vertices of this part belong to R. Therefore, there is a unique k-cut of type-i in the digraph induced by the second and third part. The last part of the partition has the property that the number of vertices in this part is "small". For the digraph induced by this part, we will get the desired bound by using Lemma 1.4 on this digraph.
The proof of Lemma 3.1 is deferred for later. We will now proceed towards the proof of Lemma 1.5 using Lemma 3.1 and induction on α. At any inductive step we use the partition of Lemma 3.1 and bound the number of k-cuts of type-i in the digraph induced on each part of the partition, thereby bounding the number of k-cuts in D because of Observation 1.
Proof of Lemma 1. 5 We prove the lemma using induction on α. For any positive integer α, let us denote the bound of Lemma 1.4 on the number of k-cuts in D ∈ D α , on at most d + α(2k + 1) vertices, by η(α, d, k). That is, η(α, d, k) = 2 c √ k log k · (d + 1 + α(2k + 1)) 2α √ k · log(d + α(2k + 1)), where c is the absolute constant hidden in the O notation of the expression in Proposition 2. Let N k (n, α, d) denote the maximum number of k-cuts in D for any digraph D ∈ D α,d on n vertices. We claim that for any positive integers n, d and α > 1, N k (n, 1, d) ≤ 1 + n · η (1, d, k) and N k (n, α, d) ≤ 1 + N k (n, α − 1, d) · η(α, d, k) · n. Solving the recurrence, we will get the desired bound on the number of k-cuts in D.
Let us first prove that for any positive integers n and d, N k (n, 1, d) ≤ 1 + n · η (1, d, k). If the independence number of the digraph D is 1, then from Lemma 3.  η(1, d, k). From Observation 1, we conclude that the number of k-cuts in D is at most 1 + η(1, d, k) · n.
By induction hypothesis, let us assume that for any positive integers n, d and for all α < α, the number of k-cuts in any digraph D ∈ D α ,d on at most n vertices is  N k (n, α , d). We will now prove that the number of k-cuts in the digraph  η(α, d, k). Thus, the number of k-cuts of type-i in D is at most N k (n, α − 1, d) · η(α, d, k). From Observation 1, we conclude that the number of k-cuts in D is at most 1 + N k (n, α − 1, d) · η(α, k, d) · n.
Proof of Partitioning Lemma. We start by a lemma that gives an upper bound on the size of a digraph in D α when every vertex has small out-degree.

Lemma 3.2 For any digraph D ∈ D α and a positive integer k, if for all
2α ≤ k, thereby implying that n ≤ α(2k + 1). To prove the above-mentioned claim, we invoke Turan's Theorem ( [9]), which states that for any graph G and integer r , if G does not contain a clique of size r + 1, . Let G be the underlying undirected graph of D. LetḠ be the complement graph of G. Since D ∈ D α ,Ḡ does not contain a clique of size α+1. Thus, by Turan's Theorem, |E(Ḡ)| ≤ (1 − 1 α ) · n 2 2 . SinceḠ is the complement graph of G, |E(G)| ≥ n(n−1) 2α . Intuitive Ideas for the proof Lemma 3.1. Let us begin by recalling that (v 1 , . . . , v n ) is a d-out-degeneracy sequence of D. Also recall that, the aim of proving Lemma 3.1 is to be able to use it to bound the number of k-cuts in D of type-i. Consider any k-cut in D of type-i. By definition, v i ∈ L and for all j > i, v j ∈ R. Thus, v i ∈ V forceL and {v j : j > i} ⊆ V forceR . Thus, to prove Lemma 3.1, we essentially need to partition the vertices that appear before v i in (v 1 , . . . , v n ). Consider the non-neighbors of v i . They induce a digraph whose independence number is strictly less than the independence number of D. Thus, they go to V induct . Thus, we are now left with the goal of partitioning the set of neighbors of v i that appear before v i in (v 1 , . . . , v n ). Since (v 1 , . . . , v n ) is a d-out-degeneracy sequence of D, the number of out-neighbors of v i that appear before v i in (v 1 , . . . , v n ) is at most d. This set of neighbors goes to the set V small . Finally, we are left with the set, say X , of vertices that appear before v i in (v 1 , . . . , v n ) and are in-neighbors of v i . Here, we observe that, if any vertex v ∈ X has out-degree at least k + 1 in D[X ], then there are at least k + 1 arc-disjoint paths from v to v i in D[X ∪ {v i }], and hence in D. Thus, such a vertex v should always belong to same part as v i in any k-cut. Thus, such vertices goes to V forceL . Finally, the remaining vertex set, say X , has the property that each vertex in X has out-degree at most k. By Lemma 3.2, in such a case the size of X is at most α(2k + 1), and hence X goes to V small . We are now ready to prove Lemma 3.1 formally. (v 1 , . . . , v n )

Proof of Lemma 3.1 Let
We have the following from the above discussion.
We now claim that the desired partition S . An illustration depicting this partitioning can be found in Fig. 2. Let us now prove that the sets V induct , V forceL , V forceR and V small satisfy the desired properties.
1. V induct : Observe that when α = 1, that is, when D is a tournament, V N N P = ∅. Therefore, in this case, has an independent set, say X , of size α then X ∪ {v i } is an independent set in D of size α + 1, which contradicts the fact that the size of any independent set in D is bounded by α.
We will now show that for any v j ∈ V I N P,L , there exists k and V I N P is the set of in-neighbors of v i in V P . Consider the set of out-neighbours of v j in D . Since the number of such out-neighbors is at least k + 1 and each of these out-neighbors is an in-neighbor of v i , we conclude that there are at least k + 1 arc-disjoint paths from v j to v i . 3. V forceR : By the definition of type-i cut, V S ⊆ R, for any type-i cut (L, R).
We need to show that |V I N P,S | ≤ α(2k + 1). Recall that, as observed before, for each . This proves that |V I N P,S | ≤ α(2k + 1). This concludes the proof.

Sub-exponential FPT Algorithms for DFAS, DIRECTED CUTWIDTH
and OLA for Digraphs in DĮ n this section, we will give sub-exponential FPT algorithms for DFAS, Directed Cutwidth and Optimal Linear Arrangement when the input graph belongs to D α , for some positive integer α. All these algorithms are based on a three step procedure. The first is observing that the digraphs that are Yes-instances of these problems have sub-exponential FPT many k-cuts. The proofs for DFAS and Directed Cutwidth are based on showing that the digraph in the Yes-instances of the problems are k-out-degenerate, and hence, the bounds follow from Lemma 1.5. For OLA, we show that if there is an ordering of the vertex set of a digraph of cost at most k then the cutwidth of this digraph is O(k 2/3 ). Hence, from the results for Directed Cutwidth, the number of k-cuts in the Yes-instances of OLA is also bounded. The second step is a procedure to enumerate all k-cuts of the input digraph. And the third is to do some dynamic programming procedure over these enumerated cuts to solve the respective problems. The last part of the algorithm (doing dynamic programming over k-cuts) is standard and is identical to the algorithm given by Fomin and Pilipczuk [13]. Proofs are given for completeness.
Before proceeding further, we make a small remark that the proofs of Lemmas 1.4 and 1.5 can be made constructive by using the constructive versions of the Gallai-Milgram's Theorem, Chromatic Coding lemma and a polynomial time procedure to output a d-out-degeneracy sequence of a digraph. Thus, one can actually enumerate all the k-cuts in the input digraphs of these Lemmas using our algorithm. However, for the sake of completeness, we state in Lemma 4.1, a different procedure that using a standard branching, enumerates all the k-cuts in any digraph with polynomial delay.

Sub-exponential Algorithm for Directed Feedback Arc Set
We begin by recalling the problem definition.
that is, the number of backward arcs in this ordering is at most k. Next we bound the number of k-cuts in the Yes-instances of DFAS.

Lemma 4.2 If (D, k) is a Yes-instance of DFAS and D ∈ D α , then the number of k-cuts in D is at most
Proof Since (D, k) is a Yes-instance of DFAS, there exists an ordering, say Thus, (v 1 , . . . , v n ) is a k-out-degeneracy sequence of V (D). Therefore, the bound follows from Lemma 1.5.

Directed Cutwidth
Parameter: k Input: A digraph D and an integer k. Question: Is ctw(D) ≤ k?
Next we bound the number of k-cuts in the Yes-instances of DFAS. (D, k) is a Yes-instance of Directed Cutwidth and D ∈ D α , then the number of k-cuts in D is at most 2 c(α+1)   Thus, for the ordering (v 1 , . . . , v )

Lemma 4.3 If
We will now prove that if there exists an ordering of L, say (v 1 , . . . , v ), such that Thus, T ((L, R)) = 1. This concludes the proof.

Sub-exponential Algorithm for Optimal Linear Arrangement
Let D be a digraph. For an ordering that is, every arc directed backward in the ordering contributes a cost that is equal to the length of this arc, which is the distance between the end-points of this arc in the ordering. Recall that [i > j], evaluates to 1 if i > j, to 0 otherwise.

Optimal Linear Arrangement (OLA)
Parameter: k Input: A digraph D and an integer k. Question: Is there an ordering of V (D) of cost at most k?
The following proposition gives an alternate definition of the cost of an ordering. Lemma 4.4 shows a relation between the cost of an ordering and its width. Note that this lemma was already proved in [13], but the authors state the result for the case when the input digraph is a semi-complete digraph. We observe that the same proof works for any digraph. For the sake of completeness, we give the same proof here.  For any arc (v q , v p ) ∈ E(D), such that p < q, the length of the arc (v q , v p ) is equal to q − p. Observe that, for any r , the number of arcs of length exactly r with tail in {v i+1 , . . . , v n } and head in {v 1 , . . . , v i } is at most r . Thus, for any r , the total number of arcs of length at most r , with tail in {v i+1 , . . . , v n } and head in {v 1 , . . . , v i }, is at most r (r +1) 2 . In particular, the number of arcs of length at most √ − 1, with tail in {v i+1 , . . . , v n } and head in {v 1 , . . . , v i } is at most Since |E({v i+1 , . . . , v n }, {v 1 , . . . , v i }) 2 3 . Thus, (v 1 , . . . , v n ) is an ordering of cutwidth at most (2k) 2 3 .
Next we bound the number of k-cuts in the Yes-instances of OLA.

Conclusion
In this paper, we designed sub-exponential time parameterized algorithms for DFAS, Directed Cutwidth and OLA on digraphs of bounded independence number. We thus significantly generalized known results for the restricted case of input digraphs that are tournaments. Towards this, we obtained an upper bound on the number of k-cuts in digraphs in D α . This bound is our main contribution, which we believe to find further implications in the future, and to be of independent interest. We conclude with an open problem: Do DFAS, Directed Cutwidth and OLA admit polynomial kernels on D α ? Funding Open Access funding enabled and organized by Projekt DEAL.
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