Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence (AAAI-10)

Fast Local Search Algorithm for Weighted in Tournaments

Fedor V. Fomin and Daniel Lokshtanov Venkatesh Raman and Saket Saurabh Department of Informatics, The Institute of Mathematical Sciences, University of Bergen, Norway. Chennai, India. {fedor.fomin|daniello}@ii.uib.no {vraman|saket}@imsc.res.in

Abstract search algorithms: “Look at all solutions in the neighbor- hood of the current solution ...”) that finding an improved We present a fast local search algorithm that finds an solution in a k-exchange neighborhood, necessarily requires improved solution (if there is any) in the k-exchange brute-force search of the neighborhood; therefore, verify- neighborhood of the given solution to an instance of k nk WEIGHTED FEEDBACK ARC SET IN TOURNAMENTS. ing -optimality requires Ω( ) time (see, e.g. (Aarts and More precisely, given an arc weighted tournament T on Lenstra 1997) p. 339 or (Kleinberg and Tardos 2005) p. 680). n vertices and a feedback arc set F (a set of arcs whose Surprisingly, in some cases the k-exchange neighbor- deletion from T turns it into a ), o(k) hood can be searched significantly faster. In particular, on our algorithm decides in time O(2 n log n) if there sparse structures like planar graphs or graphs not contain- is a feedback arc set of smaller weight and that differs ing some fixed graph as a minor, finding a k-local improve- from F in at most k arcs. To our knowledge this is the ment for many natural problems including , first algorithm searching the k-exchange neighborhood ODD TRANSVERSAL,DOMINATING SET, and r- of an NP-complete problem that runs in (parameterized) c CENTRE can be done in time f(k) · n , where c is a fixed subexponential time. Using this local search algorithm k for WEIGHTED FEEDBACK ARC SET IN TOURNA- constant independent of (Fellows et al. 2009). Similar MENTS, we obtain subexponential time algorithms for results were also obtained for sparse instances of SATISFI- a local search variant of KEMENY RANKING – a prob- ABILITY (Szeider 2009). Algorithms that have time com- c lem in social choice theory and of ONE-SIDED CROSS plexity of the form f(k) · n are said to be fixed parameter MINIMIZATION – a problem in . tractable algorithms. For an introduction to the field Param- eterized Complexity, and more recent developments, see the I – Introduction books (Downey and Fellows 1999; Flum and Grohe 2006; Niedermeier 2006). However, all the known local search Local search algorithms, also known as neighborhood algorithms strongly exploit the sparsity of the structure, in search algorithms, constitute a large class of improvement particular, the fact that in these structures a ball of radius algorithms. To perform local search, a problem specific k induces a subgraph (or substructure) of treewidth f(k), neighborhood distance function is defined on the solution for some function k. So the techniques from (Fellows et al. space and a better solution is searched for in the neigh- 2009; Szeider 2009) do not seem to work for dense struc- borhood of the current solution. In particular, many lo- tures. cal search algorithms are based on searching in the k- In this work, we use completely different techniques to exchange neighborhood. This is the set of solutions that k can be obtained from the current solution by exchanging show that fast search of the -exchange neighborhood is also possible for dense structures. A tournament is a di- at most k elements. A classical example is the TRAVEL- rected graph where every pair of vertices is connected by ING SALESPERSON problem, where the neighborhood of a tour T can be defined as the set of all tours that differ exactly one arc, and a feedback arc set is a set of arcs whose from T in at most k edges, this is called the k-exchange removal makes the graph acyclic (without any directed cy- neighborhood (Ahuja et al. 2002; Lin and Kernighan 1973; cle). Feedback arc sets in tournaments are well studied, Papadimitriou and Steiglitz 1977). from the combinatorial (Erdos¨ and J.W.Moon 1965; Fernan- As a rule of thumb, the chances of finding a better solu- dez de la Vega 1983; Jung 1970; K.B.Reid 1969; K.D.Reid tion grow with k. However, for inputs of size n,ana¨ıve and E.T.Parker 1970; Seshu and Reed 1961; Spencer 1971; brute-force search of the k-exchange neighborhood requires 1980; Younger 1963), statistical (Slater 1961), and algo- rithmic (Ailon, Charikar, and Newman 2005; Alon 2006; nO(k) time, which is not practical even for very small val- Kenyon-Mathieu and Schudy 2007; van Zuylen 2005; van ues of k. It has been generally assumed until very recently, Zuylen et al. 2007) points of view. The problem has several (perhaps because of the typical algorithmic structure of local applications - in psychology it occurs in relation to rank- Copyright c 2010, Association for the Advancement of Artificial ing by paired comparisons: here we wish to rank items Intelligence (www.aaai.org). All rights reserved. by an objective, but we don’t have access to the objective

65 function, but only to pairwise comparisons of the objects in • T \ F  is acyclic. question. An example for this setting is measuring people’s Alon et al. (Alon, Lokshtanov, and Saurabh 2009) intro- preferences for food. The weighted generalization of the duced the method of chromatic coding, a variant of classical problem, WEIGHTED FEEDBACK ARC SET IN TOURNA- color-coding (Alon, Yuster, and Zwick 1995) and obtained MENTS is applied in rank aggregation: Here we are given the first parameterized subexponential time algorithm for k- several rankings of a set of objects, and we wish to produce √ O( k log k)nO(1) a single ranking that, on average, is as consistent as pos- FAST running in time 2 . In fact this was sible with the given ones, according to some chosen mea- the first non-trivial parameterized subexponential time algo- sure of consistency. This problem has been studied in the rithm for a problem on dense graphs. Recently Feige (Feige k context of voting (Borda 1781; Condorcet 1785), machine 2009) has obtained√ a faster algorithm for -FAST running O k O learning (Cohen, Schapire, and Singer 1997), and search in time 2 ( )n (1). In this paper we use the method of engine ranking (Dwork et al. 2001a; 2001b). A natural chromatic coding and show that k-LSFAST is fixed param- consistency measure for rank aggregation is the number of eter tractable when parameterized by k, the number of arcs pairs that occur in different order in the two rankings. This we are allowed to exchange from the current solution. In leads to Kemeney-Young rank aggregation (Kemeny 1959; particular we obtain the following results: Kemeny and Snell 1962), a special case of WEIGHTED • k there is a randomized√ algorithm for -LSFAST running FEEDBACK ARC SET IN TOURNAMENTS. The problem of in time 2O( k log k)n; and finding a feedback arc set of minimum size in an unweighted • k tournament is NP-hard (Alon 2006). However, even the there is a deterministic√ algorithm for -LSFAST that runs weighted version of the problem admits a polynomial time in time 2O( k log k)n log n. approximation scheme (Kenyon-Mathieu and Schudy 2007) To our knowledge this is the first algorithm searching the and it was shown to be fixed parameter tractable (Raman k and Saurabh 2006). In this paper we consider a local search -exchange neighborhood of an NP-complete problem that variant of the following problem: runs in (parameterized) subexponential time. Using the local search algorithm for k-LSFAST, we obtain subex- k-WEIGHTED FEEDBACK ARC SET IN TOURNA- ponential time algorithms for a local search variant of k MENTS ( -FAST) KEMENY RANKING – a problem in social choice theory Instance: A tournament T =(V,A), a weight function (Conitzer, Davenport, and Kalagnanam 2006; Davenport w : A → N, where N is a set of positive integers and and Kalagnanam 2004; Ephrati and Rosenschein 1993) and k an integer . of ONE-SIDED CROSS MINIMIZATION – a problem in Question : Is there an arc set S ⊆ A such that graph drawing (Eades and Wormald 1994; Dujmovic´ et al. w e ≤ k T \ S e∈S ( ) and is acyclic? 2008; Dujmovic,´ Fernau, and Kaufmann 2008). To define the local search variant of the problem we need Our algorithm is based on the ideas of the k-FAST algo- some definitions. We use to denote the set of feasible so- rithm of Alon et al (Alon, Lokshtanov, and Saurabh 2009), lutions to a problem P . For edge/arc subset (or vertex sub- but differs significantly from it in a crucial step. The algo- set) problems, is a collection of subsets of edges/arcs and rithm presented in (Alon, Lokshtanov, and Saurabh 2009) a natural neighborhood function is obtained by exchanging k starts by preprocessing the instance and obtains an equiv- elements of the current solution. The neighborhood function alent instance with at most O(k2) vertices in polynomial in which we are interested is called k-exchange neighbor- time. That is, given a tournament T and a positive inte- hoods (k-ExN). We elaborate this further. Let w : A → N ger k, in polynomial time the preprocessing algorithm ei- be a weight function. Then the cost function w : → N ther concludes that T does not have a feedback arc set of   is defined as w(v) for all s ∈ . We say that s weight at most k or finds a new tournament T with O(k2) v∈s  is neighbor of s with respect to k-ExN if |s \ s|≤k and vertices and k ≤ k such that the original tournament T  k |s \ s|≤k Nk s s has a feedback arc set of weight at most , if and only if . Let ( ) denote the set of neighbors of with   respect to k-ExN. T has a feedback arc set of weight at most k . This pre- processing allows them to assume that the instance where k-LOCAL SEARCH WEIGHTED FEEDBACK ARC SET k they actually apply the subexponential time algorithm is of IN TOURNAMENTS ( -LSFAST) size O(k2) only, which is integral to their time analysis. For Instance T =(V,A) : A tournament , a feedback arc k-LSFAST, and in fact for most local search problems, one F ⊆ A T w : A → N set of , a weight function and an can show using standard techniques (Bodlaender et al. 2008; integer k.  Fortnow and Santhanam 2008) that such a pre-processing Question F ∈Nk(F ) : Does there exist a such that step would imply that coNP is a subset of NP/poly, an event w(F )

66 √ ors then√ the probability that G is properly colored is at least t =4 k T 1. Let . Color the vertices of uniformly at e − q/8 random with colors from {1,...,t}. (2 ) . 2. Let Ac be the set of arcs whose endpoints have differ- Lemma 1 implies that if we randomly color the vertices ent colors. Find a minimum weighted feedback arc set √ √  of T with t = 8 · 2k =4 k colors then the probability F ∈Nk(F ) such that Sin and Sout are contained in  that the subgraph T =(V,Sin ∪ Sout) is properly colored, Ac. Sin Sout Ac or equivalently√ that √ and are subsets of is at least (2e)− 2k/8 =2−c k for some fixed constant c,asT  has Figure 1: Outline of the algorithm for k-LSFAST. at most 2k arcs. Next we show how to implement the second step of our algorithm using dynamic programming over the colored instance. and 0 otherwise. Given a D =(V,A) and a set F of arcs in A, define D{F } to be the directed graph ob- tained from D by reversing all arcs of F . In our arguments Solving a Colored Instance we will need the following characterization of minimal feed- back arc sets in directed graphs. The second part of our algorithm takes a t-colored tourna- T V ∪ V ∪ ... ∪ V ,A Proposition 1 [(Gallai 1968)] Let D =(V,A) be a directed ment =(1 2 t ) and a feedback arc set F of T as input, and produces a weighted feedback graph and F be a subset of A. Then F is a minimal feedback  arc set F ∈Nk(F ) such that Sin and Sout are contained arc set of D if and only if F is a minimal set of arcs such that  in Ac and w(F ) is minimized. Now, Sin ∪ Sout ⊆ Ac D{F } is a directed acyclic graph.  implies that F ∩ (A \ Ac)=F ∩ (A \ Ac) and hence  Given a minimal feedback arc set F of a tournament T , T {F }[Vi]=T {F }[Vi] must be an acyclic tournament for the ordering σ corresponding to F is the unique topological every i. Let σold = v1v2 ...vn be the ordering of V cor- ordering of T {F }. Conversely, given an ordering σ of the responding to the given feedback arc set F of T and let vertices of T , the feedback arc set F corresponding to σ is σnew = u1u2 ...un be the ordering of V corresponding  the set of arcs whose endpoint appears before their startpoint to the feedback arc set F of T . Let ni = |Vi| for every in σ. The cost of an arc set F is e∈F w(e) and the cost i and let nˆ be the vector [n1,n2 ...,nt]. For every color σ V T vi vi ...vi of a vertex ordering is the cost of the feedback arc set class i of , let 1 2 ni be the order in which the ver- corresponding to σ. tices of Vi appear according to σold. Observe that since  For a pair of integer vectors pˆ =[p1,...,pt], qˆ = T {F }[Vi]=T {F }[Vi] and every acyclic tournament has [q1,...,qt] we say that pˆ ≤ qˆ if pi ≤ qi for all i. The t- a unique topological ordering we have that the vertices of V T σ sized vector eˆ is [1, 1,...,1], 0ˆ is [0, 0,...,0] and eˆi is the every color class i of appear in the same order in old σ vi vi ...vi t-sized vector with all entries 0 except for the i’th which is and new, that is, 1 2 ni . We exploit this fact to give a 1. For any positive integer m let [m]={1, 2,...,m}. For a dynamic programming algorithm for the problem. given subset V  ⊆ V of a digraph D =(V,A) by D[V ] we V  denote the induced√ subgraph on . For simplicity√ whenever Lemma 2 Given a t-colored tournament T and a feedback log n n log n  n we say or we mean and . arc set F , we can find a minimum weight feedback arc set  2 t+1 F ∈Nk(F ) such that Sin,Sout ⊆ Ac in O(n t(2k+1) ) III - Main Algorithm time and O((2k +1)t+1n) space. In this section we give our algorithm, argue about its cor- x ≥ Si {vi ...,vi } rectness and analyze its . We start with the Proof: For an integer 1 define x = 1 x and S Si ∅ p t description of our algorithm. 0 = 0 = . Given an integer vector ˆ of length in which i n S p S1 ∪S2 ...∪ the ’th entry is between 0 and i, let (ˆ)= p1 p2 Algorithm Description St T p T S p A p T p pt , (ˆ) be [ (ˆ)] and (ˆ) be the arc set of (ˆ). For a F T F p F ∩ A p Our algorithm consists of two steps. In the first step√ we ran- feedback arc set of we set (ˆ)= (ˆ). Observe σ v v ...v V domly color the vertices of our graph with t =4 k colors, that for any ordering = 1 2 n of corresponding to F  T F  \ A F \ A and define the arc set Ac to be the set of arcs whose end- the feedback arc set of such that c = c and x p {v ...,v } points have different colors. In the next step of the algorithm for any integer there exists a ˆ such that 1 x =  S p we find a minimum weighted feedback arc set F ∈Nk(F ) (ˆ). such that Sin and Sout are contained in Ac. A summary of Fixing the tournament T and a feedback arc set F of the algorithm is given in Figure 1. T we define LSFAS(ˆp, k1,k2), to be the weight of a min-  To analyze the first step the algorithm we use the follow- imum weight feedback arc set F of T (ˆp) such that (a) |F p \ F |≤k |F  \ F p |≤k F  \ A ing lemma of Alon et al. (Alon, Lokshtanov, and Saurabh (ˆ) 1, (b) (ˆ) 2 and (c) c = F p \ A k < k < p, k ,k 2009). (ˆ) c.If 1 0 or 2 0 then LSFAS(ˆ 1 2) is set to ∞. We call a feedback arc set F  satisfying these Lemma 1 [(Alon, Lokshtanov, and Saurabh 2009)]√ If a conditions a (k1,k2)-constrained feedback arc set of T (ˆp). graph G with q edges is colored randomly with 8q col- We proceed to prove that the following recurrence holds for

67 LSFAS(ˆp, k1,k2). is sufficient to upper bound the number of legal vectors, that is pˆ that satisfy |δ+(ˆp) ∩ F | + |δ−(ˆp) \ F |≤2k. LSFAS(ˆp, k1,k2)= Let σold = v1 ...vn be the ordering of V corresponding to F , pˆ be a legal vector and S = S(ˆp). Now, let x be the min LSFAS pˆ − eˆi,k1,k2 (1) i largest integer such that vx ∈ S. We prove that for every y 0 k = Here the minimum is taken over , such that , 1 k z k −|{uvi ∈ F (ˆp)}| and k = k −|{vi u ∈ A(ˆp)\F (ˆp)}|. Since there are at least 2 choices for this implies that 1 pi 2 2 pi | δ+ p ∩ F ∪ δ− p \ F | > k p First we prove that the left hand side is at most the right hand ( (ˆ) ) ( (ˆ) ) 2 , contradicting that ˆ is side. Let i be the integer that minimizes the right hand side. legal. To uniquely determine pˆ it is enough to uniquely de- Taking the ordering of S(ˆp−eˆi) corresponding to an optimal   S x vx ∈ S y>x (k ,k ) T (ˆp − eˆi) termine .If is given then , for all , 1 2 -constrained feedback arc set of and ap- v ∈/ S pending it with vi gives an ordering σ of T (ˆp) correspond- y and the argument in the above paragraph implies pˆi {v ,...,v }⊂S S k ,k T p 1 x−2k−1 . Thus, to specify it is sufficient ing to a ( 1 2)-constrained feedback arc set of (ˆ) with x S ∩ C C {v ,...v } p − e ,k ,k w∗ vi ,u to specify and where = x−2k x−1 . Let cost at most LSFAS(ˆ ˆi 1 2)+ u∈S(ˆp) ( pi ). j j j Cj = C ∩ Vj = {va ,v ,...v }. Then S ∩ C = To prove that the right hand side is at most the left j aj +1 bj S ∩ C ∪ S ∩ C ...∪ S ∩ C j hand side, let σ be an ordering of T (ˆp) corresponding to ( 1) ( 2) ( t) and for each there  are at most 2k +1possible choices of S ∩Cj since S ∩Cj is a (k1,k2)-constrained feedback arc set F of T (ˆp) and let i j j j  v ∈ Vi be the last vertex of this ordering. Then v = vp either ∅ or {va ,va ,...v  } for some bj ≤ bj. Thus i j j +1 bj and σ restricted to V (T (ˆp − eˆi)) is an ordering of T (ˆp − eˆi) n k t−1 S  there are (2 +1) choices for and hence at most corresponding to the feedback arc set F ∩ A(ˆp − eˆi).Now, n(2k +1)t−1k2 ≤ n(2k +1)t+1 entries in the memoiza- |F p \ F |≤k v |{uvi ∈ F p }| (ˆ) 1 and is incident to pi (ˆ) edges tion table. Since each recursive step takes O(nt) time the   t+1 of F (ˆp)\F . Similarly |F \F (ˆp)|≤k2 and v is incident to total running time bound becomes n(2k +1) · nt = |{vi u ∈ A p \ F p }| F  \ F p F  \{v} t+1 2 pi (ˆ) (ˆ) edges of (ˆ). Thus O((2k +1) n t).    is a (k ,k )-constrained feedback arc set of T (ˆp − eˆi). The 1 2 In fact, the algorithm provided in Lemma 2 can be total weight of the edges with startpoint in v and endpoint in ∗ i made to run slightly faster by pre-computing the value of V (T (ˆp − eˆi)) is exactly w (v ,u). Thus the  u∈V (T (ˆp)) pˆi w∗(vi ,u)), k and k for every pˆ and i using dy- cost of σ is at least the value of the right hand side of the u∈S(ˆp) pˆi 1 2 inequality, completing the proof. namic programming, and storing it in a table. This would Recurrence 1 naturally leads to a memoization based al- let us reduce the time to compute a table entry using Recur- rence 1 from O(nt) to O(t) yielding an algorithm that runs gorithm for the problem. We compute LSFAS(ˆp, k, k) by O(n·t·(2k+1)t+2) applying Recurrence 1, returning ∞ whenever k < 0 or in time time. This gives us the following 1 theorem. k2 < 0 and storing the output of every recursive call except k < k < k the ones with 1 0 or 2 0 in a table. If at a later stage Theorem√ 1 -LSFAST√ can be solved in expected time of the algorithm a recursive call is made with the same pa- 2O( k log k)n and 2O( k log k)n space. rameters we return the corresponding table entry instead of recomputing the function. To bound the running time it is Proof: Our algorithm proceeds by repeating the steps de- sufficient to bound the size of the memoization table and the scribed in Figure 1. We use Lemma 2 to find a minimum  ∈N F S ,S ⊆ A running time required to compute each entry from the other weight arc set F k( ) such that in out c and w F 

68 randomized coloring step in the algorithm for k-LSFAST. Theorem√ 3 k-LSFKOA can be solved in time Towards this we utilize the following result. 2O( k log k)n log n. Proposition 2 [(Alon, Lokshtanov, and Saurabh 2009)] Our other application stems from graph-drawing and the For any n>√ 10k2 there exists an explicit univer- problem itself is called k-ONE SIDED CROSS MINIMIZA- n, k, O k F |F| ≤ TION (k-OSCM). The problem consists of a bipartite graph sal √( ( ))-coloring family of size O k k G =(V ,V ,E), a permutation π of V , and a positive inte- 2 ( log ) log n. 1 2 1 ger k and the objective is to check whether there is a permu- Finally, combining the algorithm from Theorem 1 with tation πm of V2 such that, when the vertices of V1 are placed the universal coloring families given by Proposition 2 on a line (also called a layer) in the order induced by π and k yields a deterministic subexponential time algorithm for - the vertices of V2 are placed on a second layer (parallel to the LSFAST. first one) in the order induced by πm, then drawing straight- k line segment for each edge in E will introduce no more than Theorem√ 2 -LSFAST can be solved in time k 2O( k log k)n log n. (pairwise) edge crossings. It has been brought to our at- tention by a colleague working in graph drawing that even Other Applications this problem can be modelled as weighted feedback arc set in multi-tournaments, almost similar to the modelling done Our subexponential algorithm for k-LSFAST is useful in for k-KOA. This modelling allows us to prove that the local obtaining subexponential time algorithm for local search search variant obtained by viewing the problem as weighted variants of various other problems arising in different do- feedback arc set in multi-tournaments can be solved in pa- mains. The first one is related to KEMENY RANKING that rameterized subexponential time. arises in social choice theory. Preference lists are typical examples of elements in social science surveys and voting IV – Conclusion systems. In several such cases we wish to combine var- In this paper we obtained a parameterized subexponen- ious lists into one which reflects the opinion of the sur- k veyed group as much as possible. The Kemeny aggrega- tial time algorithm for a local search variant of the - tion problem was introduced by Kemeny (Kemeny 1959; WEIGHTED FEEDBACK ARC SET IN TOURNAMENTS Kemeny and Snell 1962) to abstract out problem of combin- problem. This algorithm was also useful in designing subex- m ponential time algorithm for local search variants of k-KOA ing preference lists into one. Given a set of permutations k (called votes) over a set of n alternatives (called candidates), and -ONE SIDED CROSS MINIMIZATION. It is natural to ask whether the local search variant of weighted feedback the k-KEMENY-OPTIMAL AGGREGATION (k-KOA) prob- lem asks for a permutation of candidates, called an optimal arc set is fixed parameter tractable for all digraphs. Unfor- aggregation, such that the sum of the τ-distance from the tunately, it can easily be shown that it is not the case unless votes is at most k. The τ-distances between two permuta- some unlikely collapse happens in parameterized complex- ity theory. A natural open question is whether we can solve tion π1 and π2 is the number of pairs of candidates that are √ O( k) O(1) ordered differently in the two permutations and is denoted k-LSFAST in time 2 n like k-FAST. Other ques- by τ(π1,π2). Let L(π) denote the set of pairs of candidates tions which remain open are whether we can use chromatic- (a, b) such that there exist a vote πj such that (a, b) is or- coding or color-coding to solve local search variant of other dered differently in π and πj. problems. The problems which might be feasible using this The k-local search variant of this problem, denoted by approach are the local search variants of complete version k-LSKOA, is that given an input consisting of m votes of BETWEENNESS and MINIMUM QUARTET INCONSIS- (say V ), that is m permutations {π1,...,πm}, over n can- TENCY. didates (say C) and a permutation π, to check whether π∗ m τ π∗,π < there is another permutation such that i=1 ( i) References m τ π, π |L π \ L π∗ |≤k |L π∗ \ L π |≤ i=1 ( i), ( ) ( ) and ( ) ( ) Aarts, E. H. L., and Lenstra, J. K. 1997. Local Search in k. It is well known that k-KOA (Dwork et al. 2001a; Combinatorial Optimization. Princeton University Press. 2001b) can be modelled as solving a weighted feedback arc Ahuja, R. K.; Ergun, O.;¨ Orlin, J. B.; and Punnen, A. P. set problem in multi-tournaments (tournaments where be- 2002. A survey of very large-scale neighborhood search {u, v} u, v ∈ A tween every pair of vertices there is either ( ) techniques. Discrete Applied Mathematics 123(1-3):75– v, u ∈ A u, v , v, u ∈ A or ( ) or both ( ) ( ) ). The modelling es- 102. sentially does the following: Given an instance (C, V, k) of Ailon, N.; Charikar, M.; and Newman, A. 2005. Aggre- k-KOA, constructs a multi-tournament T such that (C, V, k) gating inconsistent information: ranking and clustering. In is an YES instance of k-KOA if and only if T has a feed- ACM Symposium on Theory of Computing (STOC), 684– back arc set of weight at most k. Using this transformation 693. one can give a reduction from k-LSKOA to k-LSFAST. Lemma 2 can also be proved for multi-tournaments and Alon, N.; Lokshtanov, D.; and Saurabh, S. 2009. Fast FAST. hence Theorems 1 and 2 can be generalized for multi- In ICALP (1), volume 5555 of Lecture Notes in Computer tournaments. This together with the equivalence between Science, 49–58. k-LSKOA and k-LSFAST (in multi-tournaments) allow us Alon, N.; Yuster, R.; and Zwick, U. 1995. Color-coding. J. to conclude the following theorem. Assoc. Comput. Mach. 42(4):844–856.

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