Sitting closer to friends than enemies, revisited Marek Cygan∗ Marcin Pilipczuk† Micha l Pilipczuk‡ Jakub Onufry Wojtaszczyk§ Abstract Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly 2 usedtomodelrelationshipsinsocialnetworks. Recently,KermarrecandThraves[11]initiatedthestudy 1 of the problem of appropriately visualising the network: They asked whether any signed graph can 0 be embedded into the metric space Rl in such a manner that every vertex is closer to all its friends 2 (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, n embeddability into R1 can be expressed as a purely combinatorial problem. In this paper we pursue a a deeperstudy of this particular case, answering several questionsposed by Kermarrec and Thraves. J First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by 9 showing that the problem is closely related to the recognition of proper interval graphs. Second, we provethat thegeneral case, whose polynomial-time tractability remained open, is in fact NP-complete. ] S Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that D the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming . s approach gives a runningtime single-exponential in thenumberof vertices. c [ 1 1 Introduction v 9 Undirected graphs with edges labelled positively (by a +) and negatively (by a −), called signed graphs, in 6 many applications serve as a very simple model of relationships between a group of people, e.g., in a social 8 network. Sign labels can express in a simplified way mutual relations, like staying in a relationship, family 1 . bonds or conflicts, by classifying them either as friendship (+ edge), hostility (− edge) or ambivalence (no 1 edge). In particular, much effort has been put into properly understanding and representing the structure 0 2 of the network, balancing it or naturally partitioning into clusters [1, 3, 5, 12, 13, 14, 15, 17]. One of the 1 problems is to visualize the model graph properly, i.e., in such a way that positive relations tend to make : vertices be placed close to each other, while negative relations imply large distances between vertices. v i Intheirrecentwork,KermarrecandThraves[11]formalizedthisproblemasfollows: Considerthemetric X space Rl with the Euclidean metric denoted by d. Given a signed graph G, is it possible to embed the r verticesof Gin Rl so thatfor any positive edge uu and negativeedge uu itholds that d(u,u )0 such that no algorithmsolving the 3-CNF-SAT problem in O(2Cn) exists, where n denotes the number of variables in the input formula. Combinatorial problem statement. In [11], Kermarrec and Thraves work with the metric definition of the problem: Given a signed graph G = (V,E+,E−) a feasible embedding of G in the Euclidean space Rl is such a function f : V → Rl that for all u ,u ,u, if u u ∈ E+ and u u ∈ E−, then d(f(u ),f(u)) < 1 2 1 2 1 d(f(u ),f(u)) (recall that d stands for the Euclidean distance in Rl). However, for the 1-dimensional case 2 they have in essence proved the following result: Theorem 2.1 (Lemmata 3 and 4 of [11], rephrased). A signed graph G = (V,E+,E−) has a feasible embedding in a line iff there is an ordering π of V such that for every u∈V: (i) there are no u < u < u such that u u∈E+ and u u∈E−; 1 π 2 π 1 2 (ii) there are no u > u > u such that u u∈E+ and u u∈E−. 1 π 2 π 1 2 We will jointly call conditions (i) and (ii) the condition imposed on u. Somewhat abusing the notation, the ordering π will also be called an embedding of G into the line. Therefore, from now on we are working with the following combinatorial problem that is equivalent to the version considered by Kermarrec and Thraves: Line Cluster Embedding Input: A signed graph G=(V,E+,E−). Task: Does there exist an ordering π on V such that for every u∈V: (i) there are no u < u < u such 1 π 2 π that u u∈E+ and u u∈E−; (ii) there are no u > u > u such that u u∈E+ and u u∈E−. 1 2 1 π 2 π 1 2 3 The complete signed graph case In their work,Kermarrecand Thraves [11] announceda polynomial-time algorithm solving the Line Clus- ter Embeddingproblemin the case where the input signedgraphis complete. Their line ofreasoningwas essentiallyasfollows: ifasignedgraphcanbe embeddedinto aline, thenitspositive parthastobe chordal. However,foraconnectedchordalgraphwithatleast4verticesthatactuallyisembeddableintoaline,every perfect elimination ordering of the graph is a feasible solution. Therefore, having checked that the graph is chordal and computed a perfect elimination ordering of every connected component, we can simply verify whether the obtained ordering is a correct line embedding. WerefinetheapproachofKermarrecandThravesbyshowingthatacompletegraphhasalineembedding if and only if its positive part is a proper interval graph. Recall that proper interval graphs are a subclass of chordal graphs; therefore, the result nicely fits into the picture of Kermarrec and Thraves. Moreover, the theory of proper interval graphs is well-studied, so many results from that area can be immediately translatedto our setting. Forinstance,many NP-completeproblems become solvableinpolynomialtime on proper interval graphs (e.g., [2, 7, 10, 16]), and the linear-time algorithmof Corneil et al. [4] for recognizing proper intervalgraphsimmediately solves the Line Cluster Embedding problemin linear time in case of a complete signed graph. Theorem 3.1. A complete signed graph G=(V,E+,E−) is embeddable in R1 if and only if G+ =(V,E+) is a proper interval graph. Moreover, having a feasible ordering π of vertices of V, a proper interval model of G+ sorted with respect to the left ends of the intervals can be computed in linear time; conversely, having a proper interval model of G+ sorted with respect to the left ends of the intervals, we can compute a feasible ordering π in linear time. 3 Proof. First, let us assume that G+ is a proper intervalgraph,andlet I ={I :v ∈V} be a proper interval v modelofG+. Noticethatasnointervaliscontainedinanother,wehaveanaturalorderonI —orderingthe intervalswithrespecttotheleftends(or,equivalently,therightends). Weclaimthatπ isafeasiblesolution for the Line Cluster Embedding instance G = (V,E+,E−). Take any u ∈ V. Assume that there were some u < u < u such that u u ∈ E+ and u u ∈ E−; this implies intervals I and I would overlap. 1 π 2 π 1 2 u1 u This, in turn, means that the right end of interval I would be on the right of the left end of interval I . u1 u Therefore,the left andrightends of I are ondifferentsides ofthe left endof I ,as u < u andu < u , u2 u 2 π 1 π 2 so I and I overlap. This is a contradiction with u u∈/ E+. A symmetrical argument for the second case u2 u 2 finishes the proof in this direction. Nowletus assumethatGisembeddable inthe line andletπ,anorderingofV, beasolution. Moreover, let v← be the first (with respect to π) vertex in the closed neighbourhood of v in G+, while let v→ be the last. Of course, v← ≤ v ≤ v→. Let us define a family of intervals on R: let I = π(v),π(v→)+ π(v) π π v h |V|+1i for v ∈ V and I = {I : v ∈ V}. Observe that intervals I have nonempty interior and pairwise different v v endpoints. Now,weprovethat(1)noI isfullycontainedinsomeotherI and(2)forallv,w ∈V,vw ∈E+ v w if and only if I ∩I 6=∅. This suffices to show that I is a proper interval model for G+. v w In order to establish (1), let us assume the contrary: there exists a pair of vertices v,w such that π(v) > π(w) and π(v→)+ π(v) < π(w→)+ π(w) . Then π(v→) < π(w→). Therefore, by definition of v→, |V|+1 |V|+1 vw→ ∈E−. On the other hand, ww→ ∈E+ and w < v < w→, a contradiction with the assumption that π π π was a proper embedding. Nowweproceedtotheproofof(2). Takeanytwodistinctverticesv,w, withoutlosinggeneralityassume that v < w. If vw ∈E+, then π(v)<π(w) and π(w)≤π(v→)<π(v→)+ π(v) , so I and I overlap. On π |V|+1 v w the other hand if vw ∈ E−, then from the condition imposed on v it follows that w > v→. Consequently, π π(w)>π(v→) and, as π(v)<|V|+1, also π(w)>π(v→)+ π(v) . Therefore, in this case I and I do not |V|+1 v w overlap. This proves I is in fact a proper interval model for G+. The algorithmic part of the theorem statement follows directly from the presented constructions. Let us recall the result of Corneil et al. [4], which states that proper interval graphs can be recognized in linear time and the algorithm can also output an ordering of the vertices with respect to the left ends of intervals in some model. We can pipeline this routine with Theorem 3.1 in order to obtain the following corollary: Theorem 3.2. Assuming the input graph is complete and given as the set of positive edges, Line Cluster Embedding can be solved in O(|V|+|E+|) time complexity. Moreover, the algorithm can produce a feasible ordering of the vertices in the same time, if such an ordering exists. 4 The general case 4.1 NP-completeness of the general case In [11] Kermarrec and Thraves asked whether the Line Cluster Embedding problem is also polynomial- time solvable in the case where the input is not restricted to complete graphs. In this section we show that this is unlikely: in fact, the problem becomes NP-complete. The proof consists of two steps. First, using a reduction from the Set Splitting problem we show that an auxiliary problem, called Acyclic Digraph Partition, is NP-complete. Next, we reduce Acyclic Digraph Partition to Line Cluster Embedding. We believe that the Acyclic Digraph Partition canturnout to be a useful pivotproblem also in other hardness reductions. Acyclic Digraph Partition Input: A directed graph D =(V,A). Task: Is it possible to partition V into two sets V and V , so that both D[V ] and D[V ] are directed 1 2 1 2 acyclic graphs (DAGs)? 4 Let us also recall the definition of the NP-complete Set Splitting problem [6]. Set Splitting Input: A set system (F,U), where F ⊆2U. Task: Does there exist a subset X ⊆U such that each set in F contains both an element from X and an element from U \X? Lemma 4.1. There exists a polynomial-time algorithm that given an instance (F,U) of Set Splitting outputsanequivalentinstanceG=(V,A)ofAcyclicDigraphPartition,forwhich|V|=|U|+ |F| PF∈F and |A|=3 |F|. PF∈F Proof. We construct the directed graph D = (V,A) as follows. For every set F ∈ F and every u ∈ F we build a vertex cF and connect all the vertices corresponding to the same set F into a directed cycle in any u order. For every element u ∈U we build a vertex d and for every vertex of the form cF we introduce two u u arcs: (d ,cF) and (cF,d ). This concludes the construction; it is easy to verify the claimed sizes of V and u u u u A. Let us formally prove that the instances are equivalent. Let X be any solution to the (F,U) instance of Set Splitting. Let V ={d :u∈X}∪{cF :u∈U\X}andV ={d :u∈U\X}∪{cF :u∈X}. As X 1 u u 2 u u splits everyset F ∈F, none of the cycles formedby verticescF for fixed F is entirely containedin either V u 1 or V . Also, for every element u the vertex d becomes isolated in the corresponding graph D[V ], as all his 2 u i neighbours belong to V . Therefore,both D[V ] and D[V ] are collections of isolated vertices and directed 3−i 1 2 paths and (V ,V ) is a solution to the Acyclic Digraph Partition instance. 1 2 In the other direction, let (V ,V ) be a solution to the instance of Acyclic Digraph Partition. Let 1 2 X ={u:d ∈V }⊆U, we claim that X is a solution to the instance of Set Splitting. Take any F ∈F. u 1 As the cycle formed by vertices cF is not entirely contained in any of the graphs D[V ],D[V ], there exist u 1 2 someu suchthatcF ∈V andu suchthatcF ∈V . Asthecyclesformedbypairs{d ,cF }and{d ,cF } 1 u1 1 2 u2 2 u1 u1 u2 u2 are also not entirely contained in D[V ] nor in D[V ], d ∈V and d ∈V . Consequently, u ∈U \X and 1 2 u1 2 u2 1 1 u ∈X and the set F is split. 2 Lemma 4.2. There exists a polynomial-time algorithm that given an instance D = (V,A) of Acyclic Digraph Partition outputs an equivalent instance H = (V′,E+,E−) of Line Cluster Embedding, such that |V′|=|V|+|A|+1, |E+|=2|A| and |E−|=|A|+|V|. Proof. We construct the graph H as follows: The set of vertices, V′, consists of: • a special vertex s; • for every e∈A, a checker vertex c ; e • for every v ∈V, an alignment vertex a . v We construct the edges of the signed graph as follows: • for every e∈A, we introduce a positive edge sc ; e • for every v ∈V, we introduce a negative edge sa ; v • for every arc (v,w)∈A, we introduce a positive edge c a and a negative edge c a . (v,w) v (v,w) w This concludes the construction; it is easy to verify the claimed sizes of V′,E+,E−. Let us now formally prove equivalence of the instances. Let π, an ordering of V′, be a solution of the Line Cluster Embedding instance (V′,E+,E−). As the special vertex s is adjacent via positive edges to all the checker vertices, and via negative edges to all the other, alignment, vertices, in the ordering π the checker vertices together with the special vertex have to form an interval. Let V be the set of those v ∈V 1 for which a is to the left of this interval, whereas V is the set of those v ∈ V for which a is to the right v 2 v of this interval. Formally, V = {v ∈ V : a ≤ s} and V = {v ∈ V : a ≥ s}. We claim that (V ,V ) is 1 v π 2 v π 1 2 5 a feasible solution of the Acyclic Digraph Partition instance (V,A). Consider any arc (v,w) such that v,w ∈ V . As a ≤ c , a ≤ c , c a ∈ E+ and c a ∈ E−, then it follows that a ≤ a . 1 v π (v,w) w π (v,w) (v,w) v (v,w) w w π v Thus, π has to induce a reversetopologicalorderingon the vertices of D[V ] and, therefore, D[V ] has to be 1 1 acyclic. Symmetrically, D[V ]hasto be acyclicaswell,whichconcludesthe proofof(V ,V )beingafeasible 2 1 2 solution. Now take any solution (V ,V ) of Acyclic Digraph Partition instance (V,A). Let π be any topo- 1 2 1 logical ordering of D[V ] and π be any topological ordering of D[V ], by which we mean that if (u,v) is an 1 2 2 arc of D[V ], π (u)<π (v), and the same holds for π . Let us construct an ordering π of V′ as follows: 1 1 1 2 • first, we place all the vertices a for v ∈V in the reverse order induced by π ; v 1 1 • then, we place all the checker vertices c for which v ∈V and w ∈V , in any order; (v,w) 1 2 • then,weplaceallthecheckerverticesc forwhichv,w∈V ,inreverselexicographicorderimposed (v,w) 1 by π on pairs (v,w); 1 • then, we place the special vertex s; • then, we place all the checker vertices c for which v,w ∈V , in lexicographic order imposed by π (v,w) 2 2 on pairs (v,w); • then, we place all the checker vertices c for which v ∈V and w ∈V , in any order; (v,w) 2 1 • finally, we place all the vertices a for v ∈V in the order induced by π . v 2 2 We claim that such π is a feasible solution to Line Cluster Embedding instance (V′,E+,E−). Notethatthepositiveneighboursofthespecialvertexsformaninterval,thereforetheconditionimposed on this vertex is satisfied. Now consider a checker vertex c . If v,w belong to different sets V ,V , then (v,w) 1 2 the only negative neighbour of c is the first or the last of his closed neighbourhood with respect to π, (v,w) thus satisfying the condition imposed on c . In case when v,w ∈V or v,w∈V this is also true, due to (v,w) 1 2 π ,π being topological orderings of D[V ], D[V ] respectively. 1 2 1 2 Now take any vertex a , by symmetry assume v ∈V . We need to prove that the condition imposed on v 1 a is satisfied as well. The neighbours of v consist of: v 1. positive neighbours c(v,v′), such that v′ ∈V2; 2. positive neighbours c(v,v′), such that v′ ∈V1; 3. negative neighbours c(v′,v), such that v′ ∈V1; 4. negative neighbours c(v′,v), such that v′ ∈V2. We now verify that by the construction of π the neighbours of a lie in this very order with respect to π. v Clearly,theorderinwhichweplacedthecheckersinπensuresthattheneighboursfrom(1)areplacedbefore the neighbours from (2) and that the neighbours from (3) are placed before the neighbours from (4). Thus theonlynon-trivialcheckiswhethertheverticesfrom(2)liebeforetheverticesfrom(3). Assumeotherwise, that there are some v1′,v2′ such that (v,v1′)∈A, (v2′,v) ∈A, but c(v,v1′) >π c(v2′,v). But then v2′ <π1 v as π1 is a topological ordering of D[V ], so the pair (v′,v) is lexicographically smaller than the pair (v,v′). Thus 1 2 1 c(v,v′) >π c(v′,v) is a contradiction with the construction of π. 1 2 We have verified that for all the vertices the conditions imposed on them are satisfied, so the instances are equivalent. The NP-completeness of the Set Splitting problem [6], together with Lemmata 4.1, 4.2 and a trivial observation that Line Cluster Embedding is in NP, gives us the following theorem. Theorem 4.3. The Line Cluster Embedding problem is NP-complete. 6 As mentioned before, the question of finding the smallest dimension of the Euclidean space, into which the given graph can be embedded, clearly generalizes testing embeddability into a line. Therefore, we have the following corollary. Corollary 4.4. It is NP-hard to decide the smallest dimension of the Euclidean space, into which a given signed graph can be embedded. 4.2 Lower bound on the complexity In this subsection we observe that the presented chain of reductions in fact enables us also to establish a lower bound on the complexity of solving Line Cluster Embedding under ETH. Firstly, let us complete the chain of the reductions. Lemma 4.5. There exists a polynomial-time algorithm that given an instance ϕ of 3-CNF-SAT with n variables and m clauses, outputs an equivalent instance (U,F) of Set Splitting with |U| = 2n+1 and |F|=2n+4m. PF∈F Proof. We construct the instance (U,F) as follows. The universe U consists of one special element s and two literals x,¬x for every variable x of ϕ. The family F includes • for every variable x, a set F ={x,¬x}; x • for every clause C, a set F consisting of s and all the literals in C. C ItiseasytochecktheclaimedsizesofU,F. WeclaimthattheinstanceofSetSplitting(U,F)isequivalent to the instance ϕ of 3-CNF-SAT. Assume that ψ is a boolean evaluation of variables of ϕ that satisfies ϕ. We construct a set X ⊆ U as follows: X consists of all the literals that are true in ψ. Now, every set F is split, as exactly one of x the literals is true and one is false, whereas every set F is split as well, as it contains a true literal, which C belongs to X, and the special element s, which does not. Now assume that X ⊆ U is a solution to the Set Splitting instance (U,F). As taking U \X instead of X also yields a solution, without losing generalitywe can assume that s∈/ X. Every set F is split by X; x therefore, exactly one literal of every variable belongs to X and exactly one does not. Let ψ be a boolean evaluation of variables of ϕ such that it satisfies all the literals belonging to X. Observe that ψ satisfies ϕ: for every clause C the set F has to be split, so, as s ∈/ X, one of the literals of C belongs to X and, thus, C is satisfied by ψ. Note that by pipelining Lemmata 4.5, 4.1 and 4.2, we obtain a reduction from 3-CNF-SAT to Line Cluster Embedding, where the output instance has a number of vertices and edges bounded linearly in the number of variables and clauses of the input formula. This observation, together with the key tool used in proving complexity lower bounds under Exponential Time Hypothesis, namely the SparsificationLemma [9], gives us the following theorem. Theorem 4.6. Unless ETH fails, there is a constant δ > 0 such that there is no algorithm that given a (V,E+,E−) instance of Line Cluster Embedding problem, solves it in O(2δ(|V|+|E+|+|E−|)) time. Proof. Let us begin by recalling the Sparsification Lemma. Lemma 4.7 (Sparsification Lemma, Corollary 1 of [9]). For all ε>0 and positive k, there is a constant C so that any k-SAT formula Φ with n variables can be expressed as Φ= t Ψ , where t≤2εn and each Ψ is a k-SAT formula with at most Cn clauses. Moreover, this disjunctioWni=ca1n bie computed by an algorithmi running in time O⋆(2εn). Let us now assume that for all δ > 0 there exists an algorithm solving Line Cluster Embedding in O(2δ(|V|+|E+|+|E−|)) time complexity. We now show an algorithm solving 3-CNF-SAT in O⋆(2εn) time for every ε>0, where n is the number of variables,thus contradicting the ETH. Indeed, having fixed ε we can: 7 • take an instance of 3-CNF-SAT and using Sparsification Lemma in O⋆(2εn/2) time express it as a disjunction of at most 2εn/2 3-CNF-SAT instances, each containing at most Cn clauses for some constant C; • reduce each instance in polynomial time via Set Splitting and Acyclic Digraph Partition to Line Cluster Embedding, thus obtaining at most 2εn/2 instances of Line Cluster Embedding, each having |V|,|E+|,|E−|≤C′n for some constant C′; • in each of the instances run the assumed algorithm for Line Cluster Embedding, running in O(2δ(|V|+|E+|+|E−|)) time, for δ = ε . 6C′ 4.3 A single-exponential algorithm for Line Cluster Embedding Note that the trivial brute-force algorithm for Line Cluster Embedding checks all possible orderings, workingin O⋆(n!) time. Tocomplete the picture ofthe complexity ofLine Cluster Embedding,we show that a simple dynamic programming approach can give single-exponential time complexity. This matches the lower bound obtained from under Exponential Time Hypothesis (up to a constant in the base of the exponent). Before we proceed with the description of the algorithm, let us state a combinatorial observation that will be its main ingredient. Let (V,E+,E−) be the givenLine Cluster Embedding instance. For X ⊆V and v ∈/ X we will say that v is good for the set X iff • no vertex w ∈ X that is adjacent to v via a negative edge is simultaneously adjacent to some vertex from V \(X ∪{v}) via a positive edge; • no vertex w ∈ V \(X ∪{v}) that is adjacent to v via a negative edge is simultaneously adjacent to some vertex from X via a positive edge. Lemma 4.8. An ordering π is a feasible solution of (V,E+,E−) if and only if every vertex v ∈ V is good for the set {u:u< v}. π Proof. One direction is obvious: if π is a feasible solution, then every vertex v has to be good for the set {u: u< v}. If v would not be good for {u:u< v}, there would exist a vertex w certifying that v is not π π good, and the condition imposed upon w would be not satisfied. Now assume that every vertex v ∈ V is good for {u : u < v} and take an arbitrary vertex v ∈ V. If π there were vertices u < u < v such that u v ∈E+ while u v ∈E−, then u would not be good for the 1 π 2 π 1 2 2 set {u:u< u }, a contradiction. Similarly, if there were vertices u > u > v such that u v ∈E+ while π 2 1 π 2 π 1 u v ∈ E−, then u would not be good for the set {u : u < u }, a contradiction as well. Therefore, the 2 2 π 2 condition imposed on v is satisfied for an arbitrary choice of v. We are now ready to provide the details of the algorithm. Theorem 4.9. Line Cluster Embedding can be solved in O⋆(2n) time and space complexity. Moreover, the algorithm can also output a feasible ordering of the vertices, if it exists. Proof. Let(V,E+,E−)bethegivenLineClusterEmbeddinginstance. LetW ={(v,X):v is good for X}. Let us construct a directed graph D =(W,F), where ((v,X),(v′,X′))∈F if and only if X′ =X ∪{v}. As recognizing being good is clearly a polynomial time operation, the graph D can be constructed in O⋆(2n) time and has that many vertices and edges. Observe that by Lemma 4.8 there is a feasible ordering π if and only if some sink (v,V \{v}) is reachable from some source (u,∅); indeed, such a path corresponds to introducingtheverticesofV onebyoneinsuchamannerthateachofthemisgoodfortherespectiveprefix. Reachabilityofanysink fromanysourcecanbe,however,testedintime linearinthe sizeofthe graphusing a breadth-first search. The search can also reconstruct the path in the same complexity, thus constructing the feasible solution. 8 5 Conclusions Inthis paperweaddressedanumberofproblemsraisedbyKermarrecandThravesin[11]forembeddability of a signed graph into a line. We refined their study of the case of a complete signed graph by showing relationwith proper intervalgraphs. 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