Fixed-Parameter Algorithms for Cluster Vertex Deletion

Transcription

1 Fixed-Parameter Algorithms for Clster Vertex Deletion Falk Hüffner Christian Komsieicz Hannes Moser Rolf Niedermeier Institt für Informatik, Friedrich-Schiller-Uniersität Jena, Ernst-Abbe-Platz 2, D Jena, Germany Abstract We initiate the first systematic stdy of the NP-hard Clster Vertex Deletion (CVD) problem (neighted and eighted) in terms of fixed-parameter algorithmics. In the neighted case, one searches for a minimm nmber of ertex deletions to transform a graph into a collection of disjoint cliqes. The parameter is the nmber of ertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly ne iteratie compression techniqe. Moreoer, e stdy the ariant of CVD here the maximm nmber of cliqes to be generated is prespecified. Here, e exploit connections to fixed-parameter algorithms for (eighted) Vertex Coer. 1 Introdction Graph modification problems form a core topic in algorithmic graph theory ith many applications. In particlar, clster graph modification problems [36] hae recently receied considerable interest. Here, the basic problem is, gien an ndirected graph G, to find a minimm nmber of editing operations that transform G into a collection of disjoint complete sbgraphs, a clster graph. Each of these disjoint complete sbgraphs is called a clster. Herein, the three An extended abstract of this paper appeared in Proceedings of the 8th Latin American Theoretical Informatics Symposim (LATIN 2008), April 7 11, 2008, Armação dos Búzios, Brazil, olme 4957 in Lectre Notes in Compter Science, pages , Springer, Spported by the Detsche Forschngsgemeinschaft, Emmy Noether research grop PIAF (fixed-parameter algorithms), NI 369/4, and the Edmond J. Safra fondation. Spported by a PhD felloship of the Carl-Zeiss-Stiftng. Spported by the Detsche Forschngsgemeinschaft, project ITKO (iteratie compression for soling hard netork problems), NI 369/5, and project AREG (algorithms for generating qasi-reglar strctres in graphs), NI 369/9. 1

2 standard editing operations are adding edges, deleting edges, and deleting ertices. For instance, Clster Editing asks hether a graph can be transformed into a clster graph by altogether at most k edge additions and edge deletions. Clster Editing is NP-complete; it recently has shon particlarly sefl for clstering biological data [10, 33]. Whereas also a factor-2.5 polynomialtime approximation for Clster Editing is knon [3, 4, 38], in practical applications fixed-parameter algorithms (combined ith some heristics) proiding optimal soltions seem to dominate [5, 6, 10, 33]. For a backgrond on fixed-parameter algorithmics e refer to [12, 15, 29]. Parameterized complexity stdies for Clster Editing ere initiated by Gramm et al. [21] and hae been frther prsed in a series of papers [5, 6, 10, 13, 20, 22, 32, 33]. A preiosly shon bond of O(1.92 k + n 3 ) for an n-ertex graph [20] can be improed by combining a linear-time problem kernelization algorithm [13] that yields an instance ith O(k 2 ) ertices ith the crrently best claimed rnning time of O(1.82 k +n 3 ) [6] to get an algorithm ith rnning time O(1.82 k +n+m), here m is the nmber of edges in the graph. Moreoer, problem kernels, based on efficient data redction rles, ith only O(k) ertices are knon [13, 22], the best pper bond crrently being 4k [22]. Whereas Clster Editing has been sbject to intensie research, its sister problem Clster Vertex Deletion so far has been idely neglected. Here, e aim at finding a ertex set of minimm eight sch that its deletion transforms a gien graph into a clster graph. 1 Weighted Clster Vertex Deletion Instance: An ndirected graph G = (V, E), a ertex eight fnction ω : V [1, ), and a nonnegatie nmber k. Qestion: Is there a ertex set X V ith X ω() k sch that deleting all ertices in X from G reslts in a clster graph (i. e., a graph here eery connected component forms a complete graph)? The neighted ersion asks hether there exists a sbset X V sch that X k (in other ords, all ertices hae eight exactly one). We call a set of ertices hose deletion prodces a clster graph a CVD set. Motiation. Like Clster Editing, Clster Vertex Deletion may find applications in graph-modeled data clstering: Assme that e hae a nmber of samples, some of hich are eqialent (e. g., DNA samples, some of hich are from the same species) and a method to test to samples for eqialence. A graph is formed here each ertex corresponds to a sample and an edge beteen to ertices is added hen their samples are tested as eqialent. In the absence of errors, the reslting graph is a clster graph, here each connected component corresponds to an eqialence class (e. g., a species). Hoeer, an nknon sbset of samples may be contaminated and can prodce npredictable comparisons to other samples. An optimal soltion for neighted Clster Vertex 1 Parameterized problems (as follos) sally are formlated as decision problems all or algorithms ill also sole the corresponding optimization problem ithin the same time bonds. 2

3 Deletion, that is, a minimm-cardinality set of ertices hose deletion prodces a clster graph, then proides the most parsimonios explanation for the data nder this model. This clearly extends to the eighted case. Clster Vertex Deletion can also be seen as the problem of making a symmetric relation transitie by omitting a minimm nmber of elements. The related problem of making an antisymmetric relation transitie by omitting a minimm nmber of elements is also knon as Feedback Vertex Set in tornaments (see, e. g., [11, 34] for fixed-parameter tractability reslts). A frther application for Clster Vertex Deletion is the comptation of a maximm cliqe of a graph ith the help of a CVD set. Sppose a CVD set X has been compted. Clearly, a maximm cliqe is at least as large as the largest clster of the remaining clster graph G[V \ X]. If there is a cliqe that is een larger, then it mst contain some ertices of X. The ertex set S X that is contained in the maximm cliqe mst indce a complete graph. For each sch ertex set S, e can find the maximm cliqe that contains S and ertices of G by finding the clster in G that contains the most ertices that are neighbors of S. Therefore, e can find a maximm cliqe of a graph by enmerating all sbsets S of X, checking hether the respectie sbset indces a complete graph, and then finding the clster in G[V \ X] that contains the most ertices that are neighbors of all ertices in S. A cliqe that has maximm size of all cliqes ths fond is a maximm cliqe of G. A similar approach can be sed to find the maximm independent set of G. Therefore, the problem of finding a maximm cliqe or a maximm independent set of a graph G can be parameterized by the size of the CVD set of G, or by the size of the CVD set of the complement graph Ḡ, respectiely. Finally, in comparison to Clster Editing, a small parameter ale k (that is, the nmber of editing operations) appears een more likely for Clster Vertex Deletion, since the ertex deletion operation is more poerfl, making a parameterized approach particlarly meaningfl here. Knon reslts. By general reslts for ertex deletion problems for hereditary graph properties [27], it follos that already neighted Clster Vertex Deletion is NP-complete. The optimization ersion is MaxSNP-hard [28]. Only fe specific reslts for (neighted) Clster Vertex Deletion are knon. 2 These are based on the simple obseration that a graph is a clster graph if and only if it does not contain an indced P 3, a path on three ertices. 3 Using this characterization, one directly obtains fixed-parameter tractability [7] as ell as a factor-3 polynomial-time approximation algorithm. Gramm et al. [20] sed an elaborate case distinction fond ith compter help to derie a 2 Jansen et al. [26] stdied the closely related problem of finding d pairise disjoint cliqes ith maximm oerall nmber of ertices, motiated by applications in schedling. Note that, other than in Clster Vertex Deletion, they alloed to hae edges beteen cliqes. Jansen et al. gae polynomial-time algorithms for special graph classes, contrasting the NP-complete general case. 3 In the remainder of this ork, hen riting of containment of a P 3 in a graph e refer to an indced P 3. 3

4 search tree algorithm rnning in O(2.26 k m) time for an m-edge graph. This can be improed to O(2.08 k + n 3 ), n denoting the nmber of ertices, by sing a straightforard redction of neighted Clster Vertex Deletion to the 3-Hitting Set problem (transforming each indced P 3 into a three-element set) and employing a sophisticated algorithm for 3-Hitting Set [37]. Moreoer, kernelization reslts for 3-Hitting Set [1] also imply an O(k 2 )-ertex problem kernel for neighted Clster Vertex Deletion, hich can be fond in O(n 3 ) time. A eighted Clster Vertex Deletion instance can be easily transformed into a eighted 3-Hitting Set instance. With this transformation, an O(k 3 )-ertex problem kernel reslt for eighted 3-Hitting Set [2] can be adapted to eighted Clster Vertex Deletion. Moreoer, eighted 3-Hitting Set possesses an elaborate search tree algorithm based on case distinction [14], implying an O(2.25 k + n 3 ) rnning time for eighted Clster Vertex Deletion. Iteratie compression, as first described by Reed et al. [35], is a techniqe for the deelopment of fixed-parameter algorithms. For an introdction and a srey on state of the art reslts that employ iteratie compression e refer to Go et al. [23]. Hüffner [24] gies experimental ealations of some iteratie compression algorithms, demonstrating their seflness on real-orld and synthetic data. Ne reslts. One of or main reslts is an elegant iteratie compression algorithm for eighted Clster Vertex Deletion sing matching techniqes, rnning in O(2 k k 9 + nm) time. 4 We extend or stdies to the (also NP-hard) case here the nmber of clsters to be generated is limited by a second parameter d. Sch stdies hae also been ndertaken for Clster Editing [19, 22, 36], bt note that for Clster Editing clearly d 2k. By ay of contrast, since ertex deletion is a stronger operation than edge deletion, in the case of Clster Vertex Deletion also d > 2k is possible. Obsere that d = 1 yields the Cliqe problem; therefore, a parameterization only ith respect to the parameter d is meaningless. Considering the combined parameter (d, k), hoeer, e can proide frther fixed-parameter tractability reslts. First, e nontriially extend the kernelization reslt for eighted Clster Vertex Deletion to a problem kernel for eighted d-clster Vertex Deletion, again achieing an O(k 3 )-ertex problem kernel. Based on this, e deelop three fixed-parameter algorithms for eighted d-clster Vertex Deletion ith the folloing rnning times: O(2 k k 9 +nm), O(1.40 k k 3d +nm), and O(1.84 k+d +nm). Depending on the ale of d, each of these algorithms may be preferable in certain constellations. In the latter to algorithms, fixed-parameter algorithms for eighted Vertex Coer play a decisie role. Notation. In this paper, for a graph G = (V, E) and a ertex set S V, let G[S] be the sbgraph of G indced by S and G \ S := G[V \ S], and 4 A similar algorithm for Clster Vertex Deletion later has also been described by Fomin et al. [16]. 4

5 CompressCVD(G, X) 1 X X 2 for each S X: 3 if G[S] is a clster graph: 4 G G \ (X \ S); R V (G \ S) 5 G RedctionRle1(G, S) 6 G RedctionRle2(G, S) 7 G RedctionRle3(G, S) 8 Classify each ertex in R according to N() S 9 H axiliary graph 10 M maximm eight matching in H 11 Delete all ertices not in a class corresponding to an edge in M 12 D ertices deleted in lines 4 7 and if ω(d) < ω(x ): 14 X D 15 retrn X Figre 1: Psedo-code for CompressCVD, here ω(a) := A ω() for A V. let N() := { V {, } E}. We rite V (G) to denote the ertex set of G. 2 Iteratie compression for Clster Vertex Deletion We no describe a noel iteratie compression algorithm for eighted Clster Vertex Deletion. General considerations abot iteratie compression algorithms can be fond in [23, 25] and [29, Chapter 11]. We first describe ho to employ a compression rotine, and then the compression rotine itself. The general idea behind or iteratie compression is as follos. We start ith V = and X = ; clearly, X is a CVD set for G[V ]. Iterating oer all graph ertices, step by step e add one ertex / V from V to both V and X. Then X is still a CVD set for G[V ], althogh possibly not a minimm one. We can, hoeer, obtain a minimm one by applying the compression rotine CompressCVD. It takes a graph G and a CVD set X for G, and retrns a minimm CVD set for G. Therefore, it is a loop inariant that X is a minimm-size CVD set for G[V ]. Since eentally V = V, e obtain an optimal soltion for G once the algorithm retrns X. In the rest of this section, e describe the compression rotine Compress- CVD folloing the psedo-code in Figre 1. For this, consider a smaller CVD set X as a modification of the larger CVD set X. This modification retains some ertices Y X, hile the other ertices S := X \ Y are replaced by at most S 1 ne ertices from V \X. The idea is to try by brte force all 2 X 1 5

6 S S (a) CVD Compression instance (b) After Redction Rle 1 S S (c) After Redction Rle 2 (d) After Redction Rle 3 Figre 2: Data redction in the compression rotine. partitions of X into sch sets Y and S (line 2). For each sch partition, the ertices from Y are immediately deleted, since e already decided to take them into the CVD set. In the reslting instance G = (V, E ) := G \ Y, it remains to find an optimal CVD set that is disjoint from S. This is a mch easier task than finding a CVD set in general; in fact, it can be done in polynomial time sing data redction and maximm matching. First, e discard partitions here S does not indce a clster graph (line 3); these cannot lead to a soltion, since e determined that none of the ertices in S old be deleted. Frther, R := V \ S also indces a clster graph, since R = V \X and X is a CVD set. Therefore, the folloing problem remains: CVD Compression Instance: An ndirected graph G = (V, E), a ertex eight fnction ω : V [1, ), and a ertex set S V sch that G[S] and G \ S are clster graphs. Task: Find a ertex set X V \ S sch that G \ X is a clster graph and X ω(x) is minimm. An example instance is shon in Figre 2a. The instance can no be simplified by a series of data redction rles. The reslts are shon in Figres 2b d. Redction Rle 1. Delete all ertices in R := V \S that are adjacent to more than one clster in G[S]. Proof of correctness. If a ertex R is adjacent to ertices and in different clsters in S, then indces a P 3 that can only be remoed by deleting (becase the ertices in S cannot be deleted). 6

7 S S (a) Classification (b) The assignment problem S (c) Final reslting clster graph Figre 3: Assignment problem in the iteratie compression, neighted case. Redction Rle 2. Delete all ertices in R that are adjacent to some, bt not all ertices of a clster in G[S]. Proof of correctness. If a ertex R is adjacent to a ertex, bt not to a ertex in a clster in S, then indces a P 3, hich can only be remoed by deleting. Redction Rle 3. Remoe connected components that are complete graphs. Proof of correctness. No optimal soltion deletes ertices in sch components. After Redction Rles 1 3 hae been applied, the instance is mch simplified (Figre 2d). In each clster in G[R], e can diide the ertices into eqialence classes according to their neighborhood in S; each class then contains either ertices adjacent to all ertices of a particlar clster in G[S], or the ertices adjacent to no ertex in S (see Figre 3a). This classification is sefl becase of the folloing lemma. Lemma 1. In an optimal CVD compression soltion, for each clster in G[R], the ertices of at most one class are present. Proof. Clearly, it is neer sefl to delete only some, bt not all ertices of a class, since if that led to an optimal soltion, e cold alays re-add the deleted ertices ithot introdcing ne P 3 s. Frther, if R is adjacent to some S, and is a ertex from the same clster as, bt from a different class, then is a P 3 ; therefore, e cannot keep ertices from to different classes ithin a clster. 7

8 Becase of Lemma 1, the remaining task is an assignment of each clster in G[R] to one of its classes (corresponding to the preseration of this class, and the deletion of all other classes ithin the clster) or to nothing (corresponding to the complete deletion of the clster). Hoeer, e cannot do this independently for each clster; e mst not choose to classes from different clsters in G[R] hich are adjacent to the same clster in G[S], since that old create a P 3. This can be modelled as a eighted bipartite matching problem in an axiliary graph H, here each edge corresponds to a possible choice. The graph H is constrcted as follos (see Figre 3b): Add a ertex for eery clster in G[R] (hite ertices). Add a ertex for eery clster in G[S] (black ertices in S). For a clster C S in G[S] and a clster C R in G[R], add an edge beteen the ertex for C S and the ertex for C R if there is a class in C R adjacent to C S. This edge corresponds to choosing this class for C R and is eighted ith the total eight of the ertices in this class. Add a ertex for each class in a clster C R that is not adjacent to a clster in G[S] (black ertices otside S), and connect it to the ertex representing C R. Again, this edge corresponds to choosing this class for C R and is eighted ith the total eight of the ertices in this class. Since e only added edges beteen a black and a hite ertex, H is bipartite. The task is no to find a maximm-eight bipartite matching, that is, a set of edges of maximm eight here no to edges hae an endpoint in common. This allos any choice for a clster, as long as no to clsters share edges to the same clster in G[S]. The folloing lemma shos that this is a alid approach: Lemma 2. A maximm-eight bipartite matching in H proides an optimal CVD compression soltion. Proof. Each edge in a matching corresponds to a class in a clster of G[R]. The CVD compression soltion is to delete all ertices in R bt those of the selected classes. The matching cannot select to classes ithin the same clster, since the corresponding edges hae an endpoint in common; similarly, it cannot select to classes that share a connection to the same clster in G[S]. Therefore, a matching yields a feasible soltion. By Lemma 1, an optimal CVD compression soltion corresponds to an assignment of each clster to one of its classes or to nothing, and therefore, it corresponds to a matching. Finally, the eight of a matching corresponds to the eight of the ertices not deleted from R, and therefore a maximm-eight matching corresponds to an optimal CVD compression soltion. Figre 3c shos the reslting clster graph for or example after deleting the ertex sets corresponding to edges that are not selected by the maximmeight matching shon in Figre 3b by bold edges. Note that the size of the soltion can be pper-bonded by k + 1, since V : ω() 1. Altogether, e obtain 8

9 Proposition 1. Weighted Clster Vertex Deletion can be soled in O(2 k n 2 (m + n log n)) time. Proof. The correctness of the algorithm has already been shon. It remains to bond the rnning time. We can find clsters in S and R in O(m) time by depth-first search ithin G[S] and G[R]. Redction Rle 1 can then be exected in O(m) time. If Redction Rle 1 has been applied, Redction Rle 2 can be exected in O(m) time by examining the degree of each ertex in R. Redction Rle 3 can be exected in O(m) time. Finally, e need to find a maximm eight matching in a bipartite graph ith at most n ertices and at most m edges, hich can be done in O(n(m + n log n)) time [17]. Therefore, e can sole CVD Compression in the same time. The nmber of ertices in an intermediary soltion X to be compressed is bonded by k + 1, becase any sch X consists of an optimal soltion for a sbgraph of G pls a single ertex. In one compression step, CVD Compression is ths soled O(2 k ) times, and there are n compression steps, yielding a total of O(2 k n 2 (m+n logn)) time. For the neighted case, e can get better rnning times, since integer eighted matchings can be fond faster than general eighted ones. Theorem 1. Uneighted Clster Vertex Deletion can be soled in O(2 k km nlog n) time. Proof. For each matching instance, e can se an algorithm for integer eighted matching ith a maximm eight of C = n [18], yielding a rnning time of O(m nlog(nc)) = O(m nlog n). Frther, in iteratie compression, e can sae some iteration ronds by starting ith a large sbgraph for hich e can still find in polynomial time a soltion of size at most k. For Clster Vertex Deletion, e can find a soltion of size at most 3k by simply repeatedly finding a P 3 and then taking all three of its ertices. We can then start the iteration ith G lacking these at most 3k ertices and an empty CVD set, after hich e ill need only 3k iteration ronds. Using problem kernelization, e can reach an additie rnning time of the form O(f(k) + poly(n)). First, e improe the rnning time bond on the enmeration of P 3 s, compared to the triial rnning time bond of O(n 3 ). Proposition 2. All P 3 s of a graph can be enmerated in O(nm) time. Proof. We can enmerate the P 3 s by enmerating for each ertex V the P 3 s in hich has degree to. This is done by scanning throgh s adjacency list. For each ertex in the adjacency list, e check for all ertices that appear after in the adjacency list of hether and are adjacent. If this is not the case, then e hae fond a P 3 and otpt it. The rnning time of this approach can be bonded as follos: for each ertex, e spend O(deg() deg()) time, since e scan throgh its adjacency list, and for each position in the adjacency list, e scan once again throgh the part of the list after this position. Testing adjacency can be performed in constant time ia an adjacency 9

10 matrix. Therefore, the total rnning time can be bonded as O( V deg() deg()) = O( V deg() n) = O(nm). Applying this algorithm for enmeration of P 3 s together ith a kernelization for 3-Hitting-Set [1] leads to the folloing rnning time. Theorem 2. Uneighted Clster Vertex Deletion can be soled in O(2 k k 6 log k + nm) time. Proof. In O(nm) time, e enmerate the P 3 s of G. We then apply the lineartime kernelization by Ab-Khzam and Ferna [2], hich gies s an instance ith O(k 3 ) ertices. To this instance, e apply the kernelization algorithm that yields a kernel ith O(k 2 ) ertices (rnning in O(k 9 ) time) [1]. Finally, e apply Theorem 1. Criosly, e can se this neighted algorithm as a sbrotine to speed p the eighted case: if e hae a soltion for an neighted instance, e can get an optimal eighted soltion by execting the compression rotine once. This orks becase the compression only reqires that the set X to compress is a CVD set, and does not make any assmptions abot its eight. Theorem 3. Weighted Clster Vertex Deletion can be soled in O(2 k k 9 + nm) time. Proof. Using the kernelization by Ab-Khzam and Ferna [2], e can in O(nm) time first shrink the instance to O(k 3 ) ertices. Then e can se the algorithm for the neighted problem for all compression steps except the last one. For a kernel of size k 3, this takes O(2 k k 9 + nm) time sing Theorem 1. A single compression for the eighted problem takes O(2 k n(m + n log n)) time, giing a time of O(2 k (k 9 + k 6 log k)) = O(2 k k 9 ) for the kernelized instance. In total, e arrie at the claimed bond. In fact, e een hae a stronger parameterization in Theorem 3 hen compared to Proposition 1: as parameter k, e can se the nmber of ertices in an optimal neighted soltion, hich is less than or eqal to the nmber of ertices in an optimal eighted soltion, hich in trn is less than or eqal to the minimm eight of a eighted soltion. Since the matching sbproblem is the bottleneck of the algorithm, it old be nice to replace it ith something simpler. Hoeer, it is straightforard to sho that the assignment problem in the last step of the compression rotine is as hard as the task of finding a maximm eight matching in a bipartite graph, een after applying Redction Rles 1 3. This indicates that the bottleneck of compting the maximm eight matching might actally be ery difficlt to oercome ith or approach. 10

11 3 Clster Vertex Deletion ith a limited nmber of clsters In clstering applications, it is often desirable to limit the nmber of otpt clsters, for example to simplify manal erification of the soltion. The deletion of ertices shold then prodce a d-clster graph, that is, a graph comprising at most d clsters. Weighted d-clster Vertex Deletion Instance: An ndirected graph G = (V, E), a ertex eight fnction ω : V [1, ), and a nonnegatie nmber k. Qestion: Is there a ertex set X V ith X ω() k sch that deleting all ertices in X from G reslts in a d-clster graph? Since the property of being a d-clster graph is hereditary, it follos that d- Clster Vertex Deletion is NP-complete [27]. We can sho that slightly modifying the kernelization approach from Section 1 that makes se of a reslt for 3-Hitting Set [2] leads to a problem kernel. Theorem 4. Weighted d-clster Vertex Deletion admits a problem kernel containing O(k 3 ) ertices, and it can be fond in O(nm) time. Proof. Since eery ertex eight in a eighted d-clster Vertex Deletion instance is at least 1, any soltion set has size at most k. Therefore, e can se the kernelization algorithm for 3-Hitting Set by Ab-Khzam and Ferna [2] that prodces a kernel that contains O(k 3 ) elements. This kernelization algorithm remoes elements for to reasons. First, it remoes all elements from the eighted 3-Hitting Set instance that appear in too many sbsets, becase they mst belong to any 3-hitting set. Hence, these elements are added to the soltion. Second, it remoes elements that do not appear in any sbset, becase they mst not belong to any minimal 3-hitting set. For details, e refer to [2]. Next, e describe or kernelization algorithm and bond the kernel size and the rnning time of the procedre. Let G be the graph of the eighted d-clster Vertex Deletion instance. First, e enmerate the P 3 s of G in O(nm) time (see Proposition 2) and create a 3-Hitting-Set instance in hich eery P 3 corresponds to a sbset of size 3. Since the minimm ertex eight is 1, the soltion has size at most k. Then, e perform all redction rles of Ab-Khzam and Ferna [2] except the deletion of elements that do not appear in any sbset. Let S be the set of elements after e hae performed these redction rles. Clearly, S contains O(k 3 ) elements that appear in sbsets and an nbonded nmber of elements that do not appear in any sbset. We then transform the eighted 3-Hitting Set instance back into a eighted d-clster Vertex Deletion instance by creating the graph G[S]. This graph has O(k 3 ) ertices that appear in indced P 3 s and an nbonded nmber of ertices that do not appear in P 3 s. The ertices that do not appear in any indced P 3 are part of isolated clsters. In Clster 11

12 Vertex Deletion, e can remoe all of these ertices from the graph since adding these ertices to a CVD set is neer optimal. Hoeer, in eighted d-clster Vertex Deletion, e might hae to add some of these ertices to the d-cvd set in order to delete srpls clsters. We no describe ho e can remoe some of the isolated clsters from the graph ithot remoing any clsters that e might hae to add to the d-cvd set. Clearly, e either hae to delete all ertices of a clster or none: a d-cvd set that deletes some bt not all ertices of a clster is non-optimal since e cold reinsert these deleted ertices into the graph ithot increasing the nmber of clsters and obtain a better soltion. Frthermore, e cannot delete any clsters in hich the sm of the ertex eights is more than k. Hence, e can remoe each of these clsters from the graph, decreasing d by 1, becase any sch clster is already fixed as one clster of the final d-clster graph. All remaining clsters contain at most k ertices. Clearly, e can delete at most k clsters. Therefore, e only hae to keep the k clsters that hae the loest eight. All others can be remoed ithot decreasing k, decreasing d by 1 for each remoed isolated clster. The remaining graph then contains at most k isolated clsters ith at most k ertices each, reslting in O(k 2 ) ertices that are part of isolated clsters. Together ith the O(k 3 ) ertices that are not part of isolated clsters, the graph then has O(k 3 ) ertices oerall. Clearly, all steps can be performed in O(nm) time. The kernelization reslt implies that d-clster Vertex Deletion is fixed-parameter tractable ith respect to the parameter k. A straightforard approach to sole d-clster Vertex Deletion is to apply a search tree algorithm that branches on the different cases to destroy a P 3 by ertex deletion, deleting a different ertex in each branch. Since the minimm ertex eight is 1, the parameter is redced by at least 1 in each search tree branch. Let k be the sm of the eights of the ertices that may still be deleted at a gien search tree node. Branching is performed as long as the graph contains a P 3 and k 1. If k < 1, and the graph still contains a P 3, then e hae not fond a d-cvd set of eight at most k and e cannot delete frther ertices. If otherise the graph is P 3 -free, then it is a clster graph albeit one that might contain more than d clsters. Therefore, e delete the clster of minimm eight ntil either G is a d-clster graph, hich means that e hae fond a soltion, or k < 1, hich means that no soltion as fond in this search tree branch. Clearly, this search tree procedre finds a d-cvd set of minimm eight, since it explores all possibilities to destroy the P 3 s of the graph and afterards optimally remoes srpls clsters. A straightforard combination of search tree, kernelization and the interleaing techniqe [30] leads to the folloing rnning time. Proposition 3. Weighted d-clster Vertex Deletion can be soled in rnning time O(3 k + nm). In the remainder of this section, e describe three different algorithms that improe on this triial search tree algorithm. The first algorithm is a modifi- 12

13 cation of the iteratie compression algorithm from Section 2 and also ses the parameter k. The second and third algorithm se the combined parameter (d, k) bt are preferable if d is small compared to k. 3.1 An iteratie compression algorithm for d-clster Vertex Deletion In the folloing, e describe ho to modify the algorithm from Section 2 in order to ensre that the graph remaining after the remoal of the CVD set comprises at most d clsters. The only part of the algorithm that has to be modified is the compression rotine. The problem that has to be soled by this compression rotine can be formlated as follos: d-cvd Compression Instance: An ndirected graph G = (V, E), a ertex eight fnction ω : V [1, ), and a ertex set S V sch that G[S] and G \ S are d-clster graphs. Task: Find a ertex set X V \ S sch that G \ X is a d-clster graph and X ω(x) is minimm. The main otline of the procedre remains the same, that is, e first perform data redction, and sbseqently sole the remaining problem ith a matching algorithm. Recall that R := V \ S denotes the set of ertices that may still be deleted in order to obtain a d-clster graph. We can apply Redction Rles 1 and 2 of the original algorithm, since in these redction rles e only delete ertices that appear in a P 3 together ith to ertices of S, and hence hae to be deleted in order to destroy this P 3. In contrast, e cannot apply Redction Rle 3, since this redction rle remoes isolated clsters from the graph. Hoeer, e might hae to delete some of these clsters in order to obtain a graph comprising at most d clsters. After performing Redction Rles 1 and 2, e diide the ertices of each clster in G[R] into eqialence classes according to their neighborhood in S in the same ay as it as done in the algorithm from Section 2. This sitation is depicted in Figre 4a. From these eqialence classes and the clsters in G[S], e create an axiliary graph H, in hich each ertex represents an eqialence class, the difference being that no this graph contains isolated ertices, representing the isolated clsters that ere not remoed. An example of this graph is shon in Figre 4b. We partition the ertices of H as follos: the set S in H contains the ertices that correspond to clsters in G[S], the ertices in T correspond to sets of ertices in G that are adjacent to ertices in S, the ertices in U correspond to sets of ertices in G that are not adjacent to ertices in S, and the ertices in I correspond to isolated clsters in G. Not deleting a set of ertices that corresponds to a ertex in U means that e create an additional clster. Clearly, if S + U + I d, then e alays obtain a d-clster graph, and hence e can remoe the isolated ertices from this axiliary graph H and sole the matching problem as in Section 2. Otherise, e transform H in or- 13

14 S (a) A graph after application of Redction Rles 1 and 2. S T U I 2 1 (b) The initial axiliary graph H S T U I (c) The modified axiliary graph H. Figre 4: Example of the compression procedre for 3-Clster Vertex Deletion. der to obtain a graph hose maximm eight matching proides a minimm eight d-cvd set. Since e cannot delete any ertices from S, e can create at most d := d S additional clsters. First, e apply a data redction rle: Redction Rle 4. If d > U, then remoe the ertex corresponding to the isolated clster of maximm eight from I and set d := d 1. Proof of correctness. Een if all clsters corresponding to ertices from U are not deleted, there is at least one isolated clster that e do not need to delete, and it is optimal to not delete the isolated clster of maximm eight. After this redction rle has been exhastiely applied, e can assme that d U. We no modify the axiliary graph H as follos: Connect each ertex in I ith each ertex in U. To each ne edge, assign a eight that eqals the eight of the clster that corresponds to the ertex in I. Add U d additional ertices to H. Let I be the set of these additional ertices. Connect each ertex in I to all ertices in U, and assign a eight that is larger than the sm of all other eights in H to eery ne edge. The goal of these modifications is to ensre that the clster graph corresponding to the maximm eight matching has at most d clsters. A matching edge beteen a ertex from U to a ertex from I models that the clster corresponding to the ertex of U mst be deleted, since e decided to keep the clster corresponding to the ertex of I. With the additional ertices I, e model that at least U d clsters corresponding to ertices in U mst be deleted. In the folloing lemma, e proe the correctness of this approach. Lemma 3. Let H be an axiliary graph constrcted as described. A maximm eight matching of H proides an optimal d-cvd Compression soltion. 14

15 Proof. We sho that each maximm matching proides a d-cvd soltion. The optimality of the soltion can be demonstrated in complete analogy to Lemma 2. In order to ensre that the reslting graph is a d-clster graph, the nmber of clsters that do not contain ertices in S can be at most d = d S. We hae to create an additional clster heneer a ertex of U is matched ia an edge beteen T and U, or ia an edge beteen a ertex in I and U. An edge beteen a ertex in U and a ertex in I does not create an additional clster. Each of the U d ertices in I is matched in a maximm eight matching, becase of or choice of eights. Therefore, at most d of the ertices in U are matched ia edges that are not incident to ertices in I. The matching ths contains at most d edges that create ne clsters. Since there ere d d clsters in G[S], the graph corresponding to the matching is a d-clster graph. The modifications that hae to be applied to the axiliary graph can be performed in O(k 6 ) time, since this graph contains O(k 3 ) ertices. We can ths bond the rnning time of the algorithm as in Section 2. Theorem 5. Weighted d-clster Vertex Deletion can be soled in O(2 k k 9 + nm) time, and neighted d-clster Vertex Deletion can be soled in O(2 k k 6 log k + nm) time. 3.2 Soling d-clster Vertex Deletion ia redction to Vertex Coer Here, e present an algorithm that soles eighted d-clster Vertex Deletion ia the comptation of minimm eight ertex coers. 5 Compared to the fixed-parameter tractability reslt from Section 3.1, hich is only based on the parameter k, e no employ the combined parameter (d, k). The idea is to try all independent sets of size at most d and to sole eighted d-clster Vertex Deletion for the case that these ertices are not deleted from the graph. Since in a d-clster graph any set of ertices from different clsters forms an independent set, at least one of the independent sets of size at most d mst be a set of ertices that remain in the graph. Sppose that sch an independent set D of size at most d is gien. We call the ertices in D permanent. In the folloing, e describe ho to compte the minimm eight d-cvd set of sch a graph; an example is shon in Figre 5. First, e perform the folloing redction rle. Redction Rle 5. Delete all ertices from the graph that are not adjacent to any ertex in the independent set D and all ertices that are adjacent to more than one ertex in D. The correctness of Redction Rle 5 is obios; an example of its application is gien in Figre 5b. For each deleted ertex, e decrease k by ω(). Let G be a graph ith a size-d independent set of permanent ertices after application 5 A ertex coer of a graph is a set C of graph ertices sch that eery graph edge has at least one endpoint in C. 15

16 (a) The original graph G ith to non-adjacent permanent ertices. (b) After Redction Rle 5. (c) The graph G ith a ertex coer X (marked ith circles). (d) The 2-clster graph after the remoal of X. Figre 5: Example of the algorithm for 2-Clster Vertex Deletion hen a size-2 independent set of ertices that cannot be deleted is gien. Black ertices are permanent. of Redction Rle 5. All non-permanent ertices of G are adjacent to exactly one permanent ertex. To prodce a clster graph, e also hae to ensre that all neighbors of a permanent ertex are adjacent, and neighbors of different permanent ertices are non-adjacent. These to attribtes can be encoded into a graph G sch that a ertex coer of G is a ertex set hose remoal establishes the attribtes in G. We constrct the graph G from G as follos: For any pair, of non-permanent ertices that is adjacent to the same permanent ertex, e do the folloing: remoe the edge {, }, if and are adjacent; insert the edge {, }, otherise. Frthermore, e remoe all permanent ertices. After this, e hae obtained G (for an example of this constrction see Figre 5c). In the folloing lemma, e sho that a ertex coer of G is a d-cvd set of G; an example of this eqialence is shon in Figres 5c and 5d. Lemma 4. Let G be a graph ith a size-d independent set of permanent ertices that is redced ith respect to Redction Rle 5 and G a graph constrcted as described aboe. Then, a ertex set X is a ertex coer of G if and only if X is a d-cvd set of G. Proof. Let X be a ertex coer of G, and let and be to non-permanent ertices in V \ X. We sho that and are adjacent in G if and only if they are neighbors of the same permanent ertex. Since G is redced ith respect to Redction Rle 5, this implies that G \ X is a d-clster graph. Clearly, and are nonadjacent in G. If and are both adjacent to the same permanent ertex p in G, then they hae to be adjacent in G, becase 16

17 of the ay e constrcted G. Hence, the neighborhood of any permanent ertex p D in G \ X is a cliqe. Otherise, if and are adjacent to different permanent ertices in G, then they cannot be adjacent in G, since in the constrction of G from G e did not modify any edges beteen ertices that are neighbors of different permanent ertices. Hence, there are no edges beteen neighbors of different permanent ertices in G \ X. Therefore, G \ X comprises exactly d clsters, hich means that X is a d-cvd set of G. The same reasoning can be applied to sho that a d-cvd set of G that does not delete any permanent ertices is a ertex coer of G. We no bond the rnning time of compting a d-cvd set of a graph, once an independent set of size at most d that may not be deleted is gien. It fndamentally relies on a fixed-parameter algorithm for eighted Vertex Coer [31]. Lemma 5. Let G = (V, E) be a graph and D V an independent set of size at most d. A minimm eight d-cvd set of G of eight at most k that does not delete any ertex D can be compted in O(1.40 k + n 2 ) time. Proof. We bond the rnning time of the algorithm that as described; the correctness follos from its description. Applying Redction Rle 5 can be performed in O(n 2 ) time. Constrcting G also rns in O(n 2 ) time. The comptation of minimm eight ertex coers of eight at most k can be done in O(1.40 k +kn) time [31]. Oerall, this amonts to the claimed rnning time. Combining this approach ith the kernelization algorithm from Theorem 4, e achiee the folloing rnning time. Theorem 6. Weighted d-clster Vertex Deletion can be soled in rnning time O(1.40 k k 3d + nm). Proof. The kernelization algorithm rns in O(nm) time. After kernelization the graph comprises O(k 3 ) ertices. Hence, there are ( ) k 3 d = O(k 3d ) possible choices for independent sets of size d. By Lemma 5, e can find a minimm eight d-cvd set of a graph of size O(k 3 ) in hich an independent set D of size at most d cannot be deleted in O(1.40 k +k 6 ) = O(1.40 k ) time. This is done for all O(k 3d ) sets, and the best soltion is stored. Clearly, the oerall rnning time is O(1.40 k k 3d + nm). For the neighted case, e can apply the crrently fastest algorithm for neighted Vertex Coer by Chen et al. [9] in combination ith the size O(k 2 ) kernel for neighted 3-Hitting Set, yielding an improed rnning time. Theorem 7. Uneighted d-clster Vertex Deletion can be soled in rnning time O(1.28 k k 2d + nm). 17

18 3.3 Combining branching and redction to Vertex Coer Clearly, the algorithm from Theorem 6 is only feasible for small ales of d. In this section, e describe an algorithm that is sefl if neither d nor k is ery small, and combines an improed branching strategy ith the algorithm from Theorem 6. First, e apply the kernelization algorithm from Theorem 4. Next, e perform a search tree algorithm that branches on forbidden sbgraphs. For a ertex in a forbidden sbgraph, e hae to choices: either e hae to delete, or is one of the remaining ertices in the d-clster graph. If is deleted, the combined parameter k + d decreases by ω() 1. Explicitly not deleting means that e assign a clster to. In this case, e call permanent. If does not hae any permanent neighbors, then e hae assigned a ne clster. Hence, k + d also decreases by 1. Let k be the sm of the eights of the ertices that may still be deleted at a gien search tree node and d the nmber of clsters that may still be assigned. Before branching, e perform the folloing data redction rle. Redction Rle 6. If G contains a P 3 ith to permanent ertices, and one non-permanent ertex, then delete from G and set k := k ω(). Proof of correctness. Since e may not delete any of the to permanent ertices and, e hae to delete ertex in order to eliminate the P 3. Clearly, if k < 1, then e cannot delete any ertices and either the graph is already a d-clster graph or this particlar branch of the search tree is a dead end. Frthermore, if d = 0, then e cannot assign frther clsters. This means that there is an independent set of d permanent ertices. By Lemma 5, e can find a d-cvd set of sch a graph in O(1.40 k + k 6 ) = O(1.40 k ) time. In the folloing, e describe the branching rles for the case k 1 and d > 0. After application of Redction Rle 6, eery P 3 contains at most one permanent ertex. First, e branch on P 3 s that consist of ertices that are not adjacent to permanent ertices. If sch a P 3 does not exist, then e branch on P 3 s that contain a permanent ertex that is not the middle ertex of the P 3. Finally, e sho that if none of the other cases applies, then e can find a minimm eight d-cvd set of the graph by compting a minimm eight ertex coer. In the folloing, e describe the branching rles in detail; each of them is also shon in Figre 6. Case 1: There is a P 3 sch that,, and are not adjacent to permanent ertices. We branch into three cases. In the first case, e delete ; the parameter k + d decreases by at least 1. In the second case, e mark as permanent and delete ; the parameter k + d decreases by at least 2 (one ne assigned clster and one deleted ertex). In the third case e mark and as permanent and delete ; the parameter k +d decreases by at least 3 (to ne assigned clsters and one deleted ertex). 18

19 (a) Case 1. (b) Case 2.1. x x s s x x x x s s (c) Case 2.2. s s (d) Case 2.3. Figre 6: Branching rles on P 3 s that are not adjacent to permanent ertices (Case 1) and P 3 s that contain one permanent non-middle ertex (Cases ). Dashed ertices and edges are deleted, black ertices are permanent. Case 2: There is a P 3 sch that is permanent and and are non-permanent. We consider seeral sbcases of this case. Case 2.1: Vertex has no permanent neighbors. We branch into to cases. In the first branch, e delete ; the parameter decreases by at least 1. In the second branch, e mark as permanent and ths delete ; the parameter decreases by at least 2 (one ne assigned clster and one deleted ertex). Case 2.2: Vertex has a permanent neighbor x and (or x) has a non-permanent neighbor s that is not adjacent to (or ). Withot loss of generality assme that has a neighbor s that is not adjacent to. We branch into to cases. In the first branch, e delete ; the parameter decreases by at least 1. In the second branch, e mark as permanent and may ths delete s and ; the parameter decreases by at least 2. Case 2.3: Vertex has a permanent neighbor x and (or ) has a non-permanent neighbor s that is not adjacent to (or x). Withot loss of generality assme that has a neighbor s that is not adjacent to. We branch into to cases. In the first branch, e delete ; the parameter decreases 19

20 by at least 1. In the second branch, e mark as permanent and may ths delete s and ; the parameter decreases by at least 2. Case 2.4: Otherise. If none of the other sbcases of Case 2 applies, then the folloing mst hold: Clearly, ertex has a permanent neighbor x. Otherise, Case 2.1 old apply. Frthermore, is adjacent to all non-permanent neighbors of, since Case 2.2 does not apply. Also, is adjacent to all nonpermanent neighbors of other than, since Case 2.3 does not apply. Finally, there is no permanent neighbor of that is not adjacent to and ice ersa, since e performed Redction Rle 6. Hence, N[] = N[] {}. Using the same argments e can sho that N[] = N[x] {}. We delete that ertex of and hich has loer eight; no branching takes place. For the correctness, consider the folloing: Clearly, e hae to delete at least one of and. Let be the ertex of loer eight of and. Sppose that there is a d-cvd set S that contains, here ω() ω(). In case S, e can reinsert into the graph ithot iolating the d-clster property (becase is only adjacent to neighbors of x and to ). Obiosly, the eight of the soltion decreases. In case / S, e can delete from the graph and reinsert into the graph, again ithot iolating the d-clster property. Therefore, deleting the ertex that has loer eight is optimal. Case 3: Otherise. The folloing mst hold: Since Case 1 does not apply, eery P 3 contains at least one ertex that is adjacent to a permanent ertex. Frthermore, if a P 3 contains a permanent ertex, then this permanent ertex is. Otherise, Case 2 old apply. Therefore, eery connected component either contains a permanent ertex or it is an isolated clster. Also, if a non-permanent ertex is adjacent to a permanent ertex, then all of s neighbors mst be adjacent to. Otherise, there old be a P 3 in hich the permanent ertex is not. If the graph contains more than d connected components, then e mst remoe isolated clsters to decrease the nmber of clsters (all other connected components contain a permanent ertex and ths cannot be completely deleted). We remoe isolated clsters of minimm eight ntil either k < 1 or the graph contains exactly d connected components. In the first case, e cannot obtain a d-clster graph, in the second case, e can mark the remaining isolated clsters as permanent. Then, there is an independent set of permanent ertices of size at most d and e can ths compte a d-cvd set ith eight at most k in O(1.40 k ) time (Lemma 5). Theorem 8. Weighted d-clster Vertex Deletion can be soled in rnning time O(1.84 k+d + nm). Proof. We proe the theorem by bonding the rnning time of the described algorithm. The correctness of the algorithm follos from the gien description. As shon in Theorem 4, kernelization can be done in O(nm) time. We bond the size of the search tree by analyzing the branching ectors and their 20

21 branching nmber; for details on this type of analysis, e refer to [29, Chapter 8]. In the search tree, the branching ector ith the largest branching nmber is (1, 2, 3) from Case 1. The branching nmber of this ector is Hence, the search tree has size O(1.84 k+d ). The comptation of minimm eight ertex coers is also performed by a search tree procedre, hose largest branching nmber is Hence, applying the fixed-parameter algorithm for eighted Vertex Coer in case d = 0 does not increase the orst-case search tree size. Eery step at a search tree node can be compted in polynomial time. By interleaing kernelization and branching, the rnning time for the search tree can be improed from O(1.84 k+d poly(k)) to O(1.84 k+d ) [30]. Together ith the time needed for the kernelization (O(nm)), e arrie at the claimed oerall rnning time bond. 4 Otlook It is open to improe the triial factor-3 approximation for Clster Vertex Deletion. Cai et al. [8] improed the approximation factor for (neighted) Feedback Vertex Set in tornaments, hich is also characterized by a 3- ertex forbidden sbgraph, from 3 to 2.5; perhaps similar techniqes are applicable here. Moreoer, the exponential pper bonds for or search-tree based algorithms shold be improable. More importantly, for the neighted case of Clster Editing, O(k)-ertex problem kernels are knon [13, 22], hereas correspondingly for Clster Vertex Deletion only an O(k 2 )-ertex kernel is knon. Also, improing the O(k 3 )-ertex problem kernel for the eighted case old be desirable. Frther problem ariants, for example a ariation of d-clster Vertex Deletion that specifies the exact nmber of desired clsters cold be practically releant. While the algorithms from Sbsections 3.2 and 3.3 can be applied to this problem, the method of iteratie compression is not applicable, since the property of comprising a fixed nmber of clsters is not hereditary. Finally, all or reslts are orst-case estimates. Practical tests based on algorithm engineering seem promising. Acknoledgments. We are gratefl to anonymos referees for pointing ot some inconsistencies in the manscript of the sbmitted conference ersion and for other comments that hae improed the presentation. References [1] F. N. Ab-Khzam. Kernelization algorithms for d-hitting set problems. In Proc. 10th WADS, olme 4619 of LNCS, pages Springer, [2] F. N. Ab-Khzam and H. Ferna. Kernels: Annotated, proper and indced. In Proc. 2nd IWPEC, olme 4169 of LNCS, pages Springer, [3] N. Ailon, M. Charikar, and A. Neman. Aggregating inconsistent information: Ranking and clstering. In Proc. 37th STOC, pages ACM,