Graph-Modeled Data Clustering: Fixed-Parameter Algorithms for Clique Generation

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1 Graph-Modeled Data Clstering: Fied-Parameter Algorithms for Cliqe Generation Jens Gramm Jiong Go Falk Hüffner Rolf Niedermeier Wilhelm-Schickard-Institt für Informatik, Universität Tübingen, Sand 13, D Tübingen, Fed. Rep. of Germany Abstract We present efficient fied-parameter algorithms for the NP-complete edge modification problems Clster Editing and Clster Deletion. Here, the goal is to make the fewest changes to the edge set of an inpt graph sch that the new graph is a verte-disjoint nion of cliqes. Allowing p to k edge additions and deletions (Clster Editing), we solve this problem in O(2.27 k + V 3 ) time; allowing only p to k edge deletions (Clster Deletion), we solve this problem in O(1.77 k + V 3 ) time. The key ingredients of or algorithms are two easy to implement bonded search tree algorithms and an efficient polynomial-time redction to a problem kernel of size O(k 3 ). This improves and complements previos work. Finally, we discss frther improvements on search tree sizes sing compter-generated case distinctions. Keywords. NP-complete, graph problems, edge modification problems, data clstering, correlation clstering, fied-parameter tractability, eact algorithms. A preliminary version of this paper was presented at the 5th Italian Conference on Algorithms and Compleity (CIAC 2003), Springer-Verlag, LNCS 2653, pages , held in Rome, Italy, May 28 30, Spported by the Detsche Forschngsgemeinschaft (DFG), research project OPAL (optimal soltions for hard problems in comptational biology), NI 369/2. Spported by the Detsche Forschngsgemeinschaft (DFG), jnior research grop PIAF (fied-parameter algorithms), NI 369/4. 1

2 1 Introdction Motivation and problem definition. There is a hge variety of clstering algorithms with applications in nmeros fields (cf., e.g., [15, 17]). Here, we focs on problems closely related to algorithms for clstering gene epression data (cf. [29] for a recent srvey) and so-called correlation clstering (with applications in docment clstering and agnostic learning [3]). In this contet, Shamir et al. [27] recently stdied two NP-complete graph problems called Clster Editing and Clster Deletion. 1 These are based on the notion of a similarity graph whose vertices correspond to data elements and in which there is an edge between two vertices iff the similarity of their corresponding elements eceeds a predefined threshold. The goal is to obtain a clster graph by as few edge modifications (i.e., edge deletions and additions) as possible; a clster graph is a graph in which each of the connected components is a cliqe. Ths, we arrive at the edge modification problem Clster Editing which is central to or work: Inpt: An ndirected graph G = (V, E), and a nonnegative integer k. Qestion: Can we transform G, by deleting and adding at most k edges, into a graph that consists of a disjoint nion of cliqes? Clster Deletion is the special case where edges can only be deleted. All these problems belong to the class of edge modification problems, see Natanzon et al. [23] for a recent srvey. Previos work. The most important reference point to or work is the paper of Shamir et al. [27]. Among other things, they showed that Clster Editing is NP-complete and that there eists some constant ɛ > 0 sch that it is NP-hard to approimate Clster Deletion to within a factor of 1 + ɛ. The NP-completeness of Clster Editing, however, can already be etracted from work of Křivánek and Morávek [20] who stdied more general problems in hierarchical tree clstering. In addition, Shamir et al. stdied cases where the nmber of clsters (i.e., cliqes) is fied. Before that, Ben-Dor et al. [4] and Sharan and Shamir [28] investigated closely related clstering applications in the comptational biology contet, where they deal with modified versions of the Clster Editing problem together 1 The third problem Clster Completion (only edge additions allowed) is easily seen to be polynomial-time solvable. 2

3 with heristic polynomial-time soltions. Independently of Shamir et al. s work, Bansal et al. [3] initiated the research on correlation clstering which is motivated by docment clstering problems from machine learning. It can be easily seen that an important special case of the general problem also stdied by Bansal et al. is identical to Clster Editing. Bansal et al. mainly provide approimation reslts which partially have been improved by very recent work [6, 8, 11]. Notably, the best known approimation factor for Clster Editing is 4 [6]; moreover, it is shown to be APX-hard (meaning that a polynomial-time approimation scheme (PTAS) is nlikely) [6]. Ths, there is a strong motivation to search for efficient fied-parameter algorithms to solve Clster Editing. From the more abstract view of graph modification problems, Leizhen Cai [5] (also cf. [10]) considered the more general problem (allowing edge deletions, edge additions, and verte deletions) where the goal graphs have a forbidden set characterization with respect to hereditary graph properties and he showed that this problem is fiedparameter tractable (refer to [2, 9, 10, 12, 13] for srveys on parameterized compleity and algorithms). In particlar, Cai s reslt implies O(3 k G 4 ) time algorithms for both Clster Editing and Clster Deletion where the forbidden set consists of a P 3 (i.e., a verte-indced path consisting of three vertices). 2 Natanzon et al. [23] give a general constant-factor approimation for deletion and editing problems on bonded-degree graphs with respect to properties (sch as being a clster graph) that can be characterized by a finite set of forbidden indced sbgraphs. Kaplan et al. [18] and Mahajan and Raman [22] considered other special cases of edge modification problems with particlar emphasis on fied-parameter tractability reslts. Khot and Raman [19] recently investigated the parameterized compleity of verte deletion problems for finding sbgraphs with hereditary properties. New reslts. Following a sggestion of Natanzon et al. [23] (who note that, regarding their NP-hardness reslts for some edge modification problems,... stdying the parameterized compleity of the NP-hard problems is also of interest. ), we present significantly improved fied-parameter tractability reslts for Clster Editing and Clster Deletion. More precisely, we show that Clster Editing is solvable in O(2.27 k + V 3 ) worst-case time and that Clster Deletion is solvable in O(1.77 k + V 3 ) worst-case time. This gives simple and efficient eact algorithms for these 2 A graph is a clster graph iff it contains no P 3 as a verte-indced sbgraph. This will also be important for or work. Note that Shamir et al. [27] write P 2 -free, bt according to the graph theory literatre it shold be called P 3 -free. 3

4 NP-complete problems in case of reasonably small parameter vales k (nmber of deletions and additions or nmber of deletions only). In particlar, we present an efficient data redction by preprocessing, providing a problem kernel of size O(k 3 ). Strctre of the paper. After providing some basics in Sect. 2, in Sect. 3 we describe a set of efficient data redction rles that transform a given Clster Editing instance into a redced graph with O(k 2 ) vertices and O(k 3 ) edges. Then, in Sect. 4 and 5, we provide two depth-bonded search trees for Clster Editing and Clster Deletion. In the conclding Sect. 6, we smmarize or findings, indicate opportnities for ftre work, and point ot how the achieved search tree sizes can frther be lowered sing compter-generated case distinctions. 2 Preliminaries and Basic Notation One of the latest approaches to attack comptational intractability is to stdy parameterized compleity. For many hard problems, the seemingly navoidable combinatorial eplosion can be restricted to a small part of the inpt, the parameter, so that the problems can be solved in polynomial time when the parameter is fied. For instance, the NP-complete Verte Cover problem can be solved by an algorithm with O(1.3 k + kn) rnning time [7, 24, 26], where the parameter k is a bond on the maimm size of the verte cover set we are looking for and where n is the nmber of vertices in the given graph. The parameterized problems that have algorithms of f(k) n O(1) time compleity are called fied-parameter tractable, where f can be an arbitrary fnction depending only on k, and n denotes the overall inpt size, see [2, 9, 10, 12, 13] for details. Or bonded search tree algorithms work recrsively. The nmber of recrsions is the nmber of nodes in the corresponding tree. This nmber is governed by homogeneos, linear recrrences with constant coefficients. It is well-known how to solve them and the asymptotic soltion is determined by the roots of the characteristic polynomial (see, e.g., Kllmann [21] for more details). If the algorithm solves a problem of size s and calls itself recrsively for problems of sizes s d 1,..., s d i, then (d 1,..., d i ) is called the branching vector of this recrsion. It corresponds to the recrrence t s = t s d1 + + t s di where t s denotes the nmber of leaves in the search tree solving an instance 4

5 of size s and t j = 1 for 0 j < d with d = ma(d 1,..., d i ). This recrrence corresponds to the characteristic polynomial z d = z d d z d d i. If α is a root of the characteristic polynomial which has maimm absolte vale and is positive, then t s is α s p to a polynomial factor. We call α the branching nmber that corresponds to the branching vector (d 1,..., d i ). Moreover, if α is a single root, then t s = O(α s ); all branching nmbers that occr in this paper are single roots. The size of the search tree is therefore O(α k ), where k is the parameter and α is the largest branching nmber that will occr; in or case, for Clster Editing, it will be shown that this branching nmber is abot 2.27 and it belongs to the branching vector (1, 2, 2, 3, 3) in Sect. 4. We assme familiarity with basic graph-theoretical notations. If is a verte in an ndirected graph G = (V, E), then by N G () we denote the set of its neighbors, i.e., N G () := { v {, v} E }. With deg G () we denote the degree of a verte V, i.e. N G (). We omit the inde G if it is clear from the contet. The whole paper only works with simple graphs withot self-loops. By G we refer to the size of graph G, which is determined by the nmbers of its vertices and edges. In or algorithms, we se a table T to store annotations for the edges of the graph sch that T has an entry for every pair of vertices, v V which can be empty or take one of the following vales: permanent : In this case, {, v} E and the algorithm is not allowed to delete {, v} later on; or forbidden : In this case, {, v} / E and the algorithm is not allowed to add {, v} later on. Note that, whenever the algorithms delete an edge {, v} from E, we set T [, v] to forbidden since it wold not make sense to reintrodce previosly deleted edges. In the same way, whenever the algorithms add an edge {, v} to E, we set T [, v] to permanent. In the following, when adding and deleting edges, we assme that we make these adjstments even when not mentioned eplicitly. 5

6 3 Problem Kernel for Clster Editing A redction rle replaces, in polynomial time, a given Clster Editing instance (G, k) consisting of a graph G and a nonnegative integer k by a simpler instance (G, k ) sch that (G, k) is a yes-instance iff (G, k ) is a yes-instance, i.e., G can be transformed into disjoint clsters by deleting/adding at most k edges iff G can be transformed into disjoint clsters by deleting/adding at most k edges. An instance to which none of a given set of redction rles applies is called redced with respect to these rles. A parameterized problem sch as Clster Editing (the parameter is k) is said to have a problem kernel if, after the application of the redction rles, the reslting redced instance has size f(k) for a fnction f depending only on k. (See [7] and [1] for two recent eamples concerning the graph problems Verte Cover and Dominating Set, respectively. There, the achieved problem kernel sizes are even linear in the parameters.) We present three redction rles for Clster Editing. For each of them, we discss its correctness and give the rnning time which is necessary to eecte the rle. In or rles, we se table T as described in Sect. 2. Using the redction rles, we show, at the end of this section, a problem kernel consisting of at most 2k 2 + k vertices and at most 2k 3 + k 2 edges for Clster Editing. Althogh the following redction rles also add edges to the graph, we consider the reslting instances as simplified. The reason is that for every added edge, the parameter is decreased by one. In the following rles, it is implicitly assmed that, when an edge is added or deleted, parameter k is decreased by one. In the formlation of or rles, we se the following terminology. Given a graph G = (V, E) and a verte pair v i, v j V, we se common neighbor of v i and v j to refer to a verte z V with {z, v i } E and {z, v j } E. Similarly, a non-common neighbor of v i and v j is a verte z V with z v i and z v j sch that either {z, v i } E or {z, v j } E bt not both. Rle 1 For every pair of vertices, v V : 1. If and v have more than k common neighbors, then {, v} has to belong to E and we set T [, v] := permanent. If {, v} is not in E, we add it to E. 2. If and v have more than k non-common neighbors, then {, v} cannot belong to E and we set T [, v] := forbidden. If {, v} is in E, we delete it. 6

7 3. If and v have both more than k common and more than k noncommon neighbors, then the given instance has no soltion. Lemma 1. Rle 1 is correct. Proof. Case 1: Vertices and v have more than k common neighbors. If we did eclde {, v} from E, then we wold have to, for every common neighbor z of and v, delete {, z}, {v, z}, or both. This, however, wold reqire at least k + 1 edge deletions, a contradiction to the maimm of k edge modifications allowed. Case 2: Vertices and v have more than k non-common neighbors. If we did inclde {, v} in E, then we wold have to, for every non-common neighbor z of and v, edit one of the edges {, z} and {v, z}. Withot loss of generality, let z be a neighbor of and not a neighbor of v. Then, we wold have to either delete {, z} from E or to add {v, z} to E. With at least k + 1 non-common neighbors, this wold reqire at least k + 1 edge modifications. Case 3: Vertices and v have more than k common neighbors and more than k non-common neighbors. From the proofs for Case 1 and Case 2 it is clear that it wold reqire more than k edge modifications both when inclding {, v} in E and when eclding {, v} from E. Note that Rle 1 applies to every verte pair {, v} for which the nmber of vertices which are neighbors of or v (or both) is greater than 2k. Rle 2 For every three vertices, v, w V : 1. If T [, v] = permanent and T [, w] = permanent, then {v, w}, if not already there, has to be added to E and T [v, w] := permanent. 2. If T [, v] = permanent and T [, w] = forbidden, then {v, w}, if already there, has to be deleted from E and T [v, w] := forbidden. The correctness of Rle 2 is obvios. Regarding the rnning time, we analyze the interleaved application of Rles 1 and 2 together. Lemma 2. A graph can in O( V 3 ) time be transformed into a graph which is redced with respect to Rles 1 and 2. Proof. We present an algorithm to redce a given graph G = (V, E) to a graph G = (V, E ) with respect to Rles 1 and 2 in O( V 3 ) time. The algorithm processes Rles 1.1 and 1.2 and, with little additional effort, also Rles 1.3 and 2. Data Strctres. 7

8 Adjacency matri for G. Two V ( V 1)/2 arrays C and N where, for a verte pair {v i, v j } with i < j, C[i, j] (N[i, j]) contains the nmber of common (noncommon) neighbors of v i and v j. Linked lists to store the verte pairs that are candidates to be inserted into the edge set or to be deleted from the edge set. More precisely, we maintain lists L c,0, L c,1,..., L c,k, and L p where L c,r with 1 r k contains those verte pairs which have eactly r common neighbors. List L p contains those verte pairs which are schedled to be set to permanent de to Rle 1.1 or Rle 2 (for instance, when they have more than k common neighbors). Similarly, we maintain lists L n,0, L n,1,..., L n,k, and L f where L n,s with 1 s k contains those verte pairs which have s non-common neighbors. List L f contains those verte pairs which are schedled to be set to forbidden de to Rle 1.2 or Rle 2. Two ( V ( V 1))/2 arrays P c and P n. A verte pair {v i, v j } with i < j is contained in at most one list from L c,0, L c,1,..., L c,k. If {v i, v j } is contained in one of these lists, then array entry P c [i, j] contains a pointer to this list entry. If {v i, v j } is contained in none of these lists, then P c [i, j] contains a nll pointer. In an analogos way, P n [i, j] contains a nll pointer or a pointer to an entry of {v i, v j } in one list of L n,0, L n,1,..., L n,k. Initialization. We assme that the adjacency matri for G is given. For every verte pair v i, v j V with i < j, we initialize C[i, j] and N[i, j] by conting their common and non-common neighbors in the adjacency matri. Based on these nmbers, we add the pair {v i, v j } to the appropriate list. If v i and v j have r common neighbors for 0 r k, then {v i, v j } is added to L c,r. If v i and v j have more than k common neighbors, then {v i, v j } is added to L p. Analogosly, {v i, v j } is added to L n,s for s non-common neighbors, 0 s k, and added to L f for more than k non-common neighbors. If a verte pair is added to one of L c,0, L c,1,..., L c,k, then a pointer to that entry is stored for that verte pair in P c [i, j] (analogosly in P n [i, j] if the verte pair is added to one of L n,0, L n,1,..., L n,k ). If now or in the following algorithm, one verte pair {v i, v j } satisfies both C[i, j] > k and N[i, j] > k, then the algorithm terminates, reporting that the instance has no soltion de to Rle 1.3. Algorithm. In the following, we describe one iteration of the algorithm. The algorithm terminates when both L p and L f are empty. If at least one 8

9 of L p and L f is non-empty, the verte pairs in L p and L f are waiting to be processed. One iteration of the algorithm processes one of these verte pairs. In the following, we describe how to process a verte pair {v i, v j }, i < j, taken from L p. Processing a verte pair from L f will, then, work in an analogos way, and is omitted here. For a verte pair {v i, v j } from L p, v i and v j are schedled to be connected by a permanent edge de to Rle 1 or Rle 2. We make sre that {v i, v j } is in the edge set and we set T [v i, v j ] := permanent. If, thereby, we added a new edge to the edge set, parameter k has to be decreased by one. When adding a new edge to the edge set, (a) we have to pdate the conters of common and non-common neighbors (if we change conters, then we also have to pdate the lists) and (b) we have to test whether the added edge gives rise to an application of Rle 2. Regarding (a), we assme that we add a new edge {v i, v j }. We assme that k still has its old vale, i.e., it is not yet decreased by one. To pdate the conters and to test whether the lists have to be pdated, we consider every verte v l with v l v i and v l v j, since only for {v i, v l } and {v j, v l } the conters of common and non-common neighbors can change. Since we add a new edge, we have to consider the following sitations (withot loss of generality we assme i < j < l): Verte v l is a common neighbor of v i and v j. Then we set N[i, l] := N[i, l] 1, C[i, l] := C[i, l] + 1, N[j, l] := N[j, l] 1, and C[j, l] := C[j, l] + 1. Verte v l is a neighbor of eactly one of v i and v j. Withot loss of generality we assme that v l is a neighbor of v j bt not a neighbor of v i. Then, we set N[i, l] := N[i, l] 1, C[i, l] := C[i, l] + 1, and N[j, l] := N[j, l] + 1. Verte v l is neither a neighbor of v i nor a neighbor of v j. Then, we set N[i, l] := N[i, l] + 1 and N[j, l] := N[j, l] + 1. If the vale of an entry in C changes, we pdate lists L c,r, 1 r k, and L p. If the vale of an entry in N changes, we pdate lists L n,s, 1 s k, and L f. Updating these lists is described sing the eample of increasing C[i, l]. If C[i, l] is increased to a vale of at most k + 1, and {v i, v l } is contained in one of the lists L c,r, 1 r k, then {v i, v l } is contained in L c,c[i,l] 1 (with 9

10 respect to the new vale of C[i, l]). We remove {v i, v l } from L c,c[i,l] 1, sing the pointer stored in P c [i, l]. If the new vale C[i, l] satisfies C[i, l] = k + 1, we move the entry to the end of list L p and, otherwise, we move the entry to the end of list L c,c[i,l]. When we move {v i, v l } to L p, we delete the pointer stored in P c [i, l] since it is no longer needed. Having pdated the conters and lists in the described way, we append list L c,k (still with respect to the old vale of k) to the end of list L p since, for the verte pairs in L c,k and the new vale of k, Rle 1.1 applies. In the same way, we append list L n,k to the end of list L f de to Rle 1.2. Then, we actally decrease parameter k by 1. Regarding (b), we test whether the permanent edge {v i, v j } gives rise to an application of Rle 2 as follows. Again, we consider every verte v l with v l v i and v l v j and test whether {v i, v l } is permanent bt not {v j, v l } or vice versa. Withot loss of generality, we assme that {v i, v l } is permanent bt not {v j, v l }. If not already contained in L p or L f, we add the nonpermanent verte pair {v j, v l } to the list L p. Both tests, (a) and (b), can, by making se of the adjacency matri and the arrays P c and P n, be done in O( V ) time. This completes the description of the iteration processing verte pair {v i, v j }. When {v i, v j } is processed, its corresponding entry is removed from L p. The otlined iteration is repeated ntil k = 0 or both L p and L f are empty. If parameter k reaches 0 before L p and L f are empty, then the given instance has no soltion. If L p and L f are empty while k 0, there is no remaining verte pair for which Rle 1 or Rle 2 applies and, ths, the reslting graph is redced with respect to Rles 1 and 2. Rnning time. The initialization of the lists and arrays, i.e., to cont the nmber of common and non-common neighbors for every verte pair can be done in O( V 3 ) time: For every verte pair {v i, v j }, we consider all vertices v l with v l v i and v l v j. An entry for {v i, v j } is added to the appropriate lists in constant time. One iteration of the algorithm takes at most O( V ) time, since the list entries can be accessed and moved in constant time, by sing the pointers stored in arrays P c and P n. Beyond that, the iteration involves only a loop over all vertices in V. There are less than V 2 iterations (at most one for every verte pair) since, after a verte pair is processed, it is removed from all lists. Smmarizing, the total time to redce the given graph is O( V 3 ). The following rle completes the set of redction rles proposed in this section. 10

11 Rle 3 Delete the connected components which are cliqes from the graph. The correctness of Rle 3 is straightforward. Compting the connected components of a graph and checking for cliqes can easily be done in linear time: Lemma 3. Rle 3 can be eected in O( G ) time. Notably, for the problem kernel size to be shown, Rles 1 and 3 wold be sfficient. Rle 2 is also taken into accont since it is very easy and general and can be eected in the corse of eecting Rle 1. Ths, Rles 1 to 3 constitte a small set of general and easy redction rles which yields a problem kernel with O(k 2 ) vertices and O(k 3 ) edges and which is comptable in O( V 3 ) time. Note that the O( V 3 ) rnning time given here is only a worst-case bond and it is to be epected that the application of the rles is mch more efficient in practice. The following theorem shows that redcing a graph with respect to Rles 1 to 3 leads to a problem kernel for Clster Editing. Theorem 1. Clster Editing has a problem kernel which contains at most 2k 2 + k vertices and at most 2k 3 + k 2 edges. It can be fond in O( V 3 ) time. Proof. Let G = (V, E) be a graph which is redced with respect to Rles 1 to 3. Withot loss of generality, we assme that G is connected. For a non-connected graph G, we process every connected component separately. Since Rle 3 deletes all isolated cliqes from the given graph, G is not a cliqe and we need at least one edge modification to transform, by a minimm nmber of edge modifications, G = (V, E) into a graph G = (V, E ), consisting of disjoint cliqes. Let k be the minimm nmber of reqired edge modifications, namely k a edge additions and k d edge deletions. Under the assmption that G is redced with respect to Rles 1 to 3, we will show by contradiction that V (2k + 1) k and that E ( 2k+1) 2 k as follows. Assme that V > (2k + 1) k. We distingish two cases, namely the case that k a = 0 and the case that 1 k a k. In both cases, we show a contradiction to or assmption that the graph is redced with respect to Rle 1. (Case 1): k a = 0. We have k d edge deletions, 1 k d = k, to transform G into G. Let V C V denote the verte set of a largest cliqe in G. The vertices in V C also form a cliqe in G since k a = 0. Since G is 11

12 connected, at least one verte V C is connected to a verte v / V C. We frther distingish two sbcases: Either v is not connected to any other verte V C with or there is a V C with and {, v} E. (Case 1.1): Verte v is not connected to any other verte V C with. We can lower-bond the cliqe size by V C V /(k d + 1): By k d edge deletions, G is transformed into a graph G containing at most k d +1 cliqes and, therefore, a largest cliqe in G contains at least V /(k d + 1) vertices. Firstly, we assme that k d 2 ( ). Using or frther assmptions that V > (2k + 1) k ( ) and k d = k ( ) we obtain V C V ( ) (2k + 1) k > k d + 1 k d + 1 ( ) = 2k2 + k + k k + 1 =k2 k k2 ( ) k + 1. k + 1 Conseqently, V C k + 2 and has at least k + 1 neighbors all vertices V C with which are not neighbors of v. This contradicts the assmption that G is redced with respect to Rle 1. Secondly, assming that k d = k = 1 while V > (2k + 1) k = 3, G consists of two cliqes; either both contain at least two vertices or one of them contains at least three vertices both times, Rle 1 wold apply, a contradiction. (Case 1.2): There is a V C with and {, v} E. We can lower-bond the cliqe size by at least V /k d : G is transformed into a graph G containing at most k d cliqes and, therefore, a largest cliqe in G contains at least V /k d vertices. With the assmptions V > (2k + 1) k ( ) and k d = k ( ), we obtain V C V k d ( ) > (2k + 1) k k d ( ) = 2k + 1. Conseqently, V C contains more than 2k + 1 vertices and at most k many of them are connected to v. Therefore, has more than k + 1 neighbors in this cliqe which are not neighbors of v, contradicting the assmption that G is redced with respect to Rle 1. (Case 2): 1 k a k. We know, since k a + k d = k, that k d < k. Again, let V C V denote the verte set of a largest cliqe in G. Since G contains at most k d + 1 cliqes, we have V C V /(k d + 1). With k d < k, this yields V C V /k and, sing V > (2k + 1) k, we obtain V C > (2k + 1). (1) Since the vertices of V C form a cliqe in G and at most k many edges are added in the transformation from G to G, in G there are at most k verte pairs (v i, v j ) with i < j and v i, v j V C which are not connected by 12

13 an edge. In the following, we show that, nder the two assmptions that V C > 2(k + 1) and that {(v i, v j ) v i, v j V C, i < j, {v i, v j } / E} k, the graph cannot be redced with respect to Rle 1. To this end, we consider the cases that k a = k and that k a < k separately. (Case 2.1): k a = k. We conclde that V C = V and G consists of only one cliqe. Since k a = k 1, there are v i, v j V with i < j and {v i, v j } / E. Frther, we know that V contains more than 2k + 1 vertices and, ths, there are more than ( ) 2k+1 2 many verte pairs. Ot of these verte pairs, at most k verte pairs (inclding {v i, v j }) are not connected by an edge. By conting argments, v i and v j have at least k + 1 common neighbors in G and Rle 1 wold apply, in contradiction to or assmption that G is redced with respect to Rle 1. (Case 2.2): k a < k. We can conclde that there are V C and v / V C sch that {, v} E. De to ineqality (1), there are more than 2k + 1 vertices in V C. On the one hand, there are at least (2k + 1) k a 1 vertices V C with {, } E. On the other hand, there are at most k d 1 vertices V C with {, v} E. Conseqently, we have at least (2k + 1) (k a + k d ) = (2k + 1) k = k + 1 vertices V C with {, } E bt {, v} / E. This implies that has at least k + 1 neighbors in G which are not neighbors of v and Rle 1 applies. In both cases, for k a = k and for 1 k a < k, we obtain a contradiction to the assmption that G is redced since Rle 1 wold apply. Regarding the edge set of the connected component, we infer a contradiction from the assmption that E > ( 2k+1) 2 k in an analogos way as for the verte set: Again, we let V C be the verte set of a largest cliqe in G and we distingish between the cases k d = 0 and k d > 0. If k d = 0, then we can easily derive that V C > 2k + 1 and the contradiction follows in analogy to (Case 2.1) above. If k d > 0, we derive (omitting some details here) that V C 2k + 1. Then, the contradiction follows in analogy to (Case 2.2) above. Smmarizing, the redced graph contains at most 2k 2 +k vertices and at most ( ) 2k+1 2 k = 2k 3 + k 2 edges (otherwise, no soltion eists). The rnning time follows directly from Lemmas 2 and 3. 4 Search Tree Algorithm for Clster Editing In this section, we describe a recrsive algorithm for Clster Editing that follows the bonded search tree paradigm. The basic idea of the algorithm is 13

14 to identify a conflict triple consisting of three vertices and to branch into sbcases to repair this conflict by adding or deleting edges between the three considered vertices. Ths, we invoke recrsive calls on instances which are simplified in the sense that the vale of the parameter is decreased by at least one. Before starting the algorithm and after every sch branching step, we compte the problem kernel as described in Sect. 3. The rnning time of the algorithm is, then, mainly determined by the size of the reslting search tree. In Sect. 4.1, we introdce a straightforward branching strategy that leads to a search tree of size O(3 k ); in Sect. 4.2, we show how a more involved branching strategy leads to a search tree of worst-case size O(2.27 k ). Note that the more general reslt of Leizhen Cai [5] as discssed in the introdctory section also provides an algorithm with eponential factor 3 k. By way of contrast, however, he ses a sort of enmerative approach with more comptational overhead (concerning polynomial-time comptations). In addition, the search tree algorithm in Sect. 4.1 also lies the basis for a more refined search tree strategy with the improved eponential term 2.27 k. Since or mathematical analysis is prely worst-case, we epect that the search tree sizes wold be sally mch smaller in practical settings; this seems particlarly plasible becase or search tree strategy as discssed in Sect. 4.3 also allows nmeros heristic improvements of the rnning time and the search tree size withot inflencing the worst-case mathematical analysis. 4.1 Basic Branching Strategy Central for the branching strategy described in this section is the following lemma observed in [27]. Lemma 4. A graph G = (V, E) consists of disjoint cliqes iff there are no three vertices, v, w V with {, v} E, {, w} E, bt {v, w} / E. Lemma 4 implies that, if a given graph does not consist of disjoint cliqes, then we can find a conflict triple of vertices between which we either have to insert or to delete an edge in order to transform the graph into disjoint cliqes. In the following, we describe the recrsive procedre that reslts from this observation. Inpts are a graph G = (V, E) and a nonnegative integer k, and the procedre reports, as its otpt, whether G can be transformed into a nion of disjoint cliqes by deleting and adding at most k edges. If the graph G is already a nion of disjoint cliqes, then we are done: Report the soltion and retrn. 14

15 Otherwise, if k 0, then we cannot find a soltion in this branch of the search tree: Retrn. Otherwise, identify, v, w V with {, v} E, {, w} E, bt {v, w} / E (they eist with Lemma 4). Recrsively call the branching procedre on the following three instances consisting of graphs G = (V, E ) with nonnegative integer k as specified below: (B1) E := E {, v} and k := k 1. Set T [, v] := forbidden. (B2) E := E {, w} and k := k 1. Set T [, v] := permanent, T [, w] := forbidden, and T [v, w] := forbidden. (B3) E := E + {v, w} and k := k 1. Set T [, v] := permanent, T [, w] := permanent, and T [v, w] := permanent. Proposition 1. Clster Editing can be solved in O(3 k k 2 + V 3 ) time. Proof. The recrsive procedre sggested above is obviosly correct. Concerning the rnning time, we observe the following. The preprocessing in the beginning to obtain the redction to a problem kernel can be done in O( V 3 ) time (Theorem 1). After that, we employ the search tree with size clearly bonded by O(3 k ). Hence, it remains to jstify the factor k 2 which stands for the comptational overhead related to every search tree node. Firstly, note that in a frther preprocessing step, we can once set p a linked list of all conflict triples. This is clearly covered by the O( V 3 ) term. Secondly, within every search tree node (ecept for the root) we deleted or added one edge and, ths, we have to pdate the conflict list accordingly. De to Theorem 1, we only have O(k 2 ) graph vertices now and with little effort, one verifies that the addition or deletion of an edge can make at most O(k 2 ) new conflict triples appear and it can make at most O(k 2 ) conflict triples disappear. Using a dobly-linked list of all conflict triples, one can pdate the list, after adding or deleting an edge of the graph, in O(k 2 ) time: after adding or deleting edge {v i, v j }, v i, v j V, we iterate over all O(k 2 ) many vertices v l V, v l v i and v l v j. Only the stats of the verte triples {v i, v j, v l } can be changed by this modification, either by casing a new conflict (then, the triple has to be added to the conflict list) or by being a conflict solved by the modification (then, the triple has to be deleted from the conflict list). This pdate for one verte triple can be done in constant time, by employing a hash table or by sing a size- V 3 array to store, for every verte triple, pointers to possible entries in the conflict list. Smmarizing, the conflict list can be pdated in O( V ) = O(k 2 ) time. 15

16 In fact, it does not really matter what the polynomial factor in k is, as the interleaving techniqe of [25] can be applied improving Proposition 1: Corollary 1. Clster Editing can be solved in O(3 k + V 3 ) time. Proof. In [25], it was shown that, in case of a polynomial size problem kernel, by doing the kernelization repeatedly dring the corse of the search tree algorithm whenever possible, the polynomial factor in parameter k can be replaced by a constant factor. 4.2 Refining the Branching Strategy The branching strategy from Sect. 4.1 can be easily improved as described in the following. We still identify a conflict triple of vertices, i.e.,, v, w V with {, v} E, {, w} E, bt {v, w} / E. Based on a case distinction, we provide for every possible sitation additional branching steps. The amortized analysis of the sccessive branching steps, then, yields the better worst-case bond on the rnning time. We start with distingishing three main sitations that may apply when considering the conflict triple: (C1) Vertices v and w do not share a common neighbor, i.e. V, : {v, } E and {w, } E. (C2) Vertices v and w have a common neighbor and {, } E. (C3) Vertices v and w have a common neighbor and {, } / E. Regarding case (C1), we show in the following lemma that, here, a branching into two cases (B1) and (B2) as described in Sect. 4.1 sffices. Lemma 5. Given a graph G = (V, E), a nonnegative integer k and, v, w V with {, v} E, {, w} E, bt {v, w} / E. If v and w do not share a common neighbor besides, then branching case (B3) cannot yield a better soltion than both cases (B1) and (B2), and can therefore be omitted. Proof. Consider a clstering soltion G for G where we did add {v, w} (see Fig. 1). We se N G G (v) to denote the set of vertices which are neighbors of v in G and in G. Withot loss of generality, assme that N G G (w) N G G (v). We then constrct a new graph G from G by deleting all edges adjacent to w. It is clear that G is also a clstering soltion for G. We compare the cost of the transformation G G to that of the transformation G G : 16

17 PSfrag replacements Theory of Compting Systems, Vol. 38(4), pp , 2005 G G m v w v w N G G (v) N G G (w) Figre 1: In case (C1), adding edge {v, w} does not need to be considered. Here, G is the given graph and G is a clstering soltion of G by adding edge {v, w}. The dashed lines denote the edges being deleted to transform G into G, and the bold lines denote the edges being added. Observe that the drawing only shows that parts of the graphs (in particlar, edges) which are relevant for or argmentation 1 for not adding {v, w}, +1 for deleting {, w}, N G G (v) for not adding all edges {w, }, N G G (v), + N G G (w) for deleting all edges {w, }, N G G (w). Herein, we omitted possible vertices which are neighbors of w in G bt not neighbors of w in G becase they wold only increase the cost of transformation G G. In smmary, the cost of G G is not higher than the cost of G G, i.e., we do not need more edge additions and deletions to obtain G from G than to obtain G from G. Lemma 5 shows that in case (C1) a branching into two cases is sfficient, namely to recrsively consider graphs G 1 = (V, E {, v}) and G 2 = (V, E {, w}), each time decreasing the parameter vale by one. For case (C2), we change the order of the basic branching. In the first branch, we add edge {v, w}. In the second and third branches, we delete edges {, v} and {, w}, as illstrated by Fig. 2. Add {v, w} as labeled by 2 in Fig. 2. The cost of this branch is 1. 17

18 PSfrag replacements Theory of Compting Systems, Vol. 38(4), pp , 2005 v 1 w v w v w v w v w v w v w v w Figre 2: Branching for case (C2). Bold lines denote permanent, dashed lines forbidden edges Mark {v, w} as forbidden and delete {, v}, as labeled by 3. This creates the new conflict triple, v,. To resolve this conflict, we make a second branching. Since adding {, v} is forbidden, there are only two branches to consider: Delete {v, }, as labeled by 5. The cost is 2. Mark {v, } as permanent and delete {, }. With redction rle 2 from Sect. 3, we then delete {w, }, too, as labeled by 6. The cost is 3. Mark {v, w} as forbidden and delete {, w} ( 4 ). This case is symmetric to the previos one, so we have two branches with costs 2 and 3, respectively. In smmary, the branching vector for case (C2) is (1, 2, 3, 2, 3). For case (C3), we perform a branching as illstrated by Fig. 3: Delete {, v}, as labeled by 2. The cost of this branch is 1. Mark {, v} as permanent and delete {, w}, as labeled by 3. With Rle 2, we can additionally mark {v, w} as forbidden. We then identify a new conflict triple, v,. Not being allowed to delete {, v}, we can make a 2-branching to resolve the conflict: 18

19 PSfrag replacements Theory of Compting Systems, Vol. 38(4), pp , 2005 v 1 w v w v w v w v w v w v w v w Figre 3: Branching for case (C3) Delete {v, }, as labeled by 5. The cost is 2. Mark {v, } as permanent. This implies {, } needs to be added and {w, } to be deleted de to redction rle 2, as labeled by 6. The cost is 3. Mark {, v} and {, w} as permanent and add {v, w}, as labeled by 4. Vertices, w, and form a conflict triple. To solve this conflict withot deleting {, w}, we make a 2-branching: Delete {w, } as labeled by 7. We then also need to delete {v, }. The cost is 3. Additionally, we can mark {, } as forbidden. Add {, }, as labeled by 8. The cost is 2. Additionally, we can mark {, } and {v, } as permanent. It follows that the branching vector for case (C3) is (1, 2, 3, 3, 2). In smmary, this leads to a refinement of the branching with a worst-case branching vector of (1, 2, 2, 3, 3), yielding branching nmber Since the recrsive algorithm stops whenever the parameter vale has reached 0 or below, we obtain a search tree size of O(2.27 k ). This reslts in the following theorem. Theorem 2. Clster Editing can be solved in O(2.27 k + V 3 ) time. 19

20 4.3 Heristic Improvements The following rles do not affect the worst-case time compleity since there is no garantee that any of them ever applies; however, they might be sefl for a practical implementation, since they are fairly cheap and can help redce the size of the search tree sbstantially if they do apply Branching Rles If {, v} E and and v do not have a common neighbor, branch into two cases: either delete {, v}, or delete all edges adjacent to and v ecept {, v}, leaving {, v} as a 2-cliqe. With an argment similar to that sed in Lemma 5, it is easy to see that an optimal soltion can be fond in one of the two described sbcases. This branching is sally noticeably better than that of Theorem 2 (case (C1)); e.g., with and v both being degree 3 vertices, the branching corresponds to the branching vector (1, 4) and the branching nmber Redction Rles In some cases, no branching is needed, and an instance G with parameter k can be directly replaced with a simplified instance G with parameter k. The correctness of the following rles can be easily seen with the above branching rles and symmetry argments. Let, v, w,, y be distinct vertices. If deg() = deg(v) = 1 and N() = N(v) = {w}, then delete {, w} and set k := k 1. If deg() = 1, deg(v) = 2, N() = {v} and N(v) = {, w}, then delete {v, w} and set k := k 1. If deg() = 2, deg(v) = deg(w) = 3 and N() = {v, w}, N(v) = {, w, }, N(w) = {, v, y}, then delete {v, } and {w, y} and set k := k 2. Presmably, these rles can be frther generalized, e.g. to handle cliqes with few connections to otside vertices Bail-Ot Rles Some branches in the search tree need not be followed, since either they cannot lead to a soltion, or becase it is known that for any soltion they 20

21 might lead to, we find another soltion which is at least as good in another branch. Let G 0 be the original inpt graph and let G be the graph in the crrent state of the algorithm. If G contains a verte v with deg G (v) 2 deg G0 (v) then the crrent branch of the search tree can be omitted, since we can be certain to find an optimal soltion in another branch of the search tree. This rle is correct as can be seen as follows: for any possible clstering soltion G of G, we can constrct another clstering soltion G by removing all edges adjacent to v in G. Clearly, the cost of transforming G 0 to G is not higher than the cost of transforming G 0 to G. 5 Clster Deletion From the basic 3-branching algorithm for Clster Editing in Sect. 4.1, it is straightforward to get an O(2 k + V 3 ) time algorithm for Clster Deletion as follows. Given a conflict triple consisting of vertices, v, w V with {, v} E, {, w} E, bt {v, w} / E, the insertion of an edge is not allowed here and, ths, we only need to make a branching into two cases: Either delete edge {, v} or delete edge {, w}. In the remainder of this section, we show how the branching nmber can be improved from 2 to As in Sect. 4.2, we start with identifying, v, w V with {, v} E, {, w} E, bt {v, w} / E, and distingish the following three cases: (C1) Vertices v and w do not share any common neighbor besides, i. e., V, : {v, } E and {w, } E. (C2) Vertices v and w have a common neighbor and {, } / E. (C3) Vertices v and w have a common neighbor and {, } E. Regarding case (C1), we distingish three sbcases: (C1.1) Vertices v and w have no other neighbors besides : It is easy to observe that we do not need to make any branching, it sffices to delete one arbitrary edge from {, v} and {, w} to resolve this conflict withot branching. (C1.2) Vertices v or w have a neighbor with {, } / E. We assme that verte v has sch a neighbor, i. e., {v, } E, {w, } / E, and {, } / E. We make a branching into two cases: 21

22 Delete edge {, v}. The cost of this branch is 1. Mark edge {, v} as permanent and delete {, w}. Since {, v} is permanent and there is no edge between and, edge {v, } has to be deleted as well. Ths, the cost of this branch is 2. (C1.3) All neighbors of vertices v and w are also neighbors of. Let be sch a neighbor and assme that {v, } E, {w, } / E, and {, } E. We make a branching into two cases: Delete {, w}. The cost of this branch is 1. Mark {, w} as permanent and delete {, v}. Since edge {, w} is permanent and there is no edge between w and, edge {, } has to be deleted as well. The cost of this branch is 2. Smmarizing the three sbcases, we have for case (C1) a worst-case branching vector of (1, 2). For case (C2), we apply the following branching: Delete {, v}. The cost of this branch is 1. Mark {, v} as permanent and delete {, w}. We then identify a new conflict triple v,,. Since edge {, v} is permanent, we can only resolve this conflict by deleting {v, }. The cost of this branch is 2. Conseqently, the branching vector for case (C2) is (1, 2). For case (C3), we apply the following branching, illstrated in Fig. 4. Delete {, v} (see 2 in Fig. 4). This creates a new conflict triple,, v. To resolve this conflict, we make a 2-branching: Delete {v, } ( 3 ). The cost of this branch is 2. Mark {v, } as permanent and delete {, } ( 4 ). However, this implies that {w, } needs to be deleted. The cost is 3. Delete {, w} ( 5 ). This creates a new conflict triple,, w. To resolve this conflict, we make a 2-branching: Delete {w, } ( 6 ). The cost of this branch is 2. Mark {w, } as permanent ( 7 ). However, this implies that {, } and also {v, } have to be deleted. The cost is 3. 22

23 PSfrag replacements Theory of Compting Systems, Vol. 38(4), pp , 2005 v 1 w v 2 2 w v 5 w 3 v w v 4 3 w v 6 2 w v 7 3 w Figre 4: Branching for case (C3) of the search tree algorithm for Clster Deletion. Bold lines denote permanent, dashed lines forbidden edges It follows that the branching vector for case (C3) is (2, 3, 2, 3). In smmary, the worst case is case (C3), where the branching vector is (2, 3, 2, 3) which corresponds to branching nmber In analogy to Theorem 2, we obtain the following theorem: Theorem 3. Clster Deletion can be solved in O(1.77 k + V 3 ) time. 6 Conclsion Adopting a parameterized point of view [2, 9, 10, 12, 13], we have shed new light on the algorithmic tractability of the NP-complete problems Clster Editing and Clster Deletion. We developed efficient fied-parameter algorithms in both cases and the algorithms seem easy enogh in order to allow for efficient implementations. We feel that the whole field of data clstering problems might benefit from more stdies on the parameterized compleity of the many problems related to this field. There appear to be nmeros parameters (e.g., nmber of clsters, nmber of data cleaning operations, dimensionality of the data space) that make sense in this contet. 23

24 Ongoing and ftre work. In ongoing work, we try to provide efficient implementations of or algorithms. It has to be investigated which of the redction and branching rles are of real practical importance and which of them (althogh necessary for the theoretical worst-case analysis) only increase the administrative overhead instead of really speeding p the algorithms. Frthermore, it might be interesting to investigate how standard heristic techniqes sch as branch-and-bond or A from artificial intelligence can be sed in or approach to obtain frther speed-ps in practice. We plan to eperiment with real-world clstering data and to incorporate additional featres (sch as edge and/or verte weights) in order to deal with more realistic settings. In recent work, we started to provide a general framework for comptergenerated search trees for graph modification problems [14, 16]. Using the sheer compting power of machines and obtaining a large nmber of case distinctions, we achieved at least theoretical improvements over the search tree sizes given in this paper. More precisely, the improved search tree bonds achieved are O(1.92 k ) for Clster Editing and O(1.53 k ) for Clster Deletion. To what etent these lowered worst-case bonds also have practical significance remains an isse of ftre research. Note that the compter-generated search trees have a significantly increased nmber of branching cases which cases increased overhead in the implementation etc. Theoretical challenges. Shamir et al. [27] showed that so-called p-clster Editing is NP-complete for p 2 and p-clster Deletion is NPcomplete for p 3. Herein, p denotes the nmber of cliqes that shold be generated by as few edge modifications as possible. Hence, there is no hope for fied-parameter tractability with respect to parameter p, becase fied-parameter tractability with respect to parameter p wold ths imply P = NP. Moreover, Shamir et al. consider the p-clster Completion problem (where only edge additions are allowed) 3 and claim an algorithm rnning in O(n p ) time. 4 Frther parameterized compleity investigations concerning the parameter p in graph clstering problems seem appropriate. We conclde with two concrete open qestions concerning data redction by preprocessing, i.e., problem kernelization: 3 Observe that general Clster Completion (withot bond p on the nmber of clsters) is trivially polynomial-time solvable by simply determining all connected components of the graphs and transforming them into cliqes. 4 The algorithm described by them seems to have O(p n ) rnning time bt the O(n p ) rnning time can be shown by additional argments. 24

25 1. Is there a significantly better redction to a problem kernel for Clster Deletion than we have for Clster Editing? Note that in Sect. 5 for Clster Deletion we implicitly made se of the problem kernelization as given for Clster Editing in Sect Do Clster Editing and Clster Deletion even allow for problem kernels of linear size O(k)? For Verte Cover on general graphs [7] and Dominating Set on planar graphs [1] sch reslts are known, bt it seems hard to derive similar reslts in or setting. Acknowledgment. We thank Jochen Alber (Tübingen) and Elena Prieto- Rodrigez (Newcastle, Astralia) for inspiring discssions and two anonymos referees of Theory of Compting Systems for comments that helped improving the presentation. References [1] J. Alber, M. R. Fellows, and R. Niedermeier. Efficient data redction for Dominating Set: a linear problem kernel for the planar case. In Proc. of 8th SWAT, volme 2368 of LNCS, pages Springer, Long version to appear nder the title Polynomial-Time Data Redction for Dominating Set in Jornal of the ACM. [2] J. Alber, J. Gramm, and R. Niedermeier. Faster eact soltions for hard problems: a parameterized point of view. Discrete Mathematics, 229:3 27, [3] N. Bansal, A. Blm, and S. Chawla. Correlation clstering. In Proc. of 43rd IEEE FOCS, pages , [4] A. Ben-Dor, R. Shamir, and Z. Yakhini. Clstering gene epression patterns. Jornal of Comptational Biology, 6(3/4): , [5] Leizhen Cai. Fied-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58: , [6] M. Charikar, V. Grswami, and A. Wirth. Clstering with qalitative information. In Proc. of 44th IEEE FOCS, pages , [7] J. Chen, I. Kanj, and W. Jia. Verte Cover: frther observations and frther improvements. Jornal of Algorithms, 41: ,