Avoiding Forbidden Submatrices by Row Deletions

Transcription

1 Avoiding Forbidden Submatrice by Row Deletion Sebatian Wernicke, Jochen Alber, Jen Gramm, Jiong Guo, and Rolf Niedermeier Wilhelm-Schickard-Intitut für Informatik, niverität Tübingen, Sand 13, D Tübingen, Fed. Rep. of Germany Abtract. We initiate a ytematic tudy of the Row Deletion(B) problem on matrice: For a fixed forbidden ubmatrix B, the quetion i, given an input matrix A (both A and B have entrie choen from a finite-ize alphabet), to remove a minimum number of row uch that A ha no ubmatrix which i equivalent to a row or column permutation of B. An application of thi quetion can be found, e.g., in the contruction of perfect phylogenie. Etablihing a trong connection to variant of the NP-complete Hitting Set problem, we how that for mot matrice B Row Deletion(B) i NP-complete. On the poitive ide, the relation with Hitting Set problem yield contant-factor approximation algorithm and fixed-parameter tractability reult. 1 Introduction Forbidden ubgraph problem play an important role in graph theory and algorithm (cf., e.g., [1, Chapter 7]). For intance, in an application concerned with graph-modeled clutering of biological data [9] one i intereted in modifying a given graph by a few edge deletion a poible uch that the reulting graph conit of a dijoint union of clique (a o-called cluter graph). Exact (fixedparameter) algorithm to olve thi NP-complete problem make ue of the fact that a graph i a cluter graph iff it contain no vertex-induced path of three vertice a a ubgraph [3, 4]. There i a rich literature dealing with uch graph modification problem, cf., e.g., [6] many problem here being NP-complete. By way of contrat, in thi paper we tart the o far eemingly widely neglected invetigation of forbidden ubmatrix problem from an algorithmic point of view. Here, given an input matrix A and a fixed matrix B, the baic quetion i whether B i induced by A. Thi mean that a permutation B of B that i, B can be tranformed into B by a finite erie of row and column wapping can be obtained from A by row and column deletion. Thi work tudie correponding matrix modification problem where, given A and a fixed B, we are aked Supported by the Deutche Forchunggemeinchaft (DFG), project PEAL (parameterized complexity and exact algorithm), NI 369/1; project OPAL (optimal olution for hard problem in computational biology), NI 369/2; junior reearch group PIAF (fixed-parameter algorithm), NI 369/4.

2 to remove a few row from A a poible uch that the reulting matrix no longer induce B. Forbidden ubmatrix problem, e.g., are motivated by quetion of computational biology concerning the contruction of perfect phylogenie [8, 10]. Here, a binary input matrix A allow for a perfect phylogeny iff A doe not induce the ubmatrix B coniting of the row (1, 1), (1, 0), and (0, 1) (ee [8, 10] for detail). We initiate a ytematic tudy of matrix modification problem concerning the complexity of row deletion for forbidden ubmatrice. Our main reult i to etablih a very cloe link between many of thee problem and retricted verion of the NP-complete Hitting Set problem [2]. We decribe and analyze tructure of the forbidden ubmatrix B which make the correponding row deletion problem equivalent to particular verion of Hitting Set. On the negative ide, thi implie NP-completene for mot row deletion matrix modification problem, holding already for binary alphabet. On the poitive ide, we can alo how that approximation and fixed-parameter tractability reult for Hitting Set carry over to the correponding row deletion problem. To the bet of our knowledge, no uch ytematic tudy ha been undertaken o far. We are only aware of the related work of Klinz et al. [5] dealing with the permutation of matrice (without conidering row deletion) in order to avoid forbidden ubmatrice. There, however, they conider the cae of permuting row and column of the big matrix A to obtain a matrix A uch that A cannot be tranformed into a fixed matrix B by row and column deletion. Among other thing, they how NP-completene for the general deciion problem. The paper i tructured a follow. In Section 2, we tart with the baic definition and ome eay obervation. After that, in Section 3, the main reult of the work are preented, giving (or ketching) everal parameter-preerving reduction (the core tool of thi paper) from Hitting Set problem to Row Deletion(B) for variou type of the forbidden ubmatrix B. Then, in Section 4 we how how Row Deletion(B) can be olved uing algorithm for Hitting Set problem, again uing a parameter-preerving reduction. Finally, we end with ome concluding remark and open problem in Section 5. Due to the lack of pace, everal proof have been omitted or hortened in thi article, more detail can be found in [11]. 2 Definition and Preliminarie All matrice in thi work have entrie from an alphabet Σ of fixed ize l; we call thee matrice l-ary. Note, however, that all computational hardne reult already hold for binary alphabet. The central problem Row Deletion(B) for a fixed matrix B i defined a follow. Input: A matrix A and a nonnegative integer k. Quetion: ing at mot k row deletion, can A be tranformed into a matrix A uch that A doe not induce B? Herein, A induce B if there exit a B obtained from B through a finite erie of row and column wapping uch that B can be obtained from A by row and

3 column removal. The remaining row and column of A reulting in B are called occurrence of B in A. A matrix A i B-free if B i not induced by A. Thi work etablihe trong link between Row Deletion(B) and the d- Hitting Set problem for contant d, which i defined a follow. Input: A collection C of ubet of ize at mot d of a finite et S and a nonnegative integer k. Quetion: I there a ubet S S with S k uch that S contain at leat one element from each ubet in C? Already for d = 2, d-hitting Set i NP-complete [2]. To expre the cloene between variant of Row Deletion(B) and d- Hitting Set for variou d, we need the following trong notion of reducibility. Let (S, C, k) be an intance of d-hitting Set. We ay that there i a parameter-preerving reduction from d-hitting Set to Row Deletion(B) if there i a polynomial time algorithm that tranform (S, C) into a matrix A and (S, C, k) i a true-intance of d-hitting Set iff (A, k) i a true-intance of Row Deletion(B). The important obervation here i that the objective value parameter k remain unchanged. Thi make it poible to link approximation and exact (fixed-parameter) algorithm for both problem. Finally, for actually performing row deletion in the input matrix A of Row Deletion(B), it i neceary to find the et of row in A that induce B. A traightforward algorithm yield the following. Propoition 1 Given an n m matrix A and a fixed r matrix B (where 1 r n and 1 m), we can find all ize-r et of row in A that induce B in O(n r m r!) wort-cae time. Oberve that Propoition 1 give a pure wort-cae etimation not making ue, e.g., of the pecial tructure of the repective B. In any cae, however, for contant value of r and the running time i polynomial. 3 Computational Hardne In thi ection, we explore the relative computational hardne of Row Deletion(B) by tudying it relationhip to d-hitting Set. We point out many cae concerning the tructure of B for which Row Deletion(B) i at leat a hard a d-hitting Set for ome d depending only on B. Summary of reult. The key idea behind all reduction from d-hitting Set to Row Deletion(B) i to chooe a ymbol from the alphabet Σ and to decompoe B in a certain manner into four ubmatrice, one of them coniting only of and one of them which doe not contain. We can then ue the latter ubmatrix to encode a given d-hitting Set intance into an intance of Row Deletion(B) (i.e., a matrix A) and ue a a fill-in ymbol to prevent unwanted occurrence of B in A.

4 B non-encoding part r B permutation r r V W V W Σ, V i -free encoding part Fig. 1. General cheme for the -decompoition of a matrix B over the alphabet. We call thi pecial decompoition of the forbidden ubmatrix B a -decompoition, illutrated in Fig. 1 and formally defined a follow: Definition 1. (-decompoition) Given an l-ary r matrix B = (b ij ) over the alphabet Σ. A permutation B of B i called a -decompoition of B if there exit a Σ and there exit r, r,, with r + r = r, + = uch that (1) r > 0 and > 0, (2) 1 i r, 1 j : b ij (call thi upper left ubmatrix V ) and (3) r < i r, 1 j : b ij =. The upper right r ubmatrix (b ij ) 1 i r, <j of B i called W, the lower right r ubmatrix (b ij ) r <i r, <j i referred to a. The left part (b ij ) 1 i r,1 j of B (the one containing V ) i called the encoding part of B. The right part (b ij ) 1 i r, <j of B (the one coniting of W and ) i called the non-encoding part of B. For a given Σ, a correponding -decompoition can be eaily computed in time linearly depending on the ize of B (detail omitted). In the following hardne proof, the height of V play a crucial role. Our main hardne reult then are the following. Theorem 1. Let B be a forbidden ubmatrix of ize r with a -decompoition B where the ubmatrix V (of height r ) of B i not induced in the nonencoding part of B. Then there exit a parameter-preerving reduction from r -Hitting Set to Row Deletion(B). Hence, if r 2, Row Deletion(B) i NP-complete. Clearly, it i poible that V i induced in the non-encoding part of B. In particular, then each column vector of V i induced at leat once in the nonencoding part. If we can find one column vector of V which i induced at mot once in the non-encoding part, we again are able to achieve a hardne reult for Row Deletion(B): 1 1 Oberve that it i poible to contruct a ubmatrix B which fulfill the prerequiite of Theorem 1, but doe not fulfill the prerequiite of Theorem 2, and vice vera.

5 Theorem 2. If the r ubmatrix B ha a -decompoition B where the ubmatrix V of height r ha a column vector v that i induced at mot once in the non-encoding part of B, then r -Hitting Set i parameter-preerving reducible to Row Deletion(B). If the ubmatrix B doe not fulfill any of the two prerequiite from Theorem 1 or 2, we can determine two further ubcae for which a hardne reult can be etablihed: Theorem 3. Let B be a forbidden r ubmatrix with a -decompoition B where all entrie of are equal to and V contain r row. Then r -Hitting Set i parameter-preerving reducible to Row Deletion(B). Theorem 4. Let B be a forbidden r ubmatrix with a -decompoition B where all entrie of W are equal to and V contain r row. Then r -Hitting Set i parameter-preerving reducible to Row Deletion(B). For all other cae, i.e., if B doe not fulfill any of the prerequiite from Theorem 1 4, we are not aware of a general tatement on the complexity ) of Row Deletion(B). A an example of uch a matrix conider B = over ( the alphabet Σ = {0, 1}. It i alo clear that Row Deletion(B) i not NP-hard for all B. For example, Row Deletion(B) i olvable in polynomial time if B i a 1 1-matrix. Beide, there are alo non-trivial example for which Row Deletion(B) i olvable in polynomial time, a the matrix B = ( 1 0 ) over the alphabet Σ = {0, 1} how: Oberve that a B-free matrix A ha the property that all it column conit either olely of 1 or olely of 0, i.e., all row of A are identical. Thi implie that the minimum number of row that need to be deleted in order to make an n m matrix B-free i equal to n x, where x denote the ize of the larget et of identical row in A, which can be determined efficiently. Oberve, however, that Row Deletion(B) for B = ( 1 1 ) over the alphabet Σ = {0, 1} already i NP-complete by Theorem 1, ince B trivially ha a - decompoition with = 0, V = B, and W = =. Outline of hardne proof. A mentioned above, all hardne reult are proven by encoding an intance (S, C, k) of d-hitting Set where the value of d i determined by the forbidden ubmatrix B a an intance (A, k) of Row Deletion(B). The key idea concerning how to ue a given -decompoition of B to encode a d-hitting Set intance i illutrated by the following proof of Theorem 3. Subequently, we indicate how thi contruction can be extended to prove, in acending involvedne of the contruction, Theorem 4, Theorem 1, and Theorem 2. Note that due to lack of pace and for a better undertandability of the key idea involved, part of ome proof are only ketched; for their detail, refer to [11]. Proof (Theorem 3). Given an intance (S, C, k) of r -Hitting Set and a - decompoition B of the forbidden r ubmatrix B. Let S = {1,...,n} and

6 C = {{1,3,6},..., {3,5, n},..., {1,3,5}} 3 V W A -decompoition of the forbidden ubmatrix B S C = {{1,3,6},..., {3,5, n},..., {1,3,5}} S = {1,..., n} k An intance of 3-Hitting Set n C The 3-Hitting Set intance encoded into a Row Deletion(B) intance Fig. 2. An example reduction from 3-Hitting Set to Row Deletion(B) following from the proof of Theorem 3 (illutrated for the cae where r = r). C = {C 1,...,C m }. For now, aume that r = r r = 0 (i.e., B conit only of V and W). We generate a matrix A of ize n ( m). Each row of A correpond to an element in S. For each et C C, one occurrence of B i encoded into conecutive column of A, uing the row that correpond to the element in C. For example, conider C h C, 1 h m, with C h = {z 1,...,z r } and z 1,...,z r {1,..., n}. Then, we generate the ubmatrix (a ij ) 1 i n,(h 1) <j h of A uch that the z i th row of thi ubmatrix equal the ith row of B, for all i = 1,...,r, and all other row of thi ubmatrix are et equal to (an illutration for thi i provided in Fig. 2). In thi way, A contain m block of conecutive column where each block induce B exactly once. Thee are the only occurrence of B, ince two column from two different block cannot have r row containing no. Oberve how the property of A that B i only induced within each block and, furthermore, exactly once in each block, i due to uing a a fill-in ymbol: Since i not contained in V, we enure that, by uing a the default-entry for A, no additional occurrence of V (and therefore B) are induced in the row of a block other than the one intended. The outlined reduction can be performed in O(n m ) time. The contruction i eaily generalized for the cae where r < r, i.e., r > 0, by adding r + k row containing only at the bottom of A. Note that by thi contruction, although B i induced multiple time in each block, V i induced only once in each block. It can eaily be hown that olution to the original intance of r -Hitting Set have a 1:1-correpondence with olution to Row Deletion(B) on A: Aume that we have a olution S to the original intance of r -Hitting Set with S = k. Then, delete thoe row in A that correpond to the element

7 in S, thu obtaining A. Note that then, from the ubmatrix V of each B that wa encoded into A, at leat one row ha been deleted. Every column in A contain le than r ymbol different from. Thi directly implie that V doe not occur in A, and therefore A i B-free. We have a olution for the Row Deletion(B) intance with k row deletion. Aume that by deleting at mot k row in A we can make A B-free. Note that we cannot detroy any of the induced B in A by deleting at mot k of the bottom r + k row of A. Therefore, it mut be poible to delete at mot k of the top n row of A to make A B-free. Furthermore, from each induced B in A, at leat one row mut have been deleted. Thu, chooing the element in S that correpond to the deleted row into a et S yield a olution of ize k to the original r -Hitting Set problem. The idea of the above proof uing the ubmatrix V of B to encode an intance of d-hitting Set i employed in all of the following proof. In order to how the 1:1-correpondence of the original d-hitting Set problem and the generated Row Deletion(B) intance, mainly two condition need to be fulfilled: Condition (1): If the optimal olution to the d-hitting Set intance ha ize k, there are no cheaper olution for the generated Row Deletion(B) intance. Condition (2): If there i a olution of ize k to the original d-hitting Set intance, deleting the correponding row in A detroy all occurrence of B in A. Whilt Condition (1) i rather traightforward to meet by extending the idea of the above proof, Condition (2) i quite intricate to fulfill in general, becaue it mut be enured that part of B encoded into A due to different et in C do not induce additional occurrence of B. Proof (Theorem 4). Given an intance of r-hitting Set and a -decompoition for the forbidden ubmatrix B, the reulting matrix A of thi proof reduction i compoed of four ubmatrice: The upper left ubmatrix i generated by the encoding cheme preented in the proof of Theorem 3 uing V a the forbidden ubmatrix, the upper right and lower left ubmatrice are filled with. Two cae are ditinguihed for writing into the lower right ubmatrix of A: If doe not induce V (Cae I), the lower right ubmatrix ha ize (k+1)r (k+1) and contain k + 1 time the matrix in a diagonal cheme. If induce V (Cae II), the lower right ubmatrix ha ize (k + 1)r and contain k + 1 copie of the two cae are illutrated by Fig. 3. Oberve that the reduction for Cae I keep the right part of A V -free. Recall that a ingle by itelf cannot induce V according to the prerequiite of Cae I. However, if we would encode occurrence of one upon the other a for Cae II, on the one hand, V could be induced by row from different encoding of in the lower right part of A. On the other hand, could be induced by everal encoding of V in the upper left part of A. Conequently, we would

8 Encoding of the r -Hitting Set intance uing the reduction cheme from Theorem 3 and V a the forbidden ubmatrix Cae I: i V -free r r r (C,S,k) An intance of r -Hitting Set V A -decompoition of the forbidden ubmatrix B S (k + 1)r.... C... Encoding of the r -Hitting Set intance into a Row Deletion(B) problem (k + 1) Cae II: induce V Fig. 3. Illutration of the reduction in the proof of Theorem 4. have unwanted occurrence of B. The diagonal cheme avoid thee unwanted occurrence of V and keep in particular the right part of A B-free. In Cae II, the reduction cannot keep the right part of A V -free becaue already a ingle occurrence of induce V. However, B cannot be induced there becaue the reduction doe not provide enough column for an occurrence of B. Therefore, if we detroy all induced V in the upper left part of A, then cannot be induced in the left part of A due to the prerequiite of Cae II. Then, the matrix A can be made B-free, even if there are ome occurrence of V in the right part of A. Now, we how the 1:1-correpondence of the olution: Aume that we have a olution S to the encoded r -Hitting Set intance. Then, delete thoe of the n topmot row in A that correpond to the element in S, obtaining A. Note that thi detroy all occurrence of V in the left part of A (enuring Condition (2)). For Cae I, thi directly implie that A i V -free and therefore B-free. For Cae II, recall that B i not induced in the right part of A ince we can find no ufficiently large et of column uch that both and V are induced there. But recall that ince V i not induced in the left part of A, thi alo mean that i not induced there. Hence, A i B-free. Aume that by deleting k row, we can make A B-free. Note that by deleting one of the (k +1)r bottommot row in A, we can detroy at mot one induced B in A. Therefore, there i an optimal olution to Row Deletion(B) that involve only the deletion of k of the n topmot row of A. The ret of the argument follow from the proof of Theorem 3: If, by deleting k of the n topmot row of A, we can make A B-free, then from each V encoded into the

9 Encoding of the r -Hitting Set intance uing the cheme from Theorem 3 and V a the forbidden ubmatrix Encoding of the r -Hitting Set intance uing the cheme from Theorem 3 and W a the forbidden ubmatrix (C,S,k) An intance of r -Hitting Set S r r r V W A -decompoition of the forbidden ubmatrix B r C C C Encoding of the r -Hitting Set intance into a Row Deletion(B) problem Fig. 4. Reduction from r -Hitting Set to Row Deletion(B) ued in the proof of Theorem 1. left part of A, at leat one row mut have been deleted (enuring Condition (1)). Since each encoded V correpond directly to a et in C, chooing the element in S that correpond to the deleted row in A yield a olution to the original r -Hitting Set intance. Proof (Theorem 1). [Sketch] A illutrated in Fig. 4, thi reduction i imilar to the one ued in the proof for Cae I of Theorem 4. The reulting matrix A i again compoed of four ubmatrice, the reduction only differ in the contruction of the upper right ubmatrix of A. In the upper right ubmatrix of A, the given r -Hitting Set intance i encoded uing the cheme from the proof of Theorem 3 and W a the forbidden ubmatrix. A in the previou two proof, the parameter k i preerved and the encoding can be carried out in polynomial time with repect to the input ize. A in the proof of Theorem 4, the encoding proce enure that B i not induced in the right part of A. Thi i due to the following obervation: Aume that B i induced in the right part of A. Then, V i induced there a well. By the prerequiite of the theorem, the non-encoding part of B doe not induce V. Therefore, an occurrence of V involve column from at leat two different encoding of. However, note that any two uch column of A can due to the encoding cheme only have le than r row that do not contain a. But V ha r row with no ymbol equal to, a contradiction.

10 Now note that by deleting one of the r C bottom row in A, we can detroy at mot one induced ubmatrix B in A thi could alway be achieved by deleting one of the S topmot row of A. Therefore, every olution of the produced Row Deletion(B) intance can be tranlated to one deleting only row from the topmot S row. From the econd obervation it i clear that an optimal olution of Row Deletion(B) can only conit of the n topmot row of A. The element of S that correpond to the deleted row form a olution for the original r -Hitting Set intance, uing the ame argument a in the previou proof. Converely, if we have a olution of ize at mot k to the original r -Hitting Set intance, deleting the correponding row in A detroy all occurrence of V in A due to the firt obervation and make A B-free. Hence, every olution to the r - Hitting Set intance (S, C, k) implie a olution to the Row Deletion(B) intance (A, k) and vice vera. The cheme and idea of the above proof are alo ued in proving Theorem 2. The detail of the proof are rather involved, firtly etablihing the reult for r 2 matrice and then extending thi reult to obtain Theorem 2. We hall only preent the main idea for the reduction involved, referring to [11] for detail. Proof (Theorem 2). [Reduction cheme for r 2 matrice, key idea] For r 2 matrice that fulfill the prerequiite of the theorem, the reduction i performed a follow: Given an intance of r -Hitting Set, the matrix A i contructed jut a in the proof of Theorem 1. Then, the following algorithm i performed: A long a there are two column in the right part of A whoe upper n entrie are identical to each other, remove one of the two column from A. It i poible to how that after thi merging of column in A, the right part of A i B-free. An extenion of thi cheme from r 2 matrice to all matrice that fulfill the prerequiite of thi theorem i performed by arguing that by being able to avoid the occurrence of an r 2 ubmatrix of B in the right part of A, we are alo able to avoid the occurrence of B there altogether. 4 Algorithmic Tractability Thi ection point out an algorithmic approach to olve Row Deletion(B). To thi end, we give a parameterized reduction to r-hitting Set where r i the number of row in B. Theorem 5. Given a fixed r ubmatrix B, Row Deletion(B) i parameterpreerving reducible to r-hitting Set in O(n r m r!) time. Proof. Given an intance (A, k) of Row Deletion(B) (where A i an n m matrix), we contruct an intance (S, C, k) of r-hitting Set a follow: (1) We contruct a et S = {r 0,...,r n 1 } containing one element for every row in A. (2) We compute a et C of ubet of S: For every et R {r 0,...,r n 1 } with R = r correponding to r row in A that induce B, we add R to C. (3) The

11 r 0 r 1 r 2 r 3 r 4 r 5 c 0 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c C = {{r 0, r 1, r 3 }, {r 0, r 2, r 3 }, {r 0, r 3, r 5 }, {r 0, r 4, r 5 }, {r 1, r 3, r 4 }, {r 2, r 3, r 4 }} Fig. 5. Illutrating the reduction from Row Deletion(B) to r-hitting Set. Let B be the matrix coniting of row (1,1), (0,1), and (1,0). Given an input matrix A a hown, we generate an r-hitting Set intance coniting of a et S = {r 0, r 1,..., r 5 } and the et C a hown. The gray underlay how, a an example, how B i induced by row r 0, r 3, and r 5 in column c 1 and c 6 of A, leading to the the ubet {r 0, r 3, r 5 } in C. parameter k i directly preerved. Then, (S, C, k) i the reulting r-hitting Set intance. An example for the reduction i illutrated in Fig. 5. The direct 1:1-correpondence between the olution of the r-hitting Set intance and the Row Deletion(B) intance can be hown a follow: Let S S be a olution of ize k to the r-hitting Set intance (S, C, k). We delete the row in A that correpond to the element in S, yielding A. Aume that B were till induced in A by a et I of row. Then, the row in I did induce B in A, meaning a et containing the element correponding to thee row wa put into C. But one row of I mut then have been deleted ince S i a valid olution to (S, C, k), a contradiction. Therefore, B cannot be induced by A anymore. If, on the other hand, A can be made B-free by deleting k row, then for each occurrence of B in A by ome row, at leat one of thee row mut have been deleted. By chooing the element correponding to the deleted row a a olution S S to the generated r-hitting Set intance, we have choen at leat one element from every et in C, making S a valid olution of ize k. The running time follow with Propoition 1. Theorem 5 directly implie the following two poitive reult: The bet known polynomial-time approximation algorithm for r-hitting Set, which currently ha approximation factor r, can be ued to obtain a factor-r approximation for the correponding Row Deletion(B). r-hitting Set can be trivially olved in O(r k n r ) time, where k denote the ize of the olution. Thi mean that for contant r Row Deletion(B) i fixed-parameter tractable with repect to parameter k. See [7] for the currently bet fixed-parameter algorithm for r-hitting Set for intance, the bet exponential term for 3-Hitting Set i known to be 2.27 k intead of only 3 k.

12 5 Concluion In thi work, we have tarted a ytematic tudy on complexity of and algorithm for Row Deletion(B). Among other, we were able to how NP-completene for a number of natural cae of forbidden ubmatrice B. It remain open to generalize all pecial cae treated in thi work, e.g., by proving or diproving the following conjecture: For every forbidden ubmatrix B with at leat three row, Row Deletion(B) i NP-complete. Our work wa partially motivated by contructing perfect phylogenie from binary matrice [8, 10]. For thi pecial cae, where we have to conider a forbidden ubmatrix B coniting of the row (1, 1), (1, 0), and (0, 1), our reult yield that Row Deletion(B) i at leat a hard a 2-Hitting Set (which i the ame a the well-known Vertex Cover problem) and that it alway can be olved by tranforming it into an intance of 3-Hitting Set. Note that there remain a gap between the reult of thi work: Let an r forbidden ubmatrix B have a -decompoition with height-r ubmatrix V. Then, if r > r, we howed, on the one hand, that in certain cae Row Deletion(B) i at leat a hard to olve a r -Hitting Set and, on the other hand, that it i not harder to olve than r-hitting Set. Reference 1. A. Brandtädt, V. B. Le, and J. P. Spinrad. Graph Clae: A Survey. SIAM Monograph on Dicrete Mathematic and Application, M. R. Garey and D. S. Johnon. Computer and Intractability: A Guide to the Theory of NP-Completene. W. H. Freeman, J. Gramm, J. Guo, F. Hüffner, and R. Niedermeier. Graph-modeled data clutering: fixed-parameter algorithm for clique generation. In Proc. of 5th CIAC, volume 2653 of LNCS, page , Springer, J. Gramm, J. Guo, F. Hüffner, and R. Niedermeier. Automated generation of earch tree algorithm for graph modification problem. In Proc. of 11th ESA, volume 2832 of LNCS, page , Springer, B. Klinz, R. Rudolf, and G. J. Woeginger. Permuting matrice to avoid forbidden ubmatrice. Dicrete Applied Mathematic, 60: , A. Natanzon, R. Shamir, and R. Sharan. Complexity claification of ome edge modification problem. Dicrete Applied Mathematic, 113: , R. Niedermeier and P. Romanith. An efficient fixed-parameter algorithm for 3- Hitting Set. Journal of Dicrete Algorithm, 1:89 102, I. Pe er, R. Shamir, and R. Sharan. On the generality of phylogenie from incomplete directed character. In Proc. of 8th SWAT, volume 2368 of LNCS, page , Springer, R. Shamir, R. Sharan, and D. Tur. Cluter graph modification problem. In Proc. of 28th WG, volume 2573 of LNCS, page , Springer, R. Sharan. Graph Modification Problem and their Application to Genomic Reearch. Ph.D. Thei, School of Computer Science, Tel-Aviv niverity, S. Wernicke. On the Algorithmic Tractability of Single Nucleotide Polymorphim (SNP) Analyi and Related Problem. Diploma Thei, WSI für Informatik, niverität Tübingen, September 2003.