The Consecutive Ones Submatrix Problem for Sparse Matrices

Theoretical Computer Science, 2008

Using a dynamic programming approach, we prove that a large variety of matrix reconstruction problems from two projections can be solved in polynomial time whenever the number of rows (or columns) is fixed. We also prove some complexity results for several problems concerning the reconstruction of a binary matrix when a neighborhood constraint occurs.

Lecture Notes in Computer Science, 2011

A binary matrix has the Consecutive-Ones Property (C1P) if its columns can be ordered in such a way that all 1's in each row are consecutive. We consider here a variant of the C1P where columns can appear multiple times in the ordering. Although the general problem of deciding the C1P with multiplicity is NP-complete, we present here a case of interest in comparative genomics that is tractable.

Zeitschrift für Operations Research, 1986

This note deals with the problem of permuting elements within columns of a real matrix so as to minimize a real-valued function of row sums. The special case dealing with minimization of maximum row sum has been studied by several authors 6, recently. Here we are concerned primarily with the case in which the matrix has two columns only and the function is Schur-convex.

Discrete Applied Mathematics, 2012

Motivated by problems of comparative genomics and paleogenomics, in [6] the authors introduced the Gapped Consecutive-Ones Property Problem (k, δ)-C1P: given a binary matrix M and two integers k and δ, can the columns of M be permuted such that each row contains at most k blocks of ones and no two consecutive blocks of ones are separated by a gap of more than δ zeros. The classical C1P problem, which is known to be polynomial is equivalent to the (1, 0)-C1P problem. They showed that the (2, δ)-C1P Problem is NP-complete for all δ ≥ 2 and that the (3, 1)-C1P problem is NP-complete. They also conjectured that the (k, δ)-C1P Problem is NP-complete for k ≥ 2, δ ≥ 1 and (k, δ) = (2, 1). Here, we prove that this conjecture is true. The only remaining case is the (2, 1)-C1P Problem, which could be polynomial-time solvable.

Electronic Notes in Discrete Mathematics, 2009

Motivated by problems of comparative genomics and paleogenomics, we introduce the Gapped Consecutive-Ones Property Problem (k,δ)-C1P: given a binary matrix M and two integers k and δ, can the columns of M be permuted such that each row contains at most k sequences of 1's and no two consecutive sequences of 1's are separated by a gap of more than δ 0's. The classical C1P problem, which is known to be polynomial, is equivalent to the (1,0)-C1P Problem. We show that the (2,δ)-C1P Problem is NP-complete for δ ≥ 2. We conjecture that the (k, δ)-C1P Problem is NPcomplete for k ≥ 2, δ ≥ 1, (k, δ) = (2, 1). We also show that the (k,δ)-C1P problem can be reduced to a graph bandwidth problem parameterized by a function of k, δ and of the maximum number s of 1's in a row of M , and hence is polytime solvable if all three parameters are constant.

Computing, 1994

A General Approach to Avoiding Two by Two Submatrices. A matrix C is said to avoid a set ~ of matrices, if no matrix of~ can be obtained by deleting some rows and columns of C. In this paper we consider the decision problem whether the rows and columns of a given matrix C can be permuted in such a way that the permuted matrix avoids all matrices of a given class ~. At first an algorithm is stated for deciding whether C can be permuted such that it avoids a set ~ of 2 x 2 matrices. This approach leads to a polynomial time recognition algorithm for algebraic Monge matrices fulfilling special properties. As main result of the paper it is shown that permuted Supnick matrices can be recognized in polynomial time. Moreover, we prove that the decision problem can be solved in polynomial time, if the set ~" is sufficiently dense, and a sparse set of 2 x 2 matrices is exhibited for which the decision problem is NP-complete.

1999

A 0-1 matrix is called l-block-free if it nowhere has a pair of rows ancl a pair of columns such that the four entries thus singled out are all 1,s. What is the maximum number of 1's a 4-block-free matrix can have? We investigate a knapsack-type relaxation of a 0-1 prograrnming formulatiotr of the problcm, accotrnt for its solution properties ancl acldress issues as to t,he realizobility in terms of 0-1 matrices of the solutions found. We note frtrt,hermore that a conclusivc answer, valicl fclr all square rrratrices, would also settle the cla.ssical anrl gelrerally unanswered question as to ttle existence of a so-called finite projectiue plane of a given order. Along the way, some surprisingly simple proofs of earlier results arc provided. Also a nerv characlerization of a finite projective plane is given.

Given an m × n binary matrix A, a subset C of the columns is called t-frequent if there are at least t rows in A in which all entries belonging to C are nonzero. Let us denote by α the number of maximal t-frequent sets of A, and let β denote the number of those minimal column subsets of A which are not t-frequent (so called t-infrequent sets). We prove that the inequality α ≤ (m − t + 1)β holds for any binary matrix A in which not all column subsets are t-frequent. This inequality is sharp, and allows for an incremental quasi-polynomial algorithm for generating all minimal t-infrequent sets. We also prove that the analogous generation problem for maximal t-frequent sets is NP-hard. Finally, we discuss the complexity of generating closed frequent sets and some other related problems.

Mathematics of Computation, 2009

For an arbitrary matrix A of n × n symbols, consider its submatrices of size k×k, obtained by deleting n−k rows and n−k columns. Optionally, the deleted rows and columns can be selected symmetrically or independently. We consider the problem of whether these multisets determine matrix A. Following the ideas of Krasikov and Roditty in the reconstruction of sequences from subsequences, we replace the multiset by the sum of submatrices. For k > cn 2/3 we prove that the matrix A is determined by the sum of the k × k submatrices, both in the symmetric and in the nonsymmetric cases.

arXiv (Cornell University), 2023

A 0-1 matrix M contains another 0-1 matrix P if some submatrix of M can be turned into P by changing any number of 1-entries to 0-entries. The 0-1 matrix M is P-saturated where P is a family of 0-1 matrices if M avoids every element of P and changing any 0-entry of M to a 1-entry introduces a copy of some element of P. The extremal function ex(n, P) and saturation function sat(n, P) are the maximum and minimum possible number of 1-entries in an n × n P-saturated 0-1 matrix, respectively, and the semisaturation function ssat(n, P) is the minimum possible number of 1-entries in an n × n P-semisaturated 0-1 matrix M , i.e., changing any 0-entry in M to a 1-entry introduces a new copy of some element of P. We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-O(n d−1) d-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-O(n d−1) d-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers k, d and integer r ∈ [0, d − 1], we construct a family of d-dimensional 0-1 matrices with both extremal function and saturation function exactly kn r for sufficiently large n. We show that no family of d-dimensional 0-1 matrices has saturation function strictly between O(1) and Θ(n) and we construct a family of d-dimensional 0-1 matrices with bounded saturation function and extremal function Ω(n d−ǫ) for any ǫ > 0. Up to a constant multiplicative factor, we fully settle the problem of characterizing the semisaturation function of families of d-dimensional 0-1 matrices, which we prove to always be Θ(n r) for some integer r ∈ [0, d − 1].

We are concerned with binary matrix reconstruction from their orthogonal projections. To the basic problem we add new kinds of constraints. In the first problems we study the ones of the matrix must be isolated: All the neighbors of a one must be a zero. Several types of neighborhoods are studied. In our second problem, every one has to be horizontally not isolated. Moreover, the number of successive zeros in a horizontal rank must be bounded by a fixed parameter. Complexity results and polynomial-time algorithms are given.

2009

Motivated by problems of comparative genomics and paleogenomics, in [6] the authors introduced the Gapped Consecutive-Ones Property Problem (k, δ)-C1P: given a binary matrix M and two integers k and δ, can the columns of M be permuted such that each row contains at most k blocks of ones and no two consecutive blocks of ones are separated by a gap of more than δ zeros. The classical C1P problem, which is known to be polynomial is equivalent to the (1, 0)-C1P problem. They showed that the (2, δ)-C1P Problem is NP-complete for all δ ≥ 2 and that the (3, 1)-C1P problem is NP-complete. They also conjectured that the (k, δ)-C1P Problem is NP-complete for k ≥ 2, δ ≥ 1 and (k, δ) = (2, 1). Here, we prove that this conjecture is true. The only remaining case is the (2, 1)-C1P Problem, which could be polynomial-time solvable.

Linear Algebra and Its Applications, 1980

Let m and n be positive integers, and let R = (rl,. . . , r,) and S = (an.. . ,sn) be nonnegative integral vectors. We survey the combinatorial properties of the set of all m x n matrices of O's and l's having ri l's in row i and si I's in column i. A number of new results are proved. The results can also be formulated in terms of the set of bipartite graphs with a bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S. of Fu(R,S) . 1 d' g mc u m some recent results. In doing so, we present in some cases new proofs of theorems which may be more transparent than those in the literature. Also there appear here for the first time a number of new results, notably the solution (Theorem 6.8) of a problem posed by Ryser [56, p. 761 in 1963. Other new results include Theorems 3.10, 4.2, 4.4, 5.8, 5.9, 6.8, 6.10, 7.3, 8.3, and 8.13, and Corollaries 5.6 and 8.6. Over twenty problems are proposed. Before proceeding we give two alternative interpretations of 8(R, S). Let X=(x,,..., xm} and Y={ yr,..., y,} be disjoint sets of m and n elements, respectively. Let BG(R,S) d enote the collection of all bipartite graphs G with the following properties: (BGl) The vertices of G are xi,. . . , x,, yl,. . . , y,,. (BG2) Each edge of G joins a vertex in X to a vertex in Y. (BG3) The degree (or valency) of xi is r, for i = 1,. . . , m, and the degree of yj is si for j=l,...,n. Then there is a one-to-one correspondence between the matrices in '%(R,S) and the bipartite graphs in BG(R, S), determined as follows. If A =[a,] E

Theoretical Computer Science, 2008

ABSTRACT The paper studies the problem of reconstructing binary matrices constrained by binary tomographic information. We prove new NP-hardness results that sharpen previous complexity results in the realm of discrete tomography but also allow applications to related problems for permutation matrices. Hence our results can be interpreted in terms of other combinatorial problems including the queens’ problem.

Discrete Optimization, 2007

We investigate the computational complexity of a combinatorial problem that arises in DNA sequencing by hybridization: The input consists of an integer together with a set S of words of length k over the four symbols A, C, G, T. The problem is to decide whether there exists a word of length that contains every word in S at least once as a subword, and does not contain any other subword of length k. The computational complexity of this problem has been open for some time, and it remains open. What we prove is that this problem is polynomial time equivalent to the exact perfect matching problem in bipartite graphs, which is another infamous combinatorial optimization problem of unknown computational complexity.