Sign-Solvable Linear Complementarity Problems

Kakimura, Naonori

doi:10.1007/978-3-540-72792-7_30

Cited by 3 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we mention that there exists a line of research for sign solvability for linear systems [7,25], linear programming problem [16], and linear complementarity problem [19]. They mainly study sign patterns of the input data, that always determine sign patterns of solutions.…”

Section: The Results Obtained In This Papermentioning

confidence: 99%

Trichotomy for integer linear systems based on their sign patterns

Kimura

Makino

2016

Discrete Applied Mathematics

View full text Add to dashboard Cite

In this paper, we consider solving the integer linear systems, i.e., given a matrix A ∈ R m×n , a vector b ∈ R m , and a positive integer d, to compute an integer vector x ∈ D n such that Ax ≥ b, where m and n denote positive integers, R denotes the set of reals, andThe problem is one of the most fundamental NP-hard problems in computer science.For the problem, we propose a complexity index η which is based only on the sign pattern of A. For a real γ, let ILS = (γ) denote the family of the problem instances I with η(I) = γ. We then show the following trichotomy: In this paper, we consider the problem of computing an integer vector x ∈ D n such that Ax ≥ b, which we denote by ILS. The ILS problem is one of the most fundamental and important problems in computer science, and have been studied extensively from both theoretical and practical points of view [18,26]. It is known that the ILS problem is strongly NP-hard, and can be solved in polynomial time, if m or n are bounded by some constant [22], or A is totally unimodular and b is integral [15]. When A is quadratic (also called TVPI, i.e., each row of A contains at most two nonzero elements) or Horn (i.e., each row of A contains at most one positive element), the ILS problem is known to be weakly NP-hard, but it can be solved in time polynomial in the input length and d, and hence in pseudo-polynomial time [20,14,29]. The best known bounds for quadratic and Horn systems require O(md) time [2] and O(n 2 md) time, respectively. For unit linear systems, i.e., A ∈ {0, −1, +1} m×n , it is known that the

show abstract

Section: The Results Obtained In This Papermentioning

confidence: 99%

Trichotomy for integer linear systems based on their sign patterns

Kimura

Makino

2016

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…It should also be noted that the above algorithm for the sign-balanced 2-LCP is not obtained from the results for the other well-known subclasses of the LCP that focus on the sign pattern of M, such as the Z-LCP (i.e., the coefficient matrix is restricted to be a Z-matrix) and the sign-solvable LCP introduced by Kakimura [21].…”

mentioning

confidence: 96%

“…Then for i = 1 s, Equation (19) and w i 1 z i = 0 imply w i 1 z i = 0 1 or 1 0 . For j = 1 m, note that z k j +1 = 1 holds by (21) and w ≥ 0, and hence the complementarity condition with respect to k j + 1 means that at least one of w k j +1 1 , w k j +1 2 , and w k j +1 3 is equal to zero. Hence, if we set x i to be true if z i = 1, and false if z i = 0, then this is equivalent to satisfying clause j.…”

mentioning

confidence: 98%

The Linear Complementarity Problems with a Few Variables per Constraint

2015

Self Cite

View full text Add to dashboard Cite

In this paper, we consider the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrices are restricted to have at most k nonzero entries per row. It is known that the 1-LCP is solvable in linear time, and the 3-LCP is strongly NP-hard. We show that the 2-LCP is strongly NP-hard, and it can be solved in polynomial time if it is sign-balanced, i.e., each row of the matrix has at most one positive and one negative entry. Our second result matches the currently best-known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of the sign-balanced 2-LCP is weakly NP-hard and pseudo-polynomially solvable, and the generalized 1-LCP is strongly NP-hard.

show abstract

“…Totally sign-nonsingular matrices play an important role in the sign-solvability of linear systems of equations [2,13,14,26], linear programming [7], and linear complementarity problems [10]. Total sign-nonsingularity can be recognized in polynomial time by testing sign-nonsingularity of a related symmetric matrix [7].…”

mentioning

confidence: 99%

Matching structure of symmetric bipartite graphs and a generalization of Pólya's problem

Kakimura

2010

Journal of Combinatorial Theory, Series B

Self Cite

View full text Add to dashboard Cite

A bipartite graph is said to be symmetric if it has symmetry of reflecting two vertex sets. This paper investigates matching structure of symmetric bipartite graphs. We first apply the DulmageMendelsohn decomposition to a symmetric bipartite graph. The resulting components, which are matching-covered, turn out to have symmetry. We then decompose a matching-covered bipartite graph via an ear decomposition, which is a sequence of subgraphs obtained by adding an odd-length path repeatedly. We show that, if a matching-covered bipartite graph is symmetric, an ear decomposition can retain symmetry by adding no more than two paths. As an application of these decompositions to combinatorial matrix theory, we present a natural generalization of Pólya's problem. We introduce the problem of deciding whether a rectangular {0, 1}-matrix has a signing that is totally sign-nonsingular or not, where a rectangular matrix is totally sign-nonsingular if the sign of the determinant of each submatrix with the entire row set is uniquely determined by the signs of the nonzero entries. We show that this problem can be solved in polynomial time with the aid of the matching structure of symmetric bipartite graphs. In addition, we provide a characterization of this problem in terms of excluded minors.

show abstract

Sign-Solvable Linear Complementarity Problems

Cited by 3 publications

References 21 publications

Trichotomy for integer linear systems based on their sign patterns

Trichotomy for integer linear systems based on their sign patterns

The Linear Complementarity Problems with a Few Variables per Constraint

Matching structure of symmetric bipartite graphs and a generalization of Pólya's problem

Contact Info

Product

Resources

About