Solving linear programs from sign patterns

This paper is an attempt to provide a connection between qualitative matrix theory and linear programming. A linear program is said to be sign-solvable if the set of sign patterns of the optimal solutions is uniquely determined by the sign patterns of A, b, and c. It turns out to be NP-hard to decide whether a given linear program is sign-solvable or not. We then introduce a class of sign-solvable linear programs in terms of totally sign-nonsingular matrices, which can be recognized in polynomial time. For a linear program in this class, we devise an efficient combinatorial algorithm to obtain the sign pattern of an optimal solution from the sign patterns of A, b, and c. The algorithm runs in O(mγ) time, where m is the number of rows of A and γ is the number of all nonzero entries in A, b, and c.     

Computing the inertia from sign patterns

A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang showed that the inertia of a sign-nonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such symmetric matrices. The algorithm runs in time for a symmetric matrix of order n with m nonzero entries. In addition, it is shown to be NP-complete to decide whether the inertia of a given symmetric matrix is not determined by its sign pattern.     

A direct proof for the matrix decomposition of chordal-structured positive semidefinite matrices

Agler, Helton, McCullough, and Rodman proved that a graph is chordal if and only if any positive semidefinite (PSD) symmetric matrix, whose nonzero entries are specified by a given graph, can be decomposed as a sum of PSD matrices corresponding to the maximal cliques. This decomposition is recently exploited to solve positive semidefinite programming efficiently. Their proof is based on a characterization for PSD matrix completion of a chordal-structured matrix due to Grone, Johnson, Sá, and Wolkowicz. This note gives a direct and simpler proof for the result of Agler et al., which leads to an alternative proof of Grone et al.     

Total dual integrality of the linear complementarity problem

Annals of Operations Research - In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of...     

Packing cycles through prescribed vertices under modularity constraints

The well-known theorem of Erd?s–Pósa says that either a graph G has k disjoint cycles or there is a vertex set X   of order at most f(k)$f(k)$ for some function f   such that G?X$G?X$ is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization.     

Half-integral packing of odd cycles through prescribed vertices

The well-known theorem of Erd?s-Pósa says that a graph G has either k disjoint cycles or a vertex set X of order at most f(k) for some function f such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we discuss packing disjoint S-cycles, i.e., cycles that are required to go through a set S of vertices. For this problem, Kakimura-Kawarabayashi-Marx (2011) and Pontecorvi-Wollan (2010) recently showed the Erd?s-Pósa-type result holds. We further try to generalize this result to packing S-cycles of odd length. In contrast to packing S-cycles, the Erd?s-Pósa-type result does not hold for packing odd S-cycles. We then relax packing odd S-cycles to half-integral packing, and show the Erd?s-Pósa-type result for the half-integral packing of odd S-cycles, which is a generalization of Reed (1999) when S=V. That is, we show that given an integer k and a vertex set S, a graph G has either 2k odd S-cycles so that each vertex is in at most two of these cycles, or a vertex set X of order at most f(k) (for some function f) such that G\X has no odd S-cycle.