Naonori Kakimura

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

Packing Directed Circuits through Prescribed Vertices Bounded Fractionally

\max\{cx \mid Ax=b, x\geq 0\}

Robust Independence Systems

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.

Half-integral packing of odd cycles through prescribed vertices

Cooperative matching games have drawn much interest partly because of the connection with bargaining solutions in the networking environment. However, it is not always guaranteed that a network under investigation gives rise to a stable bargaining outcome. To address this issue, we consider a modification process, called stabilization, that yields a network with stable outcomes, where the modification should be as small as possible. Therefore, the problem is cast to a combinatorial-optimization problem in a graph. Recently, the stabilization by edge removal was shown to be NP-hard. On the contrary, in this paper, we show that other possible ways of stabilization, namely, edge addition, vertex removal and vertex addition, are all polynomial-time solvable. Thus, we obtain a complete complexity-theoretic classification of the natural four variants of the network stabilization problem. We further study weighted variants and prove that the variants for edge addition and vertex removal are NP-hard.

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

A seminal result of Reed et al. [Combinatorica, 16 (1996), pp. 535--554] says that a directed graph has either

Computing the inertia from sign patterns

k

Computing knapsack solutions with cardinality robustness

vertex-disjoint directed circuits or a set of at most

Maximizing Time-Decaying Influence in Social Networks

f(k)