In this paper, we propose a new counting scheme for m n contingency tables. Our scheme is a modification of Dyer and Greenhill's scheme for two rowed contingency tables. We can estimate not only the sizes of error, but also the sizes of the bias of the number of tables obtained by our scheme, on the assumption that we have an approximate sampler.

Sports scheduling concerns how to construct a schedule of a sports competition mathematically. Many sports competitions are held as round-robin tournaments. In this paper, we consider a particular class of round-robin tournaments. We report some properties of round-robin tournaments and enumerate home-away tables satisfying some practical requirements by computer search.

This paper deals with a strategic issue in the stable marriage model with complete preference lists (i.e., a preference list of an agent is a permutation of all the members of the opposite sex). Given complete preference lists of n men over n women, and a marriage µ, we consider the problem for finding preference lists of n women over n men such that the men-proposing deferred acceptance algorithm (Gale-Shapley algorithm) adopted to the lists produces µ. We show a simple necessary and sufficient condition for the existence of a set of preference lists of women over men. Our condition directly gives an O(n2) time algorithm for finding a set of preference lists, if it exists.

Suppose that we have a timetable of a round-robin tournament with a number of teams, and distances among their homes. The home-away assignment problem is to find a home-away assignment that minimizes the total traveling distance of the teams. We propose a formulation of the home-away assignment problem as an integer program, and a rounding algorithm based on Bertsimas, Teo and Vohra's dependent randomized rounding method [2]. Computational experiments show that our method quickly generates feasible solutions close to optimal.

An s-t flow in a directed network is called uncontrollable, when the flow is representable as a positive sum of elementary s-t path flows. In this paper, we discuss the problem Is a given flow uncontrollable?. We show that the problem is NP-complete.

The firefighter problem is used to model the spread of fire, infectious diseases, and computer viruses. This paper deals with firefighter problem on rooted trees. It is known that the firefighter problem is NP-hard even for rooted trees of maximum degree 3. We propose techniques to improve a given approximation algorithm. First, we introduce an implicit enumeration technique. By applying the technique to existing ()-approximation algorithm, we obtain -approximation algorithm when a root has k children. In case of ternary trees, k=3 and thus the approximation ratio satisfies ≥ 0.6892, which improves the existing result ≥ 0.6321. Second technique is based on backward induction and improves an approximation algorithm for firefighter problem on ternary trees. If we apply the technique to existing () -approximation algorithm, we obtain 0.6976-approximation algorithm. Lastly, we combine the above two techniques and obtain 0.7144-approximation algorithm for firefighter problem on ternary trees.

In this paper, we consider a network design problem with hub-and-spoke structure. We propose a relaxation technique for the problem where the location of hub nodes is given and decides the allocation of non-hub nodes to one of the hub nodes. We linearize the non-convex quadratic objective function of the original problem, introducing Hitchcock transportation problems defined for each pair of non-hub nodes. We provide two linear relaxation problems, one based on the Hitchcock transportation problems and the other on the dual Hitchcock transportation problems. We show the tightness of the lower bounds obtained by our formulations by computational experiences.