ポリマトロイド対の普遍基の特徴付け

Abstract

This note gives a characterization of the universal pair of bases of a pair of polymatroids as the nearest pair of bases with respect to a class of pseudo-distances including the Kullback-Leibler divergence. N. Megiddo considered the lexico-optimal flow problem in a multiterminal network N = (V, A, c; S^+, S^-) (V: vertex set, A: arc set, c: capacity, S^+: supply (source) vertices, S^-: demand (sink) vertices), which is to find a maximal flow such that the supply flow (s^+ (v) | v ∈ S^+)[resp., the demand flow (s^- (v) | v ∈ S^-)] is as proportional as possible to a given weight vector. This problem is treated by S. Fujishige as a special case of the lexico-optimal base problem for a single polymatroid. This paper considers the problem of finding a maximal flow such that the supply flow s^+ and the demand flow s^- are as "near" as possible (where a one-to-one correspondence between S^+ and S^- is assumed to be given), and generalizes it to the problem of finding a "nearest" pair of bases of a pair of polymatroids It is shown that the "nearest" pair coincides with the universal pair if either of the following criteria is adopted. (1) The f-divergence (a generalization of the Kullback-Leibler divergence) between the bases should be minimized; (2) The vector consisting of the ratios of the corresponding components of the bases should be lexicographically maximized.