Polynomial Time Algorithm for Determining Max-Min Paths in Networks and Solving Zero Value Cyclic Games

We study the max-min paths problem, which represents a game version of the shortest and the longest paths problem in a weighted directed graph. In this problem the vertex set V of the weighted directed graph G = (V, E) is divided into two disjoint subsets V-A and V-B which are regarded as positional sets of two players. The players are seeking for a directed path from the given starting position v(0) to the final position v(f), where the first player intends to maximize the integral cost of the path while the second one has aim to minimize it. Polynomial-time algorithm for determining max-min path in networks is proposed and its application for solving zero value cyclic games is developed.