A strongly polynomial time algorithm for the shortest path problem on coherent planar periodic graphs

Abstract

A periodic graph is an infinite graph obtained by copying a finite graph to each room of ℤd-lattice and connecting them regularly. Höfting and Wanke formulated the shortest path problem on periodic graphs as an integer programming and showed that it is NP-hard, together with a pseudopolynomial time algorithm for bounded d. Using Iwano and Steiglitz's result, the time complexity can be shown to be weakly polynomial on planar periodic graphs with d = 2. In this paper, we show a strongly polynomial time algorithm for the shortest path problem on coherent planar periodic graphs with d = 2. The coherence is a combinatorial property of periodic graphs introduced in this paper, which naturally holds for planar regular tilings, etc. We show that the coherence is the necessary and sufficient condition that an optimal solution to the integer programming is still optimal if the connectedness constraint is removed. Using the theory of toric ideal, we further show that the incidence-transit matrix describing the linear constraints in the integer linear programming for planar cases with d = 2 form a new class of unimodular matrices, which itself is of theoretical interest and leads to the strongly polynomial time algorithm. © Springer-Verlag 2012.