On finding shortest paths in arc-dependent networks

Abstract

This paper is concerned with the design and analysis of algorithms for the arc-dependent shortest path (ADSP) problem. A network is said to be arc-dependent if the cost of an arc a depends upon the arc taken to enter a. These networks are fundamentally different from traditional networks in which the cost associated with an arc is a fixed constant and part of the input. The ADSP problem is also known as the suffix-1 path-dependent shortest path problem in the literature. This problem has a polynomial time solution if the shortest paths are not required to be simple. The ADSP problem finds applications in a number of domains including highway engineering, turn penalties and prohibitions, and fare rebates. In this paper, we are interested in the ADSP problem when restricted to simple paths. We call this restricted version the simple arc-dependent shortest path (SADSP) problem. We show that the SADSP problem is NP-complete. We present inapproximability results and an exact exponential algorithm for this problem. Additionally, we explore the problem of detecting negative cost cycles in arc-dependent networks and discuss its algorithmic complexity.