On the Approximability of Path and Cycle Problems in Arc-Dependent Networks

Abstract

In the field of transportation planning, it is often insufficient to model transportation networks by using networks with fixed arc costs. There may be additional factors that modify the time or cost of a single trip. These include turn prohibitions, fare rebates, and transfer times. Each of these factors causes the cost of a portion of the trip to depend directly on the previous portion of the trip. This dependence can be modeled using arc-dependent networks. In an arc-dependent network, the cost of an arc a depends upon the arc used to enter a. In this paper, we study the approximability of a number of negative cost cycle problems in arc-dependent networks. In a general network, the cost of an arc is a fixed constant and part of the input. Arc-dependent networks can be used to model several real-world problems, including the turn-penalty shortest path problem. Previous literature established that corresponding path problems in these networks are NP-hard. We extend that research by providing inapproximability results for several of these problems. In [7], it was established that a more general form of the shortest path problem in arc-dependent networks, known as the quadratic shortest path problem, cannot be approximated to within a constant factor. In this paper, we strengthen that result by showing NPO PB-completeness.