The time-dependent rural postman problem: polyhedral results

Abstract

The Chinese postman problem is a famous and classical problem in graph theory. This paper introduces a new variant of this problem, namely, the time-dependent rural postman problem, which is motivated by applications, involving scheduling with time-dependent processing time. We present an arc-path formulation for the problem with a constraint set that is divided into two parts. The first part is linear and has a strong combinatorial structure, which defines the polytope of the arc-path alternation sequence (APAS), while the second part is nonlinear and is closely related to time-dependent travel time. First, we investigate the facial structure of the APAS polytope and present several facet inequalities. Second, considering the travel and service time functions as piecewise functions, we linearize the nonlinear inequalities and propose several strong, valid inequalities. Finally, we propose a cutting plane algorithm and report numerical results on several randomly generated instances.