A Cut-Based Mixed Integer Programming Formulation for the Hop-Constrained Cheapest Path Problem

Given a simple graph G = (V,E) with edge cost c ∈ℝ^|E| , a positive integer h, source s ∈ V and terminal t ∈ V , the hop-constrained cheapest path problem ( HCCPP ) seeks to find an s-t path of length at most h hops with the cheapest cost. This paper proposes a cut-based mixed integer programming (MIP) formulation, which has only one set of constraints that capture connectivity and hop constraints simultaneously, to solve the HCCPP when edge costs are nonnegative. As the only set of constraints of the formulation has exponentially many constraints, we show that their corresponding fractional separation problem is easy for h ∈{2,3} and hard for h ≥ 4 . We also propose a polytime algorithm for solving the integer separation problem. Furthermore, we show that our cut-based model is at least as strong as the jump-based model of Dahl (Operations Research Letters, 1999), which results in perfectness of our proposed model for h ∈{2,3} . We finally conduct a brief set of computational experiments to compare the performance of the cut formulation against the jump model.