A Strong Class of Lifted Valid Inequalities for the Shortest Path Problem in Digraphs with Negative Cost Cycles

Mamane Souleye Ibrahim

doi:10.5539/jmr.v7n4p162

Abstract

In this paper, we address a strong class of lifted valid inequalities for the shortest path problem in digraphs with possibly negative cost cycles. We call these lifted inequalities the $incident \ lifted \ valid \ inequalities$ ($ILI$) as they are based on the incident arcs of a given vertex. The $ILI$ inequalities are close in spirit of the so-called \textit{simple lifted valid inequalities} ($SLI$) and $cocycle \ lifted \ valid \ inequalities$ ($CLI$) introduced in Ibrahim et al. (2015). However, as we will see the $ILI$ inequalities are stronger than the first ones in term of linear relaxation strengthening. Indeed, contrary to $SLI$ and $CLI$ inequalities, consider the same instances, in a cutting plane algorithm, the computational results prove that the $ILI$ inequalities provide the optimal integer solution for all the considered instances within no more than three iterations except one case for which after the first strengthening iteration, there exists no generated inequality.

Highlights

Let G = (V, A) be a general directed graph, where V represents the vertex set and A the set of arcs with an arc cost function w : A → R
According to the results presented in Ibrahim et al (2015), with respect to the sub-digraph GX = (X, E) supporting the optimal solution of the linear relaxation, we have that yu,v ≤ k w∈S k (u,v)∈Fk is valid to the polytope PX of all the s − t elementary directed paths of the sub-digraph GX
We present a flow based single origin single destination linear formulation of the shortest path problem with possibly negative cost cycles in the considered digraphs and we introduce some lifted valid inequalities used to perform a cutting plane algorithm in view to solve efficiently the problem

Summary

Introduction

We consider the problem that consists in searching an elementary shortest path from the source vertex s to the sink vertex t (single origin - single destination shortest path problem) in digraphs containing negative cost cycles. We continue our investigation begun in Ibrahim et al (2015) about a cutting plane algorithm devised for the single origin - single destination shortest path problem with possibly negative cost cycles. We recall that the algorithm is based on a MIP formulation of the single origin - single destination shortest path problem in digraphs possibly containing negative cost cycles. We address a new class of lifted valid inequalities called the incident lifted valid inequalities (ILI) as they are based on the incident arcs of a given vertex.

A Strong Class of Lifted Valid Inequalities for P

Computational Results

Conclusion