An efficient cutting plane algorithm for the minimum weighted elementary directed cycle problem in planar digraphs

¹ Université A. Moumouni, Faculté des Sciences, BP 10.622, Niamey, Niger.
ibrahdz@yahoo.fr
² Federal University of Rio de Janeiro, Rio de Janeiro, Brazil.
maculan@cos.ufrj.br
³ Sorbonne Universités, UPMC University Paris 06, LIP6 UMR 7606, 4 place Jussieu, 75005 Paris  .
hacene.ouzia@lip6.fr

Received: 25 June 2015
Accepted: 11 November 2015

Abstract

In this paper, we study the efficiency (both theoretically and computationally) of a class of valid inequalities for the minimum weighted elementary directed cycle problem (MWEDCP) in planar digraphs with negative weight elementary directed cycles. These valid inequalities are called cycle valid inequalities and are parametrized by an integer called inequality’s order. From a theoretical point of view, we prove that separating cycle valid inequalities of order 1 in planar digraph can be done in polynomial time. From a computational point of view, we present a cutting plane algorithm featuring the efficiency of a lifted form of the cycle valid inequalities of order 1. In addition to these lifted valid inequalities, our algorithm is also based on a mixed integer linear formulation of the MWEDCP. The computational results are carried out on randomly generated planar digraph instances of the MWEDCP. For all 29 instances considered, we obtain in average 26.47% gap improvement. Moreover, for some of our instances the strengthening process directly displays the optimal integer elementary directed cycle.