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.

Keywords: Digraph, cycle, linear relaxation, valid inequality, polytope

^{1}; Maculan, Nelson

^{2}; Ouzia, Hacène

^{3}

@article{RO_2016__50_3_665_0, author = {Ibrahim, Mamane Souley and Maculan, Nelson and Ouzia, Hac\`ene}, title = {An efficient cutting plane algorithm for the minimum weighted elementary directed cycle problem in planar digraphs}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {665--675}, publisher = {EDP-Sciences}, volume = {50}, number = {3}, year = {2016}, doi = {10.1051/ro/2015052}, mrnumber = {3538846}, zbl = {1349.90815}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro/2015052/} }

TY - JOUR AU - Ibrahim, Mamane Souley AU - Maculan, Nelson AU - Ouzia, Hacène TI - An efficient cutting plane algorithm for the minimum weighted elementary directed cycle problem in planar digraphs JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2016 SP - 665 EP - 675 VL - 50 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro/2015052/ DO - 10.1051/ro/2015052 LA - en ID - RO_2016__50_3_665_0 ER -

%0 Journal Article %A Ibrahim, Mamane Souley %A Maculan, Nelson %A Ouzia, Hacène %T An efficient cutting plane algorithm for the minimum weighted elementary directed cycle problem in planar digraphs %J RAIRO - Operations Research - Recherche Opérationnelle %D 2016 %P 665-675 %V 50 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro/2015052/ %R 10.1051/ro/2015052 %G en %F RO_2016__50_3_665_0

Ibrahim, Mamane Souley; Maculan, Nelson; Ouzia, Hacène. An efficient cutting plane algorithm for the minimum weighted elementary directed cycle problem in planar digraphs. RAIRO - Operations Research - Recherche Opérationnelle, Volume 50 (2016) no. 3, pp. 665-675. doi : 10.1051/ro/2015052. http://archive.numdam.org/articles/10.1051/ro/2015052/

On the cycle polytope of a directed graph. Networks 36 (2000) 34–46. | DOI | MR | Zbl

and ,On the cycle polytope of a directed graph and its relaxations. Networks 54 (2009) 47–55. | DOI | MR | Zbl

and ,P. Bauer, A Polyhedral Approach to the Weighted Girth Problem. Dissertation, Institut fur informatik, Universitat zu Koln (1994).

Matrix multiplication via arithmetic progressions. J. Symbolic Comput. 9 (1990) 251–280. | DOI | MR | Zbl

and ,On cycle cones and polyhedra. Linear Algebra Appl. 114–115 (1989) 613–640. | DOI | MR | Zbl

and ,The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10 (1980) 111–121. | DOI | MR | Zbl

, and ,V. Kaibel and R. Stephan, On cardinality constrained cycle and path polytopes. ZIB report (2007). | Zbl

M.S. Ibrahim, N. Maculan and H. Ouzia, A new class of valid inequalities for the minimum weighted elementary directed cycle problem in digraph. Submitted (2016). | Numdam | MR

Strong flow-based formulation for the shortest path problem in digraphs with negative cycles. Int. Trans. Oper. Res. (ITOR) 16 (2009) 361–369. | DOI | MR | Zbl

, and ,A note on the NP-hardness of the separation problem on some valid inequalities for the elementary shortest path problem. Pesquisa Operacional 34 (2014) 117–124. | DOI

, and ,Valid inequalities and lifting procedures for the shortest path problem in digraphs with negative cycles. Optim. Lett. 9 (2015) 345–357. | DOI | MR | Zbl

, and ,Finding a minimum circuit in a graph. SIAM J. Comput. 7 (1978) 413–423. | DOI | MR | Zbl

and ,A fast algorithm for minimum weight odd circuits and cuts in planar graphs. Oper. Res. Lett. 33 (2005) 625–628. | DOI | MR | Zbl

and ,Efficient approximation algorithms for shortest cycles in undirected graphs. J. Inf. Process. Lett. 109 (2009) 493–498. | DOI | MR | Zbl

and ,Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems. Pesquisa Operacional 23 (2003) 161–168. | DOI

, and ,An Hierarchy of Strong Block-Decomposable Linear Relaxations for 0–1 MIPs. Discrete Appl. Math. 158 (2010) 2031–2048. | DOI | MR | Zbl

and ,L. Roditty and V.V. Williams, Minimum weight cycles and triangles: equivalences and algorithms. Proc. of the $52nd$ Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2011) 180–189. | MR | Zbl

Finding even cycles faster. SIAM. J. Discrete Math. 10 (1997) 209–222. | DOI | MR | Zbl

and ,*Cited by Sources: *