Clique-connecting forest and stable set polytopes
RAIRO - Operations Research - Recherche Opérationnelle, Tome 44 (2010) no. 1, pp. 73-83.

Let G = (V,E) be a simple undirected graph. A forest F ⊆ E of G is said to be clique-connecting if each tree of F spans a clique of G. This paper adresses the clique-connecting forest polytope. First we give a formulation and a polynomial time separation algorithm. Then we show that the nontrivial nondegenerate facets of the stable set polytope are facets of the clique-connecting polytope. Finally we introduce a family of rank inequalities which are facets, and which generalize the clique inequalities.

DOI : 10.1051/ro/2010005
Classification : 05C15, 90C09
Mots-clés : graph, polytope, separation, facet 
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     title = {Clique-connecting forest and stable set polytopes},
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     pages = {73--83},
     publisher = {EDP-Sciences},
     volume = {44},
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     doi = {10.1051/ro/2010005},
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Cornaz, Denis. Clique-connecting forest and stable set polytopes. RAIRO - Operations Research - Recherche Opérationnelle, Tome 44 (2010) no. 1, pp. 73-83. doi : 10.1051/ro/2010005. http://www.numdam.org/articles/10.1051/ro/2010005/

[1] M. Campêlo, R. Corrêa and Y. Frota, Cliques, holes and the vertex coloring polytope. Inf. Process. Lett. 89 (2004) 159-164. | Zbl

[2] D. Cornaz and V. Jost, A one-to-one correspondence beetween stables sets and colorings. Oper. Res. Lett. 36 (2008) 673-676. | Zbl

[3] E. Cheng and W.H. Cunningham, Wheel inequalitites for stable set polytopes. Math. Program. 77 (1997) 389-421. | Zbl

[4] J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices. Journal of Resarch National Bureau of Standards Section B 69 (1965) 67-72. | Zbl

[5] J. Edmonds, Matroids and the greedy algorithm. Math. Program. 1 (1971) 127-136. | Zbl

[6] M. Grötschel, L. Lovàsz and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1 (1981) 169-197. | Zbl

[7] M.W. Padberg, On the facial structure of set packing polyhedra. Math. Program. 5 (1973) 199-215. | Zbl

[8] M.W. Padberg and L.A. Wolsey, Fractional covers for forests and matchings. Math. Program. 29 (1984) 1-14. | Zbl

[9] J.-C. Picard and M. Queyranne, Selected applications of minimum cuts in networks. INFOR Can. J. Oper. Res. Inf. Process. 20 (1982) 394-422. | Zbl

[10] J.M.W. Rhys, A selection problem of shared fixed costs and network flows. Manag. Sci. 17 (1970) 200-207. | Zbl

[11] A. Schrijver, Combinatorial Optimization. Springer-Verlag, Berlin, Heidelberg (2003). | Zbl

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