Abstract
The subject of this work is the study of the Lovász-Schrijver PSD-operator \(LS_+\) applied to the edge relaxation \(\mathrm{ESTAB}(G)\) of the stable set polytope \(\mathrm{STAB}(G)\) of a graph G. We are interested in the problem of characterizing the graphs G for which \(\mathrm{STAB}(G)\) is achieved in one iteration of the \(LS_+\)-operator, called \(LS_+\)-perfect graphs, and to find a polyhedral relaxation of \(\mathrm{STAB}(G)\) that coincides with \(LS_+(\mathrm{ESTAB}(G))\) and \(\mathrm{STAB}(G)\) if and only if G is \(LS_+\)-perfect. An according conjecture has been recently formulated (\(LS_+\)-Perfect Graph Conjecture); here we verify it for the well-studied class of claw-free graphs.
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This work was supported by an ECOS-MINCyT cooperation (A12E01), a MATH-AmSud cooperation (PACK-COVER), PID-CONICET 0277, PICT-ANPCyT 0586.
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Bianchi, S., Escalante, M., Nasini, G., Tunçel, L.: Near-perfect graphs with polyhedral \(N_+(G)\). Electron. Notes Discrete Math. 37, 393–398 (2011)
Bianchi, S., Escalante, M., Nasini, G., Tunçel, L.: Lovász-Schrijver PSD-operator and a superclass of near-perfect graphs. Electron. Notes Discrete Math. 44, 339–344 (2013)
Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)
Chudnovsky, M., Seymour, P.: The structure of claw-free graph (unpliblished manuscript) (2004)
Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory (B) 18, 138–154 (1975)
Coulonges, S., Pêcher, A., Wagler, A.: Characterizing and bounding the imperfection ratio for some classes of graphs. Math. Program. A 118, 37–46 (2009)
Edmonds, J.R.: Maximum matching and a polyhedron with (0,1) vertices. J. Res. Nat. Bur. Stand. 69B, 125–130 (1965)
Edmonds, J.R., Pulleyblank, W.R.: Facets of I-matching polyhedra. In: Berge, C., Chuadhuri, D.R. (eds.) Hypergraph Seminar. LNM, pp. 214–242. Springer, Heidelberg (1974)
Eisenbrand, F., Oriolo, G., Stauffer, G., Ventura, P.: The stable set polytope of quasi-line graphs. Combinatorica 28, 45–67 (2008)
Escalante, M., Montelar, M.S., Nasini, G.: Minimal \(N_+\)-rank graphs: progress on Lipták and Tunçel’s conjecture. Oper. Res. Lett. 34, 639–646 (2006)
Escalante, M., Nasini, G.: Lovász and Schrijver \(N_+\)-relaxation on web graphs. In: Fouilhoux, P., Gouveia, L.E.N., Mahjoub, A.R., Paschos, V.T. (eds.) ISCO 2014. LNCS, vol. 8596, pp. 221–229. Springer, Heidelberg (2014)
Escalante, M., Nasini, G., Wagler, A.: Characterizing \(N_+\)-perfect line graphs. Int. Trans. Oper. Res. (2016, to appear)
Galluccio, A., Gentile, C., Ventura, P.: Gear composition and the stable set polytope. Oper. Res. Lett. 36, 419–423 (2008)
Galluccio, A., Gentile, C., Ventura, P.: The stable set polytope of claw-free graphs with stability number at least four. I. Fuzzy antihat graphs are W-perfect. J. Comb. Theory Ser. B 107, 92–122 (2014)
Galluccio, A., Gentile, C., Ventura, P.: The stable set polytope of claw-free graphs with stability number at least four. II. Striped graphs are G-perfect. J. Comb. Theory Ser. B 108, 1–28 (2014)
Giles, R., Trotter, L.E.: On stable set polyhedra for K1,3 -free graphs. J. Comb. Theory 31, 313–326 (1981)
Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1988)
Liebling, T.M., Oriolo, G., Spille, B., Stauffer, G.: On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs. Math. Methods Oper. Res. 59(1), 25–35 (2004)
Lipták, L., Tunçel, L.: Stable set problem and the lift-and-project ranks of graphs. Math. Program. A 98, 319–353 (2003)
Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)
Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)
Minty, G.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory 28, 284–304 (1980)
Oriolo, G., Pietropaoli, U., Stauffer, G.: A new algorithm for the maximum weighted stable set problem in claw-free graphs. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 77–96. Springer, Heidelberg (2008)
Pêcher, A., Wagler, A.: Almost all webs are not rank-perfect. Math. Program. B 105, 311–328 (2006)
Pêcher, A., Wagler, A.: On facets of stable set polytopes of claw-free graphs with stability number three. Discrete Math. 310, 493–498 (2010)
Pietropaoli, U., Wagler, A.: Some results towards the description of the stable set polytope of claw-free graphs. In: Proceedings of ALIO/EURO Workshop on Applied Combinatorial Optimization, Buenos Aires (2008)
Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Math. 29, 53–76 (1980)
Shepherd, F.B.: Near-perfect matrices. Math. Program. 64, 295–323 (1994)
Shepherd, F.B.: Applying Lehman’s theorem to packing problems. Math. Program. 71, 353–367 (1995)
Stauffer, G.: On the stable set polytope of claw-free graphs. Ph.D. thesis, Swiss Institute of Technology in Lausanne (2005)
Wagler, A.: Critical edges in perfect graphs. Ph.D. thesis, TU Berlin and Cuvillier Verlag Göttingen (2000)
Wagler, A.: Antiwebs are rank-perfect. 4OR 2, 149–152 (2004)
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Bianchi, S., Escalante, M., Nasini, G., Wagler, A. (2016). Lovász-Schrijver PSD-Operator on Claw-Free Graphs. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_6
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