Abstract
We study facets of the cut coneC n , i.e., the cone of dimension 1/2n(n − 1) generated by the cuts of the complete graph onn vertices. Actually, the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope. We present several operations on facets and, in particular, a “lifting” procedure for constructing facets ofC n+1 from given facets of the lower dimensional coneC n . After reviewing hypermetric valid inequalities, we describe the new class of cycle inequalities and prove the facet property for several subclasses. The new class of parachute facets is developed and other known facets and valid inequalities are presented.
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References
P. Assouad, “Sur les inégalités valides dansL 1,”European Journal of Combinatorics 5 (1984) 99–112.
P. Assouad and C. Delorme, “Graphes plongeables dansL 1,”Comptes Rendus de l'Académie des Sciences de Paris 291 (1980) 369–372.
P. Assouad and M. Deza, “Metric subspaces ofL 1,”Publications Mathématiques d'Orsay, Vol. 3 (1982).
D. Avis and Mutt, “All facets of the six point Hamming cone,”European Journal of Combinatorics 10 (1989) 309–312.
D. Avis and M. Deza, “The cut cone,L 1-embeddability, complexity and multicommodity flows,”Networks 21 (1991) 595–617.
F. Barahona, “The max-cut problem on graphs not contractible toK 5,”Operations Research Letters 2 (1983) 107–111.
F. Barahona and M. Grötschel, “On the cycle polytope of a binary matroid,”Journal of Combinatorial Theory B 40 (1986) 40–62.
F. Barahona, M. Grötschel, M. Jünger and G. Reinelt, “An application of combinatorial optimization to statistical physics and circuit layout design,”Operations Research 36(3) (1988) 493–513.
F. Barahona, M. Grötschel and A.R. Mahjoub, “Facets of the bipartite subgraph polytope,”Mathematics of Operations Research 10 (1985) 340–358.
F. Barahona, M. Jünger and G. Reinelt, “Experiments in quadratic 0–1 programming,”Mathematical Programming 44 (1989) 127–137.
F. Barahona and A. R. Mahjoub, “On the cut polytope,”Mathematical Programming 36 (1986) 157–173.
A.E. Brower, A.M. Cohen and A. Neumaier,Distance Regular Graphs (Springer, Berlin, 1989).
F.C. Bussemaker, R.A. Mathon and J.J. Seidel, “Tables of two-graphs,” TH-Report 79-WSK-05, Technical University Eindhoven (Eindhoven, Netherlands, 1979).
M. Conforti, M.R. Rao and A. Sassano, “The equipartition polytope: parts I & II,”Mathematical Programming 49 (1990) 49–70, 71–91.
C. De Simone, “The Hamming cone, the cut polytope and the boolean quadric polytope,” preprint (1988).
C. De Simone, “The cut polytope and the boolean quadric polytope,”Discrete Mathematics 79 (1989/90) 71–75.
C. De Simone, M. Deza and M. Laurent, “Collapsing and lifting for the cut cone,” Report No. 265, IASI-CNR (Roma, 1989).
M. Deza, “On the Hamming geometry of unitary cubes,”Doklady Akademii Nauk SSR 134 (1960) 1037–1040. [English translation in:Soviet Physics Doklady 5 (1961) 940–943.]
M. Deza, “Linear metric properties of binary codes,”Proceedings of the 4th Conference of USSR on Coding Theory and Transmission of Information, Moscow-Tachkent (1969) 77–85. [In Russian.]
M. Deza, “Matrices de formes quadratiques non négatives pour des arguments binaires,”Comptes Rendus de l'Académie des Sciences de Paris 277 (1973) 873–875.
M. Deza, “Small pentagonal spaces,”Rendiconti del Seminario Matematico di Brescia 7 (1982) 269–282.
M. Deza, K. Fukuda and M. Laurent, “The inequicut cone,” Research Report No. 89-04, GSSM, University of Tsukuba (Tokyo, 1989).
M. Deza, V.P. Grishukhin and M. Laurent, “The hypermetric cone is polyhedral,” to appear in:Combinatorica.
M. Deza and M. Laurent, “Facets for the cut cone II: Clique-web inequalities,”Mathematical Programming 56 (1992) 161–188, this issue.
J. Fonlupt, A.R. Mahjoub and J-P. Uhry, “Composition of graphs and the bipartite subgraph polytope,” to appear in:Discrete Mathematics.
M.R. Garey and D.S. Johnson,Computers and intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, CA, 1979).
V.P. Grishukhin, “All facets of the cut coneC n forn=7 are known,”European Journal of Combinatorics 11 (1990) 115–117.
M. Grötschel and W.R. Pulleyblank, “Weakly bipartite graphs and the max-cut problem,”Operations Research Letters 1 (1981) 23–27.
M. Grötschel and Y. Wakabayashi, “Facets of the clique partitioning polytope,”Mathematical Programming 47 (1990) 367–387.
F.O. Hadlock, “Finding a maximum cut of planar graph in polynomial time,”SIAM Journal on Computing 4 (1975) 221–225.
P.L. Hammer, “Some network flow problems solved with pseudo-boolean programming,”Operations Research 13 (1965) 388–399.
A.V. Karzanov, “Metrics and undirected cuts,”Mathematical Programming 32 (1985) 183–198.
J.B. Kelly, “Hypermetric spaces,”Lecture notes in Mathematics No. 490 (Springer, Berlin, 1975) pp. 17–31.
J.B. Kelly, unpublished.
M. Padberg, “The Boolean quadric polytope: Some characteristics, facets and relatives,”Mathematical Programming 45 (1989) 139–172.
S. Poljak and D. Turzik, “On a facet of the balanced subgraph polytope,”Casopis Pro Pestovani Matematiky 112 (1987) 373–380.
S. Poljak and D. Turzik, “Max-cut in circulant graphs,” to appear in:Annals of Discrete Mathematics.
P. Terwilliger and M. Deza, “Classification of finite connected hypermetric spaces,”Graphs and Combinatorics 3(3) (1987) 293–298.
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Deza, M., Laurent, M. Facets for the cut cone I. Mathematical Programming 56, 121–160 (1992). https://doi.org/10.1007/BF01580897
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DOI: https://doi.org/10.1007/BF01580897