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Generalized cut and metric polytopes of graphs and simplicial complexes


The metric polytope \({{\mathrm{METP}}}(K_n)\) of the complete graph on n nodes is defined by the triangle inequalities \(x(i,j)\le x(i,k) + x(k,j)\) and \(x(i,j) + x(j,k) + x(k,i)\le 2\) for all triples ijk of \(\{1,\dots ,n\}\). The cut polytope \({{\mathrm{CUTP}}}(K_n)\) is the convex hull of the \(\{0,1\}\) vectors of \({{\mathrm{METP}}}(K_n)\). For a graph G on n vertices the metric polytope \({{\mathrm{METP}}}(G)\) and cut polytope \({{\mathrm{CUTP}}}(G)\) are the projections of \({{\mathrm{METP}}}(K_n)\) and \({{\mathrm{CUTP}}}(K_n)\) on the edge set of G. The facets of the cut polytopes are of special importance in optimization and are studied here in some detail for many simple graphs. Then we define variants \({{\mathrm{QMETP}}}(G)\) for quasi-metrics, i.e. not necessarily symmetric distances and we give an explicit description by inequalities. Finally we generalize distances to m-dimensional area between \(m+1\) points and this defines an hemimetric. In that setting the generalization of the notion of graph is the notion of m-dimensional simplicial complex \({\mathcal {K}}\) for which we define a cone of hemimetric \({{\mathrm{HMET}}}({\mathcal {K}})\).

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  1. 1.

    Avis, D., Hayden, P., Wilde, M .M.: Leggett–Garg inequalities and the geometry of the cut polytope. Phys. Rev. A (3) 82(3), 030102, 4 (2010)

  2. 2.

    Avis, D., Imai, H., Ito, T.: On the relationship between convex bodies related to correlation experiments with dichotomic observables. J. Phys. A 39(36), 11283–11299 (2006)

  3. 3.

    Avis, D., Imai, H., Ito, T., Sasaki, Y.: Two-party Bell inequalities derived from combinatorics via triangular elimination. J. Phys. A 38(50), 10971–10987 (2005)

  4. 4.

    Avis, D., Meagher, C.: On the directed cut cone and polytope. J. Comb. Optim. 31, 1685–1708 (2016)

  5. 5.

    Avis, D., Mutt, : All the facets of the six-point Hamming cone. Eur. J. Comb. 10(4), 309–312 (1989)

  6. 6.

    Barahona, F.: The max-cut problem on graphs not contractible to \(K_{5}\). Oper. Res. Lett. 2(3), 107–111 (1983)

  7. 7.

    Barahona, F.: On cuts and matchings in planar graphs. Math. Program. 60(1), 53–68 (1993)

  8. 8.

    Barahona, F., Mahjoub, A.R.: On the cut polytope. Math. Program. 36(2), 157–173 (1986)

  9. 9.

    Bonato, T., Jünger, M., Reinelt, G., Rinaldi, G.: Lifting and separation procedures for the cut polytope. Math. Program. Ser. A 146, 351–378 (2014)

  10. 10.

    Bremner, D., Sikirić, M.D., Pasechnik, D.V., Rehn, T., Schürmann, A.: Computing symmetry groups of polyhedra. LMS J. Comput. Math. 17(1), 565–581 (2014)

  11. 11.

    Czap, J., Hudák, D.: 1-planarity of complete multipartite graphs. Discrete Appl. Math. 160(4–5), 505–512 (2012)

  12. 12.

    Deza, E., Deza, M., Sikirić, M.D.: Generalizations of Finite Metrics and Cuts. World Scientific Publishing Co., Pte. Ltd., Hackensack (2016)

  13. 13.

    Deza, M., Grishukhin, V.P., Laurent, M.: The symmetries of the cut polytope and of some relatives. In: Applied Geometry and Discrete Mathematics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 205–220. Amer. Math. Soc., Providence (1991)

  14. 14.

    Deza, M.-M., Rosenberg, I.G.: \(n\)-semimetrics. Eur. J. Comb. 21(6), 797–806 (2000). Discrete metric spaces (Marseille,1998)

  15. 15.

    Deza, M.-M., Rosenberg, I.G.: Small cones of \(m\)-hemimetrics. Discrete Math. 291(1–3), 81–97 (2005)

  16. 16.

    Deza, M., Deza, E.: Cones of partial metrics. Contrib. Discrete Math. 6, 26–47 (2011)

  17. 17.

    Deza, M., Dutour, M., Panteleeva, E.: Small cones of oriented semi-metrics. In: Forum for Interdisciplinary Mathematics Proceedings on Statistics, Combinatorics & Related Areas (Bombay, 2002), vol. 22, pp. 199–225 (2000)

  18. 18.

    Deza, M., Dutour, M., Maehara, H.: On volume-measure as hemi-metrics. Ryukyu Math. J. 17, 1–9 (2004)

  19. 19.

    Deza, M., Panteleeva, E.: Quasi-semi-metrics, oriented multi-cuts and related polyhedra. Eur. J. Comb. 21(6), 777–795 (2000). Discrete metric spaces (Marseille, 1998)

  20. 20.

    Deza, M., Sikirić, M.D.: Enumeration of the facets of cut polytopes over some highly symmetric graphs. Int. Trans. Oper. Res. 23(5), 853–860 (2016)

  21. 21.

    Deza, M.-M., Dutour, M.: Cones of metrics, hemi-metrics and super-metrics. Ann. Eur. Acad. Sci. 1, 141–162 (2003)

  22. 22.

    Deza, M.M., Laurent, M.: Geometry of cuts and metrics. In: Algorithms and Combinatorics, vol. 15. Springer, Heidelberg (2010). First softcover printing of the 1997 original [MR1460488]

  23. 23.

    Grishukhin, V.P.: All facets of the cut cone \({ C}_n\) for \(n=7\) are known. Eur. J. Comb. 11(2), 115–117 (1990)

  24. 24.

    Kinnewig, S.: Bell inequalities and grothendieck’s constant. Master thesis (2017)

  25. 25.

    Neto, J.: On the diameter of cut polytopes. Discrete Math. 339, 1605–1612 (2016)

  26. 26.

    Ohsugi, H.: Normality of cut polytopes of graphs is a minor closed property. Discrete Math. 310, 1160–1166 (2010)

  27. 27.

    Ohsugi, H.: Gorenstein cut polytopes. Eur. J. Comb. 38, 122–129 (2014)

  28. 28.

    Römer, T., Madani, S.S.: Retracts and algebraic properties of cut algebras (2016). arXiv:1608.04973

  29. 29.

    Seymour, P.D.: Matroids and multicommodity flows. Eur. J. Comb. 2(3), 257–290 (1981)

  30. 30.

    Sikirić, M.D.: Cut polytopes (2018).

  31. 31.

    Sikirić, M.D.: Polyhedral (2018).

  32. 32.

    Tylkin, M.E.: On Hamming geometry of unitary cubes. Soviet Phys. Dokl. 5, 940–943 (1960)

  33. 33.

    Wagner, K.: über eine Eigenschaft der ebene Komplexe. Math. Ann. 114, 570–590 (1937)

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The second author gratefully acknowledges support from the Alexander von Humboldt foundation and thanks the anonymous referees.

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Correspondence to Mathieu Dutour Sikirić.

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Deza, M., Dutour Sikirić, M. Generalized cut and metric polytopes of graphs and simplicial complexes. Optim Lett 14, 273–289 (2020).

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  • Max-cut problem
  • Cut polytope
  • Metrics
  • Graphs
  • Cycles
  • Quasi-metrics
  • Hemimetrics