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Quadratic functions of binary variables

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Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1403)

Keywords

  • Boolean Function
  • Matched Graph
  • Boolean Expression
  • Signed Graph
  • Conflict Graph

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References

  • A. V. Aho, J. E. Hopcroft, J. D. Ullman: The design and analysis of computer algorithms (Addison-Wesley, Reading, Mass., 1974).

    Google Scholar 

  • B. Aspvall, M. F. Plass, R. E. Tarjan: “A linear-time algorithm for testing the truth of certain quantified Boolean formulas”, Inform. Process. Letters 8 (1979) 121–123.

    CrossRef  Google Scholar 

  • M. L. Balinski: “Notes on a constructive approach to linear programming”, in: G. B. Dantzig and A. F. Veinott, Jr. eds., Mathematics of decision sciences, Part I. (Amer. Math. Soc., Providence, 1968) 179–256.

    Google Scholar 

  • E. M. L. Beale, M. G. Kendall, D. W. Wann: “The discarding of variables in multivariate analysis”, Biometrika 54 (1967) 357–366.

    CrossRef  Google Scholar 

  • C. Benzaken, S.C. Boyd, P. L. Hammer, B. Simeone: “Adjoints of bidirected graphs”, Congressus numerantium 39 (1983) 123–144.

    Google Scholar 

  • C. Benzaken, P. L. Hammer, B. Simeone: “Some remarks on conflict graphs of quadratic pseudo-boolean functions”, in: L. Collatz, G. Meinardus, W. Wetterling eds., Konstruktive Methoden der finiten nichtlinearen Optimierung (Birkhäuser, Basel, 1980) 9–30.

    CrossRef  Google Scholar 

  • C. Berge: Graphs and hypergraphs (North-Holland, Amsterdam, 1973).

    Google Scholar 

  • J. M. Bourjolly, P. L. Hammer, B. Simeone: “A boolean simplex method for computing lower bounds of quadratic pseudo-boolean functions”, Research Report CORR 83-29, Univ. of Waterloo (1983).

    Google Scholar 

  • J. M. Bourjolly, P. L. Hammer, B. Simeone: “Node-weighted graphs having the König-Egerváry property”, Math. Programming Studies 22(1984) 44–63.

    CrossRef  Google Scholar 

  • R. Burkard, F. Rendl: “Quadratic assignment problems”, Technical Report, Technische Universität Graz (1985).

    Google Scholar 

  • V. Chvátal, P. L. Hammer: “Aggregation of inequalities in integer programming”, Ann. Discr. Math. 1 (1977) 145–162.

    CrossRef  Google Scholar 

  • Y. Crama, P. L. Hammer: “The complexity of recognizing partition-quadratic graphs”, RRR 3-85, Rutgers University (1985).

    Google Scholar 

  • S. Even, A. Itai, A. Shamir: “On the complexity of time-table and multi-commodity flow problems”, SIAM J. on Computing 5(1976) 691–703.

    CrossRef  Google Scholar 

  • J. E. Falk, K. L. Hoffman: “A successive underestimation method for concave minimization problems”, Math. Oper. Res. 1(1976), 251–259.

    CrossRef  Google Scholar 

  • M. L. Fisher, G. L. Nemhauser, L. A. Wolsey: “An analysis of approximations for maximizing submodular set functions-I”, Math. Programming 14 (1978) 265–294.

    CrossRef  Google Scholar 

  • R. J. Freeman, D. C. Gogerty, G. W. Graves, R. B. S. Brooks: “A mathematical model of supply support for space operations”, Oper. Research 14 (1966) 1–15.

    CrossRef  Google Scholar 

  • D. R. Fulkerson, A. J. Hoffman, M. H. McAndrew: “Some properties of graphs with multiple edges”, Canad. J. of Math. 17(1965) 166–177.

    CrossRef  Google Scholar 

  • G. Gallo, P. L. Hammer, B. Simeone: “Quadratic knapsack problems”, Math. Programming Studies 12(1980) 132–149.

    CrossRef  Google Scholar 

  • M. R. Garey, D. S. Johnson: Computers and intractability: a guided tour to the theory of NP-completeness. (Freeman, S. Francisco, 1979).

    Google Scholar 

  • F. Gavril: “Testing for equality between maximum matching and minimum node covering”, Inform. Process. Letters 6(1977) 199–202.

    CrossRef  Google Scholar 

  • P. L. Hammer: “Pseudo-boolean remarks on balancing signed graphs”, Graph Theory and Integer Programming International Series of Numerical Mathematics, 36 (1977) 69–78, Birkhäuser Verlag, Basel, Switzerland.

    Google Scholar 

  • P. L. Hammer, P. Hansen: “Logical relations in quadratic 0–1 programming”, Revue Roumaine de Math. pures et appliqueés 26(1981) 421–429.

    Google Scholar 

  • P. L. Hammer, S. Rudeanu: “Boolean methods in operations research and related areas” (Springer, Berlin, 1968).

    CrossRef  Google Scholar 

  • P. L. Hammer, P. Hansen, B. Simeone: “Upper planes of quadratic 0–1 functions and stability in graphs”, in: O. L. Mangasarian, R. R. Meyer and S. M. Robinson eds., Nonlinear Programming 4. (Academic Press, New York, 1981) 395–414.

    CrossRef  Google Scholar 

  • P. L. Hammer, P. Hansen, B. Simeone: “Vertices belonging to all or to no maximum stable sets of a graph”, SIAM J. Algebr. Discr. Methods 3(1982) 511–522.

    CrossRef  Google Scholar 

  • P. L. Hammer, P. Hansen, B. Simeone: “Roof-duality, complementation and persistency in quadratic 0–1 optimization”, Math. Programming 28 (1984) 121–155.

    CrossRef  Google Scholar 

  • P. L. Hammer, B. Kalantari: “Worst-case analysis of the roof-dual gap in quadratic zero-one optimization”, Working paper, Rutgers University (1986).

    Google Scholar 

  • P. L. Hammer, U. N. Peled, S. Sorensen: “Pseudo-boolean functions and game theory, I: Core elements and Shapley value”, Cahiers du Centre d'Etudes de Rech. Operationnelle 19(1977) 159–176.

    Google Scholar 

  • P. L. Hammer, I. G. Rosenberg: “Linear decomposition of a positive group-boolean function”, in: L. Collatz, W. Wetterling, eds., Numerische Methoden bei Optimierung, vol. II. (Birkhäuser, Basel, 1974) 51–62.

    CrossRef  Google Scholar 

  • P. L. Hammer, B. Simeone: “Quasimonotone boolean functions and bistellar graphs”, Ann. Discr. Math. 9(1980) 107–119.

    CrossRef  Google Scholar 

  • P. Hansen: “Fonctions d'evaluation et pénalités pour les programmes quadratiques en variables 0–1”, in: B. Roy, ed., Combinatorial programming, methods and applications (Reidel, Dordrecht, 1975) 361–370.

    CrossRef  Google Scholar 

  • P. Hansen: “Labelling algorithms for balance in signed graphs”, in: J.-C. Bermond et al. eds., Problemes combinatoires et theorie des graphes (Editions CNRS, Paris, 1978) 215–217.

    Google Scholar 

  • P. Hansen, B. Jaumard, M. Minoux: “A linear expected-time algorithm for deriving all logical conclusions implied by a set of boolean inequalities”, Math. Programming 34 (1986) 223–231.

    CrossRef  Google Scholar 

  • P. Hansen, B. Simeone: “Unimodular functions”, Discr. Appl. Math., 14 (1986) 269–281.

    CrossRef  Google Scholar 

  • F. Harary: “On the notion of balance of a signed graph”, Michigan Math. J. 2(1953) 143–146.

    CrossRef  Google Scholar 

  • E. L. Johnson, M. W. Padberg: “Degree-two inequalities, clique facets and biperfect graphs”, Ann. Discr. Math. 16 (1982) 169–187.

    Google Scholar 

  • S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi: “Optimization by simulated annealing”, Science 220 (1983) 671–680.

    CrossRef  Google Scholar 

  • E. L. Lawler: Combinatorial optimization: networks and matroids (Holt, Rinehart and Winston, New York, 1976).

    Google Scholar 

  • S. H. Lu, B. Simeone: “On the equivalence between roof-duality and paved-duality in quadratic 0–1 optimization”, Technical Report, RRR# 22-87, Rutgers University, 1987.

    Google Scholar 

  • S. H. Lu, A. C. Williams: “Roof-duality for non-linear 0–1 programming”, RRR 2-85, Rutgers University (1985).

    Google Scholar 

  • S. Micali, Vazirani: “An O(V 1/2 E) algorithm for finding maximum matchings in general graphs”, Proc. 21st Annual Symposium on Foundations of Computer Science, Syracuse 1980 (IEEE Computer Society Press, New York, 1980) 17–27.

    Google Scholar 

  • G. L. Nemhauser, L. E. Trotter: “Vertex packings: structural properties and algorithms”, Math. Program. 8(1975) 235–248.

    CrossRef  Google Scholar 

  • R. Petreschi, B. Simeone: “A switching algorithm for the solution of quadratic boolean equations”, Inform. Process. Letters. 11(1980) 193–198.

    CrossRef  Google Scholar 

  • R. Petreschi, B. Simeone: “Numerical comparison of 2-satisfiability algorithms”, Working paper, Math. Department, University of Rome (1985).

    Google Scholar 

  • W. V. Quine: “A way of simplifying truth functions”, Amer. Math Monthly 52(1952) 627–631.

    Google Scholar 

  • J. Rhys: “A selection problem of shared fixed costs and networks”, Manag. Science 17(1970) 200–207.

    CrossRef  Google Scholar 

  • I. G. Rosenberg: “Reduction of unconstrained nonlinear 0–1 programming to the quadratic case”, Cahiers du Centre d'Etudes de Rech. Operationnelle 17(1975) 71–74.

    Google Scholar 

  • S. Rudeanu: Boolean functions and equations (North-Holland, Amsterdam, 1974).

    Google Scholar 

  • B. Simeone: “Quadratic 0–1 programming, boolean functions and graphs”, Doctoral dissertation, Univ. of Waterloo (1979).

    Google Scholar 

  • B. Simeone: “Consistency of quadratic boolean equations and the König-Egerváry property for graphs”, Ann. Discr. Math. 25(1985) 281–290.

    Google Scholar 

  • B. Simeone: “An asymptotically exact polynomial algorithm for equipartition problems”, Discr. Appl. Math., 14 (1986) 283–293.

    CrossRef  Google Scholar 

  • R. E. Tarjan: “Depth-first search and linear graph algorithms”, SIAM J. Computing 1(1972) 146–160.

    CrossRef  Google Scholar 

  • C. Witzgall: “Mathematical methods of site selection for Electronic Message Systems (EMS)”, NBS Internal Report (1975).

    Google Scholar 

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© 1989 Springer-Verlag

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Hammer, P.L., Simeone, B. (1989). Quadratic functions of binary variables. In: Simeone, B. (eds) Combinatorial Optimization. Lecture Notes in Mathematics, vol 1403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083461

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  • DOI: https://doi.org/10.1007/BFb0083461

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