Roof duality, complementation and persistency in quadratic 0-1 optimization. (English) Zbl 0574.90066

The paper is concerned with the ”primal” problem of maximizing a given quadratic pseudoboolean function. Four equivalent problems are discussed - the primal, the ”complementation”, the ”discrete Rhys LP” and the ”weighted stability problem of a SAM graph”. Each of them has a relaxation - the ”roof dual”, the ”quadratic complementation”, the ”continuous Rhys LP” and the ”fractional weighted stability problem of a SAM graph”. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ”roof-dual” define ”best” linear majorants p(x) of f, having the following persistency property: if the ith coefficient in p is positive (negative) then \(x_ i=1(0)\) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.


90C09 Boolean programming
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
Full Text: DOI


[1] B. Aspvall, M.F. Plass and R.E. Tarjan, ”A linear-time algorithm for testing the truth of certain quantified Boolean formulas”,Information Processing Letters 8 (1979) 121–123. · Zbl 0398.68042
[2] M.L. Balinski, ”Integer programming: Methods, uses, computation”, in: G.B. Dantzig and A.F. Veinott, Jr., eds.,Mathematics of the decision sciences, Part I (American Mathematical Society, Providence, 1968), pp 179–256. · Zbl 0186.24203
[3] M.L. Balinski, ”Notes on a constructive approach to linear programming”, in: G.B. Dantzig and A.F. Veinott, Jr., eds.,Mathematics of the decision sciences, Part I (American Mathematical Society, Providence, 1968) pp. 38–64. · Zbl 0187.17403
[4] S. Cook, ”The complexity of theorem proving procedures”,Proceedings of the Third Symposium of the Association of Computing Machinery on the Theory of Computing (1971) 151–158. · Zbl 0253.68020
[5] M.L. Fischer, G.L. Nemhauser and L.A. Wolsey, ”An analysis of approximations for maximizing submodular set functions–I”,Mathematical Programming 14 (1978) 265–294. · Zbl 0374.90045
[6] D.R. Fulkerson, A.J. Hoffman and M.H. McAndrew, ”Some properties of graphs with multiple edges”,Canadian Journal of Mathematics 17 (1965), 166–177. · Zbl 0132.21002
[7] G. Gallo, P.L. Hammer and B. Simeone, ”Quadratic knapsack problems”Mathematical Programming Studies 12 (1980) 132–149. · Zbl 0462.90068
[8] P.L. Hammer, ”Boolean elements in combinatorial optimization”,Annals of Discrete Mathematics 4 (1979) 51–71. · Zbl 0414.90062
[9] P.L. Hammer and P. Hansen, ”Logical relations in quadratic 0–1 programming”,Revue Roumaine de Mathématiques Pures et Appliquées 26 (1981) 421–429. · Zbl 0457.90052
[10] P.L. Hammer, P. Hansen and B. Simeone, ”Vertices belonging to all or to no maximum stables sets of a graph”,SIAM Journal on Algebraic and Discrete Methods 3 (1982) 511–522. · Zbl 0496.90056
[11] P.L. Hammer, P. Hansen and B. Simeone, ”Upper planes of quadratic 0–1 functions and stability in graphs”,Nonlinear Programming 4 (Academic Press, New York, 1981), pp. 395–414. · Zbl 0534.90062
[12] P.L. Hammer, U.N. Peled and S. Sorensen ”Pseudo-boolean functions and game theory, I, Core elements and Shapley value”,Cahiers du Centre d’Etudes de Recherche Opérationnelle 19 (1977) 159–176. · Zbl 0362.90143
[13] P.L. Hammer and I.G. Rosenberg, ”Linear decomposition of a positive group-boolean function”, in: L. Collatz and W. Wetterling, eds.Numerische Methoden bei Optimierung Vol. II (Birkhäuser, Basel and Stuttgart, 1974) pp. 51–62. · Zbl 0294.94025
[14] P.L. Hammer and S. Rudeanu,Boolean methods in operations research and related areas (Springer, Berlin, New York, 1968). · Zbl 0155.28001
[15] P. Hansen, ”Fonctions d’évaluation et pénalites pour les programmes quadratiques en variables 0–1”, in: B. Roy, ed.Combinatorial programming, methods and applications, (Reidel, Dordrecht, 1975) pp. 361–370.
[16] P. Hansen, ”Labelling algorithms for balance in signed graphs”, in: J.-C. Bermond et al., eds.,Problèmes combinatoires et théorie des graphes, Colloque International du Centre National de la Recherche Scientifique, No. 260 (Editions Centre National de la Recherche Scientifique, Paris, 1978) pp. 215–217.
[17] P. Hansen, ”Methods of nonlinear 0–1 programming”,Annals of Discrete Mathematics 5 (1979) 53–69. · Zbl 0426.90063
[18] P. Hansen and B. Simeone, ”A class of quadratic pseudoboolean functions whose maximization is reducible to a network flow problem,” CORR 79-39, Department of Combinatorics and Optimization, University of Waterloo, (Waterloo, Ontario, 1979).
[19] E.L. Lawler,Combinatorial optimization: Networks and matroids (Holt, Rinehart and Winston, New York, 1976). · Zbl 0413.90040
[20] E.L. Lawler, ”Shortest path and network flow algorithms”,Annals of Discrete Mathematics 4 (1979) 251–263. · Zbl 0453.90099
[21] G.L. Nemhauser and L.E. Trotter, ”Vertex packings: Structural properties and algorithms”,Mathematical Programming 8 (1975) 232–248. · Zbl 0314.90059
[22] J.C. Picard and M. Queyranne, ”On the integer valued variables in the linear vertex packing problem”,Mathematical Programming 12 (1977) 97–101. · Zbl 0362.90065
[23] J. Rhys, ”A selection problem of shared fixed costs and networks”,Management Science 17 (1970) 200–207. · Zbl 0203.52505
[24] I.G. Rosenberg, ”Reduction of bivalent maximization to the quadratic case”,Cahiers du Centre d’Etudes de Recherche Opérationnelle 17 (1975) 71–79. · Zbl 0302.90041
[25] B. Simeone, ”Quadratic 0–1 programming, boolean functions and graphs”, Doctoral Dissertation, University of Waterloo (Waterloo, Ontario, 1979).
[26] B. Simeone, ”A generalizaed consensus approach to non-linear 0–1 optimization”,Journal of the Association of Computing Machinery, forthcoming. · Zbl 1163.90657
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.