Abstract
This paper investigates the complexity of various recognition problems for pseudo-Boolean functions (i.e., real-valued functions defined on the unit hypercubeB n = {0, 1}n), when such functions are represented as multilinear polynomials in their variables. Determining whether a pseudo-Boolean function (a) is monotonic, or (b) is supermodular, or (c) is threshold, or (d) has a unique local maximum in each face ofB n, or (e) has a unique local maximum inB n, is shown to be NP-hard. A polynomial-time recognition algorithm is presented for unimodular functions, previously introduced by Hansen and Simeone as a class of functions whose maximization overB n is reducible to a network minimum cut problem.
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References
A. Billionnet and M. Minoux, “Maximizing a supermodular pseudoboolean function: A polynomial algorithm for supermodular cubic functions,”Discrete Applied Mathematics 12 (1985) 1–11.
W.H. Cunningham, “On submodular function minimization,”Combinatorica 5 (1985) 185–192.
W.H. Cunningham, “Minimum cuts, modular functions, and matroid polyhedra,”Networks 15 (1985) 205–215.
G. Gallo and B. Simeone, “On the supermodular knapsack problem,” preprint (1986).
M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman and Company, San Francisco, 1979).
M. Grötschel, L. Lovász and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica 1 (1981) 169–197.
P.L. Hammer, “A note on the monotonicity of pseudo-Boolean functions,”Zeitschrift für Operations Research 18 (1974) 47–50.
P.L. Hammer and S. Rudeanu,Boolean Methods in Operations Research and Related Areas (Springer, Berlin, Heidelberg, New York, 1968).
P.L. Hammer, B. Simeone, T. Liebling and D. de Werra, “From linear separability to unimodality: a hierarchy of pseudo-Boolean functions,”SIAM Journal on Discrete Mathematics 1 (1988) 174–184.
P. Hansen, “Labelling algorithms for balance in signed graphs,” in: J.-C. Bermond et al., eds,Problèmes Combinatoires et Théorie des Graphes, Colloques Internationaux du CNRS 260 (CNRS, Paris, 1978) 215–217.
P. Hansen, “Methods of nonlinear 0–1 programming,”Annals of Discrete Mathematics 5 (1979) 53–70.
P. Hansen and B. Simeone, “Unimodular functions,”Discrete Applied Mathematics 14 (1986) 269–281.
K. Hoke,Valid Numberings of the Vertices of the d-Cube, Ph.D. Thesis (University of North Carolina, Chapel Hill, NC, 1986).
N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.
L. Lovász, “Submodular functions and convexity,” in: A. Bachem et al., eds,Mathematical Programming: The State of the Art (Springer-Verlag, Berlin, 1983) 235–257.
B. Simeone, D. de Werra and M. Cochand, Combinatorial properties and recognition of some classes of unimodular functions, preprint (1985; revised 1987).
C.A. Tovey, “Hill climbing with multiple local optima,”SIAM Journal of Algebraic and Discrete Methods 6 (1985) 384–393.
C.A. Tovey, “Low order polynomial bounds on the expected performance of local improvement algorithms,”Mathematical Programming 35 (1986) 193–224.
D.J. Wilde and J.M. Sanchez-Anton, “Discrete optimization on a multivariable Boolean lattice,”Mathematical Programming 1 (1971) 301–306.
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Crama, Y. Recognition problems for special classes of polynomials in 0–1 variables. Mathematical Programming 44, 139–155 (1989). https://doi.org/10.1007/BF01587085
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DOI: https://doi.org/10.1007/BF01587085