Submodular Function Minimization and Maximization in Discrete Convex Analysis

This paper sheds a new light on submodular function minimization and maximization from the viewpoint of discrete convex analysis. L-convex functions and M-concave functions constitute subclasses of submodular functions on an integer interval. Whereas L-convex functions can be minimized efficiently on the basis of submodular (set) function minimization algorithms, M-concave functions are identified as a computationally tractable subclass for maximization.

[1]  László Lovász,et al.  Matroid matching and some applications , 1980, J. Comb. Theory, Ser. B.

[2]  Kazuo Murota,et al.  M-Convex Function on Generalized Polymatroid , 1999, Math. Oper. Res..

[3]  Satoru Iwata,et al.  A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions , 2000, STOC '00.

[4]  Kazuo Murota,et al.  Valuated Matroid Intersection II: Algorithms , 1996, SIAM J. Discret. Math..

[5]  D. M. Topkis.  Supermodularity and Complementarity , 1998 .

[6]  Gérard Cornuéjols,et al.  Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem , 1984, Discret. Appl. Math..

[7]  Walter Wenzel,et al.  Valuated matroids: a new look at the greedy algorithm , 1989 .

[8]  Daniel Lehmann,et al.  Combinatorial auctions with decreasing marginal utilities , 2001, EC '01.

[9]  Satoru Fujishige,et al.  Submodular functions and optimization , 1991 .

[10]  Zaifu Yang,et al.  A Note on Kelso and Crawford's Gross Substitutes Condition , 2003, Math. Oper. Res..

[11]  Alexander Schrijver,et al.  A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time , 2000, J. Comb. Theory B.

[12]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[13]  László Lovász,et al.  Submodular functions and convexity , 1982, ISMP.

[14]  James B. Orlin,et al.  A faster strongly polynomial time algorithm for submodular function minimization , 2007, Math. Program..

[15]  Jan Vondrák,et al.  Optimal approximation for the submodular welfare problem in the value oracle model , 2008, STOC.

[16]  Bernhard Korte,et al.  Complexity of Matroid Property Algorithms , 1982, SIAM J. Comput..

[17]  Satoru Iwata,et al.  Submodular function minimization , 2007, Math. Program..

[18]  Kazuo Murota,et al.  Conjugacy relationship between M-convex and L-convex functions in continuous variables , 2004, Math. Program..

[19]  Maxim Sviridenko,et al.  Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee , 2004, J. Comb. Optim..

[20]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[21]  Kazuo Murota,et al.  On Steepest Descent Algorithms for Discrete Convex Functions , 2003, SIAM J. Optim..

[22]  Akiyoshi Shioura,et al.  On the Pipage Rounding Algorithm for Submodular Function Maximization - a View from Discrete Convex Analysis , 2009, Discret. Math. Algorithms Appl..

[23]  Kazuo Murota,et al.  Discrete convex analysis , 1998, Math. Program..

[24]  Kazuo Murota,et al.  Notes on L-/M-convex functions and the separation theorems , 2000, Math. Program..

[25]  Kazuo Murota,et al.  Valuated Matroid Intersection I: Optimality Criteria , 1996, SIAM J. Discret. Math..

[26]  Kazuo Murota,et al.  Extension of M-Convexity and L-Convexity to Polyhedral Convex Functions , 1999, Adv. Appl. Math..

[27]  Kazuo Murota,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS Recent Developments in Discrete Convex Analysis , 2008 .

[28]  Satoru Iwata,et al.  Conjugate Scaling Algorithm for Fenchel-Type Duality in Discrete Convex Optimization , 2002, SIAM J. Optim..

[29]  András Frank,et al.  A Weighted Matroid Intersection Algorithm , 1981, J. Algorithms.

[30]  Jan Vondrák,et al.  Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract) , 2007, IPCO.

[31]  Kazuo Murota,et al.  Fundamental Properties of M-Convex and L-Convex Functions in Continuous Variables , 2004 .

[32]  P. Favati.  Convexity in nonlinear integer programming , 1990 .

[33]  K. Murota.  Convexity and Steinitz's Exchange Property , 1996 .

[34]  A. Frank.  An Algorithm for Submodular Functions on Graphs , 1982 .

[35]  Satoru Fujishige.  Theory of submodular programs: A fenchel-type min-max theorem and subgradients of submodular functions , 1984, Math. Program..

[36]  Kazuo Murota,et al.  Matrices and Matroids for Systems Analysis , 2000 .