Discrete Convexity in Joint Winner Property

Abstract In this paper, we reveal a relation between joint winner property (JWP) in the field of valued constraint satisfaction problems (VCSPs) and M ♮ -convexity in the field of discrete convex analysis (DCA). We introduce the M ♮ -convex completion problem, and show that a function f satisfying the JWP is Z-free if and only if a certain function f ¯ associated with f is M ♮ -convex completable. This means that if a function is Z-free, then the function can be minimized in polynomial time via M ♮ -convex intersection algorithms. Furthermore we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values.

[1]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

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

[3]  A. Tamura,et al.  GROSS SUBSTITUTES CONDITION AND DISCRETE CONCAVITY FOR MULTI-UNIT VALUATIONS: A SURVEY , 2015 .

[4]  Edward G. Coffman,et al.  Scheduling independent tasks to reduce mean finishing time , 1974, CACM.

[5]  Martin C. Cooper,et al.  Hybrid tractability of valued constraint problems , 2010, Artif. Intell..

[6]  W. A. Horn.  Technical Note - Minimizing Average Flow Time with Parallel Machines , 1973, Oper. Res..

[7]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[8]  Martin C. Cooper,et al.  Tractable Triangles and Cross-Free Convexity in Discrete Optimisation , 2012, J. Artif. Intell. Res..

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

[10]  K. Murota,et al.  M-convex functions and tree metrics , 2004 .

[11]  Jean-Charles Régin,et al.  A Filtering Algorithm for Constraints of Difference in CSPs , 1994, AAAI.

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

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

[14]  J. Potters,et al.  Verifying gross substitutability , 2002 .

[15]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[16]  Christian Bessiere,et al.  Specific Filtering Algorithms for Over-Constrained Problems , 2001, CP.

[17]  K. Murota.  Discrete convex analysis: A tool for economics and game theory , 2016, 2212.03598.

[18]  Stanislav Zivny,et al.  The Complexity of Valued Constraint Satisfaction Problems , 2012, Cognitive Technologies.