Hiroshi Hirai
Stanislav Živný
Abstract
A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions. An important line of VCSP research is to investigate what functions can be solved in polynomial time. Cooper and Živný classified the tractability of binary VCSP instances according to the concept of “triangle,” and showed that the only interesting tractable case is the one induced by the joint winner property (JWP). Recently, Iwamasa, Murota, and Živný made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two quadratic M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given.
In this article, we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be represented as the sum of two quadratic M-convex functions and can be solved in polynomial time via an M-convex intersection algorithm? We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case. Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.
- U. Bertelé and F. Brioschi. 1972. Nonserial Dynamic Programming. Academic Press. Google Scholar
Digital Library
- E. Boros and P. L. Hammer. 2002. Pseudo-Boolean optimization. Discrete Applied Mathematics 123 (2002), 155--225. Google ScholarDigital Library
- A. A. Bulatov. 2017. A dichotomy theorem for nonuniform CSPs. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS’17), 319--330.Google ScholarCross Ref
- J. C. Colberson and P. Rudnicki. 1989. A fast algorithm for construction trees from distance matrices. Information Processing Letters 30 (1989), 215--220. Google ScholarDigital Library
- M. C. Cooper and S. Živný. 2011. Hybrid tractability of valued constraint problems. Artificial Intelligence 175 (2011), 1555--1569. Google ScholarDigital Library
- M. C. Cooper and S. Živný. 2012. Tractable triangles and cross-free convexity in discrete optimisation. Journal of Artificial Intelligence Research 44 (2012), 455--490. Google ScholarDigital Library
- M. C. Cooper and S. Živný. 2017. Hybrid tractable classes of constraint problems. In The Constraint Satisfaction Problem: Complexity and Approximability, Dagstuhl Follow-Ups Series, Vol. 7, A. Krokhin and S. Živný (Eds.), Chap. 4, Schloss Dagstuhl -- Leibniz-Zentrum für Informatik, 113--135.Google Scholar
- Y. Crama and P. L. Hammer. 2011. Boolean Functions—Theory, Algorithms, and Applications. Cambridge University Press, Cambridge.Google Scholar
- M. M. Deza and M. Laurent. 1997. Geometry of Cuts and Metrics. Springer, Heidelberg.Google Scholar
- A. W. M. Dress and W. Wenzel. 1990. Valuated matroids: A new look at the greedy algorithm. Applied Mathematics Letters 3, 2 (1990), 33--35.Google ScholarCross Ref
- A. W. M. Dress and W. Wenzel. 1992. Valuated matroids. Advances in Mathematics 93 (1992), 214--250.Google ScholarCross Ref
- J. Edmonds. 1970. Submodular functions, matroids, and certain polyhedra. In Combinatorial Structures and Their Applications, Gordon and Breach (Eds.), New York, 69--87.Google Scholar
- H. Hirai. 2006. A geometric study of the split decomposition. Discrete and Computational Geometry 36 (2006), 331--361.Google ScholarDigital Library
- H. Hirai. 2016. Discrete convexity and polynomial solvability in minimum 0-extension problems. Mathematical Programming, Series A 155 (2016), 1--55. Google ScholarDigital Library
- H. Hirai. 2018. L-convexity on graph structures. Journal of the Operations Research Society of Japan 61 (2018), 71--109.Google ScholarCross Ref
- H. Hirai and Y. Iwamasa. 2018. Reconstructing phylogenetic tree from multipartite quartet system. In Proceedings of the 29th International Symposium on Algorithms and Computation (ISAAC’18).Google Scholar
- H. Hirai, Y. Iwamasa, K. Murota, and S. Živný. 2018. Beyond JWP: A tractable class of binary VCSPs via M-convex intersection. In Proceedings of the 35th International Symposium on Theoretical Aspects of Computer Science (STACS’18), Leibniz International Proceedings in Informatics, Vol. 96, 39:1--39:14.Google Scholar
- H. Hirai and K. Murota. 2004. M-convex functions and tree metrics. Japan Journal of Industrial and Applied Mathematics 21 (2004), 391--403.Google ScholarCross Ref
- Y. Iwamasa. 2018. The quadratic M-convexity testing problem. Discrete Applied Mathematics 238 (2018), 106--114.Google ScholarCross Ref
- Y. Iwamasa, K. Murota, and S. Živný. 2018. Discrete convexity in joint winner property. Discrete Optimization 28 (2018), 78--88.Google ScholarDigital Library
- V. Kolmogorov, A. Krokhin, and M. Rolínek. 2017. The complexity of general-valued CSPs. SIAM Journal on Computing 46, 3 (2017), 1087--1110.Google ScholarDigital Library
- V. Kolmogorov, J. Thapper, and S. Živný. 2015. The power of linear programming for general-valued CSPs. SIAM Journal on Computing 44, 1 (2015), 1--36. Google ScholarDigital Library
- B. Korte and J. Vygen. 2010. Combinatorial Optimization: Theory and Algorithms (5th ed.). Springer, Heidelberg. Google ScholarDigital Library
- A. Krokhin and S. Živný, editors. 2017. The Constraint Satisfaction Problem: Complexity and Approximability, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Vol. 7. Dagstuhl Follow-Ups Series.Google Scholar
- K. Murota. 1996. Convexity and Steinitz’s exchange property. Advances in Mathematics 124 (1996), 272--311.Google ScholarCross Ref
- K. Murota. 1996. Valuated matroid intersection, I: Optimality criteria. SIAM Journal on Discrete Mathematics 9 (1996), 545--561. Google ScholarDigital Library
- K. Murota. 1996. Valuated matroid intersection, II: Algorithms. SIAM Journal on Discrete Mathematics 9 (1996), 562--576. Google ScholarDigital Library
- K. Murota. 1998. Discrete convex analysis. Mathematical Programming 83 (1998), 313--371. Google ScholarDigital Library
- K. Murota. 2000. Matrices and Matroids for Systems Analysis. Springer, Heidelberg. Google ScholarDigital Library
- K. Murota. 2003. Discrete Convex Analysis. SIAM, Philadelphia.Google Scholar
- K. Murota. 2009. Recent developments in discrete convex analysis. In Research Trends in Combinatorial Optimization, W. Cook, L. Lovász, and J. Vygen (Eds.), Chap. 11, Springer, Heidelberg, 219--260.Google Scholar
- K. Murota. 2016. Discrete convex analysis: A tool for economics and game theory. Journal of Mechanism and Institution Design, 1, 1 (2016), 151--273.Google ScholarCross Ref
- K. Murota and A. Shioura. 2004. Quadratic M-convex and L-convex functions. Advances in Applied Mathematics 33 (2004), 318--341. Google ScholarDigital Library
- A. Schrijver. 2003. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg.Google Scholar
- C. Semple and M. Steel. 2003. Phylogenetics. Oxford University Press, Oxford.Google Scholar
- M. Steel. 1992. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification 9 (1992), 91--116.Google ScholarCross Ref
- M. S. Waterman, T. F. Smith, M. Singh, and W. A. Beyer. 1977. Additive evolutionary trees. Journal of Theoretical Biology 64 (1977), 199--213.Google ScholarCross Ref
- D. Zhuk. 2017. A proof of CSP dichotomy conjecture. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS’17), Vol. 331--342.Google ScholarCross Ref
- S. Živný. 2012. The Complexity of Valued Constraint Satisfaction Problems. Springer, Heidelberg. Google ScholarDigital Library
Index Terms
- A Tractable Class of Binary VCSPs via M-Convex Intersection
Recommendations
Discrete convexity in joint winner property
AbstractIn 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 ...
Projection and convolution operations for integrally convex functions
AbstractThis paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable ...
Using a Min-Cut Generalisation to Go Beyond Boolean Surjective VCSPs
AbstractIn this work, we first study a natural generalisation of the Min-Cut problem, where a graph is augmented by a superadditive set function defined on its vertex subsets. The goal is to select a vertex subset such that the weight of the induced cut ...
Comments