The Complexity of Valued CSPs ∗

We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems (VCSPs). We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, G.1.6 Optimization.

[1]  R. McKenzie,et al.  Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties , 2014 .

[2]  Martin C. Cooper,et al.  Generalising submodularity and horn clauses: Tractable optimization problems defined by tournament pair multimorphisms , 2008, Theor. Comput. Sci..

[3]  Brigitte Maier,et al.  Supermodularity And Complementarity , 2016 .

[4]  Rustem Takhanov.  Extensions of the Minimum Cost Homomorphism Problem , 2010, COCOON.

[5]  Anna Huber,et al.  Towards Minimizing k-Submodular Functions , 2012, ISCO.

[6]  Martin C. Cooper,et al.  Supermodular functions and the complexity of MAX CSP , 2005, Discret. Appl. Math..

[7]  Andrei A. Bulatov,et al.  Bounded relational width , 2009 .

[8]  D. SIAMJ..  BISUBMODULAR FUNCTION MINIMIZATION∗ , 2006 .

[9]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[10]  Mihalis Yannakakis,et al.  The Complexity of Multiterminal Cuts , 1994, SIAM J. Comput..

[11]  Shin-ichi Tanigawa,et al.  Polynomial combinatorial algorithms for skew-bisubmodular function minimization , 2018, Math. Program..

[12]  S. Fujishige,et al.  Minimizing Submodular Functions on Diamonds via Generalized Fractional Matroid Matchings , 2014 .

[13]  Luca Trevisan,et al.  The Approximability of Constraint Satisfaction Problems , 2001, SIAM J. Comput..

[14]  Johan Thapper,et al.  The limits of SDP relaxations for general-valued CSPs , 2017, LICS.

[15]  Ágnes Szendrei,et al.  Clones in universal algebra , 1986 .

[16]  Konstantin Makarychev,et al.  Approximation Algorithms for CSPs , 2017, The Constraint Satisfaction Problem.

[17]  Martin C. Cooper,et al.  The complexity of soft constraint satisfaction , 2006, Artif. Intell..

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

[19]  Michael I. Jordan,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008, Found. Trends Mach. Learn..

[20]  Todd Niven,et al.  A finer reduction of constraint problems to digraphs , 2014, Log. Methods Comput. Sci..

[21]  Peter Jonsson,et al.  Min CSP on Four Elements: Moving beyond Submodularity , 2011, CP.

[22]  Anna Huber,et al.  Oracle Tractability of Skew Bisubmodular Functions , 2013, SIAM J. Discret. Math..

[23]  Satoru Iwata,et al.  A simple combinatorial algorithm for submodular function minimization , 2009, SODA.

[24]  Libor Barto,et al.  The collapse of the bounded width hierarchy , 2016, J. Log. Comput..

[25]  Stanislav Zivny,et al.  The expressive power of binary submodular functions , 2008, Discret. Appl. Math..

[26]  Stanislav Zivny,et al.  The complexity and expressive power of valued constraints , 2009 .

[27]  Stanislav Zivny,et al.  The Power of Linear Programming for Valued CSPs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[28]  Hannes Uppman,et al.  The Complexity of Three-Element Min-Sol and Conservative Min-Cost-Hom , 2012, ICALP.

[29]  Libor Barto,et al.  Constraint satisfaction problem and universal algebra , 2014, SIGL.

[30]  Peter Jeavons,et al.  The Complexity of Valued Constraint Satisfaction , 2014, Bull. EATCS.

[31]  Andrei A. Krokhin,et al.  Maximizing Supermodular Functions on Product Lattices, with Application to Maximum Constraint Satisfaction , 2008, SIAM J. Discret. Math..

[32]  Stanislav Zivny,et al.  Sherali-Adams Relaxations for Valued CSPs , 2015, ICALP.

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

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

[35]  B. Larose,et al.  Bounded width problems and algebras , 2007 .

[36]  Manuel Bodirsky,et al.  Constraint Satisfaction Problems over Numeric Domains , 2017, The Constraint Satisfaction Problem.

[37]  Andrei A. Krokhin,et al.  The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231) , 2018, Dagstuhl Reports.

[38]  Pavol Hell,et al.  The Dichotomy of Minimum Cost Homomorphism Problems for Digraphs , 2012, SIAM J. Discret. Math..

[39]  藤重 悟.  Submodular functions and optimization , 1991 .

[40]  Andrei A. Bulatov,et al.  A Simple Algorithm for Mal'tsev Constraints , 2006, SIAM J. Comput..

[41]  Heribert Vollmer,et al.  Complexity of Constraints - An Overview of Current Research Themes [Result of a Dagstuhl Seminar] , 2008, Complexity of Constraints.

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

[43]  I JordanMichael,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008 .

[44]  Anna Huber,et al.  Skew Bisubmodularity and Valued CSPs , 2013, SIAM J. Comput..

[45]  Benoît Larose.  Algebra and the Complexity of Digraph CSPs: a Survey , 2017, The Constraint Satisfaction Problem.

[46]  Andrei A. Bulatov,et al.  Complexity of conservative constraint satisfaction problems , 2011, TOCL.

[47]  Marc Gyssens,et al.  Closure properties of constraints , 1997, JACM.

[48]  P. Jeavons.  Algebraic structures in combinatorial problems , 2001 .

[49]  Martin C. Cooper,et al.  An Algebraic Theory of Complexity for Discrete Optimization , 2012, SIAM J. Comput..

[50]  R. Willard,et al.  Characterizations of several Maltsev conditions , 2015 .

[51]  Gustav Nordh,et al.  MAX ONES Generalized to Larger Domains , 2008, SIAM J. Comput..

[52]  Peter Jonsson,et al.  The Approximability of Three-valued MAX CSP , 2004, SIAM J. Comput..

[53]  Peter L. Hammer,et al.  Boolean Functions - Theory, Algorithms, and Applications , 2011, Encyclopedia of mathematics and its applications.

[54]  W. Taylor.  Varieties Obeying Homotopy Laws , 1977, Canadian Journal of Mathematics.

[55]  S. Kak.  Information, physics, and computation , 1996 .

[56]  Pawel M. Idziak,et al.  Tractability and learnability arising from algebras with few subpowers , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[57]  Stanislav Zivny,et al.  The Power of Sherali-Adams Relaxations for General-Valued CSPs , 2016, SIAM J. Comput..

[58]  Manuel Bodirsky,et al.  Complexity Classification in Infinite-Domain Constraint Satisfaction , 2012, ArXiv.

[59]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[60]  M. Maróti,et al.  Existence theorems for weakly symmetric operations , 2008 .

[61]  Libor Barto,et al.  Robust satisfiability of constraint satisfaction problems , 2012, STOC '12.

[62]  Hannes Uppman.  Computational Complexity of the Extended Minimum Cost Homomorphism Problem on Three-Element Domains , 2014, STACS.

[63]  P. Jonsson,et al.  Introduction to the Maximum Solution Problem , 2007 .

[64]  Peter Jonsson,et al.  The approximability of MAX CSP with fixed-value constraints , 2006, JACM.

[65]  Vladimir Kolmogorov,et al.  The Power of Linear Programming for General-Valued CSPs , 2013, SIAM J. Comput..

[66]  Andrei A. Krokhin,et al.  Robust Satisfiability for CSPs: Hardness and Algorithmic Results , 2013, TOCT.

[67]  Dániel Marx,et al.  Fixed-parameter Approximability of Boolean MinCSPs , 2016, ESA.

[68]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[69]  Michael I. Jordan.  Graphical Models , 2003 .

[70]  Yin Tat Lee,et al.  A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[71]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[72]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraint satisfaction problems on a 3-element set , 2006, JACM.

[73]  Peter Jonsson,et al.  Approximability of the Maximum Solution Problem for Certain Families of Algebras , 2009, CSR.

[74]  Libor Barto,et al.  The Dichotomy for Conservative Constraint Satisfaction Problems Revisited , 2011, 2011 IEEE 26th Annual Symposium on Logic in Computer Science.

[75]  Stanislav Zivny,et al.  A Galois Connection for Valued Constraint Languages of Infinite Size , 2015, ICALP.

[76]  Michael Pinsker,et al.  Algebraic and model theoretic methods in constraint satisfaction , 2015, ArXiv.

[77]  Martin C. Cooper,et al.  Binarisation for Valued Constraint Satisfaction Problems , 2016, SIAM Journal on Discrete Mathematics.

[78]  Martin Grohe.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2007, JACM.

[79]  Marcin Kozik,et al.  Algebraic Properties of Valued Constraint Satisfaction Problem , 2014, ICALP.

[80]  S. Thomas McCormick,et al.  Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization , 2008, SODA '08.

[81]  Hiroshi Hirai,et al.  Discrete convexity and polynomial solvability in minimum 0-extension problems , 2013, Math. Program..

[82]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[83]  Stanislav Zivny,et al.  The complexity of finite-valued CSPs , 2013, STOC '13.

[84]  Stanislav Zivny,et al.  Minimal Weighted Clones with Boolean Support , 2016, 2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL).

[85]  Libor Barto,et al.  Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem , 2012, Log. Methods Comput. Sci..

[86]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[87]  Stanislav Zivny,et al.  The limits of SDP relaxations for general-valued CSPs , 2016, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[88]  Peter Jeavons,et al.  The Complexity of Constraint Languages , 2006, Handbook of Constraint Programming.

[89]  Vladimir Kolmogorov,et al.  The complexity of conservative valued CSPs , 2011, JACM.

[90]  Georg Gottlob,et al.  Tractable Optimization Problems through Hypergraph-Based Structural Restrictions , 2009, ICALP.

[91]  M. Siggers.  A strong Mal’cev condition for locally finite varieties omitting the unary type , 2010 .

[92]  Fredrik Kuivinen,et al.  On the complexity of submodular function minimisation on diamonds , 2009, Discret. Optim..

[93]  Libor Barto,et al.  Constraint Satisfaction Problems Solvable by Local Consistency Methods , 2014, JACM.

[94]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[95]  Martin C. Cooper,et al.  Hybrid Tractable Classes of Constraint Problems , 2017, The Constraint Satisfaction Problem.

[96]  Gregory Gutin,et al.  A dichotomy for minimum cost graph homomorphisms , 2008, Eur. J. Comb..

[97]  Ryan O'Donnell,et al.  Linear programming, width-1 CSPs, and robust satisfaction , 2012, ITCS '12.

[98]  Marcin Kozik.  Weak consistency notions for all the CSPs of bounded width∗ , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[99]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[100]  Olga Veksler,et al.  Markov random fields with efficient approximations , 1998, Proceedings. 1998 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No.98CB36231).

[101]  Rustem Takhanov,et al.  A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem , 2007, STACS.

[102]  Stanislav Zivny,et al.  On Minimal Weighted Clones , 2011, CP.

[103]  Pascal Tesson,et al.  Universal algebra and hardness results for constraint satisfaction problems , 2007, Theor. Comput. Sci..

[104]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[105]  Ferdinand Börner.  Basics of Galois Connections , 2008, Complexity of Constraints.

[106]  Jaroslav Nesetril,et al.  Colouring, constraint satisfaction, and complexity , 2008, Comput. Sci. Rev..

[107]  Shin-ichi Tanigawa,et al.  Generalized skew bisubmodularity: A characterization and a min-max theorem , 2014, Discret. Optim..

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