The power of linear programming for valued CSPs: a constructive characterization

A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. We study which classes of finite-valued languages can be solved exactly by the basic linear programming relaxation (BLP). Thapper and Zivny showed [20] that if BLP solves the language then the language admits a binary commutative fractional polymorphism. We prove that the converse is also true. This leads to a necessary and a sufficient condition which can be checked in polynomial time for a given language. In contrast, the previous necessary and sufficient condition due to [20] involved infinitely many inequalities. More recently, Thapper and Zivny [21] showed (using, in particular, a technique introduced in this paper) that core languages that do not satisfy our condition are NP-hard. Taken together, these results imply that a finite-valued language can either be solved using Linear Programming or is NP-hard.

[1]  Vladimir Kolmogorov,et al.  Convergent Tree-Reweighted Message Passing for Energy Minimization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[3]  Tomás Werner,et al.  A Linear Programming Approach to Max-Sum Problem: A Review , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  Pushmeet Kohli,et al.  Markov Random Fields for Vision and Image Processing , 2011 .

[5]  Subhash Khot.  On the Unique Games Conjecture (Invited Survey) , 2010, Computational Complexity Conference.

[6]  Justin Pearson,et al.  Closure Functions and Width 1 Problems , 1999, CP.

[7]  P. Raghavendra,et al.  Approximating np-hard problems efficient algorithms and their limits , 2009 .

[8]  Martin J. Wainwright,et al.  MAP estimation via agreement on trees: message-passing and linear programming , 2005, IEEE Transactions on Information Theory.

[9]  D. Sontag.  1 Introduction to Dual Decomposition for Inference , 2010 .

[10]  Thomas Schoenemann,et al.  Generalized sequential tree-reweighted message passing , 2012, ArXiv.

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

[12]  Ofer Meshi,et al.  An Alternating Direction Method for Dual MAP LP Relaxation , 2011, ECML/PKDD.

[13]  Martin C. Cooper,et al.  An Algebraic Characterisation of Complexity for Valued Constraint , 2006, CP.

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

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

[16]  Martin J. Wainwright,et al.  MAP estimation via agreement on (hyper)trees: Message-passing and linear programming , 2005, ArXiv.

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

[18]  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..

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

[20]  Martin C. Cooper,et al.  Soft arc consistency revisited , 2010, Artif. Intell..

[21]  Christoph Schnörr,et al.  Efficient MRF Energy Minimization via Adaptive Diminishing Smoothing , 2012, UAI.

[22]  Anna Huber,et al.  Skew Bisubmodularity and Valued CSPs , 2014, SIAM J. Comput..

[23]  Arie M. C. A. Koster,et al.  The partial constraint satisfaction problem: Facets and lifting theorems , 1998, Oper. Res. Lett..