Closure Functions and Width 1 Problems
CP '99 Proceedings of the 5th International Conference on Principles and Practice of Constraint Programming
Convergent Tree-Reweighted Message Passing for Energy Minimization
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Linear Programming Approach to Max-Sum Problem: A Review
IEEE Transactions on Pattern Analysis and Machine Intelligence
The complexity of soft constraint satisfaction
Artificial Intelligence
Approximating np-hard problems efficient algorithms and their limits
Approximating np-hard problems efficient algorithms and their limits
Soft arc consistency revisited
Artificial Intelligence
On the Unique Games Conjecture (Invited Survey)
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Markov Random Fields for Vision and Image Processing
Markov Random Fields for Vision and Image Processing
Linear programming, width-1 CSPs, and robust satisfaction
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
An algebraic characterisation of complexity for valued constraint
CP'06 Proceedings of the 12th international conference on Principles and Practice of Constraint Programming
The complexity of conservative valued CSPs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
MAP estimation via agreement on trees: message-passing and linear programming
IEEE Transactions on Information Theory
The partial constraint satisfaction problem: Facets and lifting theorems
Operations Research Letters
The Power of Linear Programming for Valued CSPs
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
The complexity of finite-valued CSPs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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 Živný 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 Živný [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.