We give a precise algebraic characterization of the power of Sherali--Adams relaxations for solvability of valued constraint satisfaction problems (CSPs) to optimality. The condition is that of bounded width, which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterization has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and well-known valued constraint languages are tractable via the third level of the Sherali--Adams relaxation. For the known languages, this is a significantly simpler algorithm than those previously obtained. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Živný [J. ACM, 60 (2013), 10], and also a ...
There are valued constraint languages that have valued relational width @math but not @math . For example, languages improved by a tournament pair fractional polymorphism @cite_40 , discussed in detail in Example in , have valued relational width @math by the results in this paper, but do not have valued relational width in @math as shown [Example 5] ktz15:sicomp using Theorem .
@cite_40
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The submodular function minimization problem (SFM) is a fundamental problem in combinatorial optimization and several fully combinatorial polynomial-time algorithms have recently been discovered to solve this problem. The most general versions of these algorithms are able to minimize any submodular function whose domain is a set of tuples over any totally-ordered finite set and whose range includes both finite and infinite values. In this paper we demonstrate that this general form of SFM is just one example of a much larger class of tractable discrete optimization problems defined by valued constraints. These tractable problems are characterized by the fact that their valued constraints have an algebraic property which we call a tournament pair multimorphism. This larger tractable class also includes the problem of satisfying a set of Horn clauses (Horn-SAT), as well as various extensions of this problem to larger finite domains.