The Power of Sherali--Adams Relaxations for General-Valued CSPs

Abstract

We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems 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 characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many 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 the previously obtained ones. 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 andŽivný [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13]. Union is not liable for any use that may be made of the information contained therein.unless we settle the CSP dichotomy conjecture [30], some additional requirement on the unary weighted relations (such as injectivity) is necessary.As a corollary of our characterisation, we give, in Section 3.4, a complete complexity classification of exact solvability of Minimum-Solution problems over arbitrary finite domains, thus improving on previous partial classifications for domains of size two [42] and three [68], homogeneous and maximal (under a certain algebraic conjecture) languages [39] and on graphs with few vertices [41]. Theorem 3.19 shows that the Minimum-Solution problem is NP-hard unless it satisfies the bounded width condition. Previous partial results included ad-hoc algorithms for various special cases. Our result shows that one algorithm, the third level of the Sherali-Adams relaxation, solves all tractable cases and is thus universal. As a matter of fact, we actually prove a complexity classification for a larger class of problems that includes Minimum-Solutions problems as a special case, as described in detail in Section 3.4. Related workThe first level of the Sherali-Adams hierarchy is known as the basic linear programming (BLP) relaxation [16]. In [63], the authors gave a precise algebraic characterisation of Γ for which any instance of VCSP(Γ) is solved to optimality by BLP, see also [44]. The characterisation proved important not only in the study of VCSPs [36] and other classes of problems [34], but also in the design of fixed-parameter algorithms [37]. In [66], it was then shown that for finite-valued CSPs, the BLP solves all tractable cases; i.e. if BLP fails to solve any instance of some finite-valued constraint language then this la...

Proof Complexity Meets Algebra

“…Therefore, Theorem 5 implies the following statement, which will be crucial for obtaining lower bounds in Section 6: One more consequence of Theorem 5 concerns proofs in Frege proof system without any bounds on the depth. Corollary 3 above immediately implies that Frege is well-behaved with respect to the classical CSP reductions, that is: In the case of algebraic proof systems, if E and E ′ are any local algebraic encoding schemes over a field F for CSP(B) and CSP(B ′ ), respectively, we show that: We point out that Theorem 7 in the case of the Sherali-Adams and Sums-of-Squares proof systems and the EQ encoding scheme can be extracted from [48] and [47].…”

On a General Framework for Network Representability in Discrete Optimization

“…We thank Hiroshi Hirai and Magnus Wahlström for careful reading and helpful comments. We also thank the referees for pointing out a similarity between extended expressive power and ppinterpretation or weighted variety and for information on the papers [20,27]. This research is supported by JSPS Research Fellowship for Young Scientists and by JST ERATO Grant Number JPMJER1305, Japan.…”

“…One of the referees pointed out a similarity between extended expressive power and primitive positive interpretation (pp-interpretation, for short). Here pp-interpretation is a well-known concept in constraint satisfaction problems, and its generalization for valued constraint satisfaction problems is defined as follows (see e.g., [27,Definition 5.3]). Let Γ D and Γ F be languages on D and on F , respectively.…”

Symmetric Polymorphisms and Efficient Decidability of Promise CSPs

“…As is well-studied in the CSP literature (e.g., [RS09,TŽ17]), we consider the canonical linear programming relaxation of a CSP, often refer to as the "Basic LP." For our CSP instance Ψ A , we represent the assignment to x i by a (rational) probability distribution of weights {w i (d)} d∈D summing to 1.…”