The power of linear programming for general-valued CSPs

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrised by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimisation problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors (arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213) to appear in SIAM Journal on Computing (SICOMP

The power of Sherali-Adams relaxations for general-valued CSPs

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 and Zivny [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].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301

Robust algorithms with polynomial loss for near-unanimity CSPs

An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a xed domain to the variables so that all constraints are satised. In the optimization version, the goal is to maximize the number of satised constraints. An approximation algorithm for CSP is called robust if it outputs an assignment satisfying an (1????g("))-fraction of constraints on any (1????")-satisable instance, where the loss function g is such that g(") ! 0 as " ! 0. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some g) have been characterised by Barto and Kozik, with the general bound on the loss g being doubly exponential, specically g(") = O((log log(1="))= log(1=")). It is natural to ask when a better loss can be achieved: in particular, polynomial loss g(") = O("1=k) for some constant k. In this paper, we consider CSPs with a constraint language having a nearunanimity polymorphism. This general condition almost matches a known necessary condition for having a robust algorithm with polynomial loss. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter k in the loss depends on the size of the domain and the arity of the relations in ????, while the other works for a special ternary near-unanimity operation called dual discriminator with k = 2 for any domain size. In the latter case, the CSP is a common generalisation of Unique Games with a xed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidenite programming relaxation for CSP

Towards a Characterization of Constant-Factor Approximable Min CSPs

We study the approximability of Minimum Constraint Satisfaction Problems (Min CSPs) with a fixed finite constraint language Γ on an arbitrary finite domain. The goal in such a problem is to minimize the number of unsatisfied constraints in a given instance of CSP(Γ). A recent result of Ene et al. says that, under the mild technical condition that Γ contains the equality relation, the basic LP relaxation is optimal for constant-factor approximation for Min CSP(Γ) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we introduce a new natural algebraic condition, stable probability distributions on symmetric polymorphisms of a constraint language, and show that the presence of such distributions on polymorphisms of each arity is necessary and sufficient for the finiteness of the integrality gap for the basic LP relaxation of Min CSP(Γ). We also show how stable distributions on symmetric polymorphisms can in principle be used to round solutions of the basic LP relaxation, and how, for several examples that cover all previously known cases, this leads to efficient constant-factor approximation algorithms for Min CSP(Γ). Finally, we show that the absence of another condition, which is implied by stable distributions, leads to NP-hardness of constant-factor approximation