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The power of linear programming for general-valued CSPs

Let $D$, called the domain, be a fixed finite set and let $Γ$, 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 $Γ$, denoted by VCSP$(Γ)$. These are minimisation problems given by $n$ variables and the objective function given by a sum of functions from $Γ$, 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 $Γ$, BLP is a decision procedure for $Γ$ if and only if $Γ$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $Γ$, BLP is a decision procedure if and only if $Γ$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $Γ$ 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.