The complexity of finite-valued CSPs

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, $\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as a sum of functions from $\Gamma$. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite size. We show that every constraint language $\Gamma$ either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or $\Gamma$ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$.

💡 Research Summary

The paper investigates the exact minimisation problem for rational‑valued discrete functions defined over a fixed finite domain, formalised as the Valued Constraint Satisfaction Problem VCSP(Γ) where Γ is a finite‑valued constraint language. The authors prove a complete dichotomy theorem that classifies every such language into one of two mutually exclusive categories, thereby resolving the long‑standing question of which VCSPs admit polynomial‑time algorithms and which are intrinsically hard. The first category is characterised by the existence of a binary symmetric fractional polymorphism for Γ. A fractional polymorphism is a probability‑weighted combination of operations that preserves the cost functions in Γ; symmetry and binary arity imply that the operation treats its two arguments interchangeably. When such a polymorphism exists, the basic linear programming relaxation (BLP) of the VCSP is provably exact: the LP optimum coincides with the true integer optimum for every instance. The proof hinges on the duality theory of linear programming and on algebraic arguments showing that the polymorphism forces the LP solution to be integral. Consequently, any instance of VCSP(Γ) can be solved in polynomial time simply by solving its BLP, which is highly practical because BLP solvers are widely available and efficient. The second category comprises languages that lack any binary symmetric fractional polymorphism. For these languages the authors identify a simple hardness condition: there must be a pair of functions in Γ that can simulate the cut function of a graph. Using these functions, they construct a polynomial‑time reduction from the classic Max‑Cut problem to VCSP(Γ). Since Max‑Cut is NP‑hard, this reduction establishes that VCSP(Γ) is also NP‑hard for such languages, and that no polynomial‑time algorithm can solve all instances unless P = NP. The reduction is careful to preserve rational costs and to avoid blow‑up of instance size, thereby demonstrating that the hardness is inherent rather than an artefact of encoding. In addition to the theoretical dichotomy, the paper provides an algorithmic procedure for testing which side of the dichotomy a given language falls on. The test checks for the presence of a suitable fractional polymorphism by solving a finite system of linear inequalities; if the system is feasible, BLP solves the VCSP, otherwise the language is declared hard. This decision procedure works for arbitrary finite domains, extending earlier results that were limited to Boolean or small domains. The work unifies and generalises several previous partial classifications, such as those for submodular functions, metric labeling, and certain classes of convex cost functions. By bridging algebraic CSP theory, linear programming duality, and classical complexity reductions, the authors deliver a powerful framework that not only settles the exact complexity of finite‑valued VCSPs but also suggests a roadmap for tackling more general valued CSPs, including those with infinite‑valued costs or approximation‑focused objectives.