The complexity of general-valued CSPs seen from the other side

TL;DR

This paper investigates the complexity boundaries of general-valued CSPs, extending known results from classical CSPs to the valued case, and explores solvability conditions and tractability in relation to algebraic and linear programming hierarchies.

Contribution

It establishes the exact complexity thresholds for polynomial-time solvability and Sherali-Adams LP hierarchy solvability of VCSPs, generalizing previous CSP results and providing new insights into tractability.

Findings

Identifies the precise borderline of polynomial-time solvability for VCSPs.

Determines the level of Sherali-Adams hierarchy needed for solving VCSPs.

Provides applications to solution finding and tractability recognition in database theory.

Abstract

The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the $k$-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory.