# The complexity of Boolean surjective general-valued CSPs

## TL;DR

This paper classifies the computational complexity of Boolean surjective VCSPs, revealing new tractable classes and providing algorithms for near-optimal solutions, thus advancing understanding of these optimization problems.

## Contribution

It establishes a complete complexity classification for Boolean surjective VCSPs, including a novel tractable class related to downsets and upsets, and introduces a polynomial-time enumeration algorithm.

## Key findings

- Complete complexity dichotomy for Boolean surjective VCSPs
- Identification of a new tractable class related to downsets and upsets
- Polynomial-time algorithm for enumerating near-optimal solutions

## Abstract

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $(\mathbb{Q}\cup\{\infty\})$-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from $D=\{0,1\}$ and an optimal assignment is required to use both labels from $D$. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for $\{0,\infty\}$-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and H\'ebrard. For the maximisation problem of $\mathbb{Q}_{\geq 0}$-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability. Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.