A Dichotomy Theorem for General Minimum Cost Homomorphism Problem

TL;DR

This paper establishes a comprehensive classification of the computational complexity for the minimum cost homomorphism problem, extending algebraic methods from CSPs to solve a broad range of related optimization problems.

Contribution

It introduces an algebraic approach to classify the complexity of MinHom problems for all predicate sets, resolving a longstanding conjecture.

Findings

Classifies MinHom complexity for all predicate sets.

Resolves a general dichotomy conjecture for MinHom.

Extends algebraic techniques from CSPs to MinHom.

Abstract

In the constraint satisfaction problem ($CSP$), the aim is to find an assignment of values to a set of variables subject to specified constraints. In the minimum cost homomorphism problem ($MinHom$), one is additionally given weights $c_{va}$ for every variable $v$ and value $a$, and the aim is to find an assignment $f$ to the variables that minimizes $\sum_{v} c_{vf(v)}$. Let $MinHom(\Gamma)$ denote the $MinHom$ problem parameterized by the set of predicates allowed for constraints. $MinHom(\Gamma)$ is related to many well-studied combinatorial optimization problems, and concrete applications can be found in, for instance, defence logistics and machine learning. We show that $MinHom(\Gamma)$ can be studied by using algebraic methods similar to those used for CSPs. With the aid of algebraic techniques, we classify the computational complexity of $MinHom(\Gamma)$ for all choices of $\Gamma$. Our result settles a general dichotomy conjecture previously resolved only for certain classes of directed graphs, [Gutin, Hell, Rafiey, Yeo, European J. of Combinatorics, 2008].