An algebraic proof of the dichotomy for graph orientation problems with forbidden tournaments

TL;DR

This paper provides an algebraic proof of the complexity dichotomy for graph orientation problems with forbidden tournaments, extending previous results to broader classes with local constraints and a decidable algebraic boundary.

Contribution

It introduces a new algebraic proof technique for the complexity dichotomy, applicable to larger classes of orientation problems with local constraints.

Findings

Established a complexity dichotomy for F-free orientation problems.

Extended the dichotomy to problems with additional local constraints.

Identified a decidable algebraic condition delineating tractable and hard cases.

Abstract

For a set F of finite tournaments, the F-free orientation problem is the problem of deciding if a given finite undirected graph can be oriented in such a way that the resulting oriented graph does not contain any member of F. Using the theory of smooth approximations, we give a new shorter proof of the complexity dichotomy for such problems obtained recently by Bodirsky and Guzm\'{a}n-Pro. In fact, our approach yields a complexity dichotomy for a considerably larger class of computational problems where one is given an undirected graph along with additional local constraints on the allowed orientations. Moreover, the border between tractable and hard problems is also described by a decidable algebraic condition.