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

2024-05-30

AI Summary

This paper investigates the *F*-free orientation problem for graphs: given a finite set *F* of tournaments, determine whether an undirected graph admits an orientation containing no member of *F* as a (not necessarily induced) subgraph. Addressing this classical problem, the paper introduces a novel algebraic approach grounded in smooth approximation theory, extending the classical complexity dichotomy theorem to generalized orientations with local constraints. The main contribution is a decidable algebraic criterion that yields an exact P vs. NP-complete classification for a broad class of orientation constraint problems—marking the first such precise characterization. Compared to traditional combinatorial proofs, the new method is more concise and yields boundary conditions that are effectively decidable. The results unify and generalize the intersection of constraint satisfaction problems and graph orientation theory, providing a robust framework for analyzing constrained orientations.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For a set F of finite tournaments, the F-free orientation problem is the problem of orienting a given finite undirected graph 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 described by a decidable algebraic condition.

Problem

• Classifying complexity of graph orientation problems with forbidden tournaments
• Extending dichotomy to problems with additional local constraints
• Providing decidable algebraic condition for tractability border

Innovation

• Smooth approximations for complexity dichotomy
• Local constraints on graph orientation
• Decidable algebraic condition for tractability