Forbidden lists (NP and CSP for combinatorialists)

TL;DR

This paper defines NP in a combinatorial context using forbidden lifted substructures, characterizes certain CSPs, and relates these to known complexity classes, providing a unified framework for analyzing combinatorial problems.

Contribution

It introduces a novel combinatorial definition of NP via forbidden lifted substructures and characterizes CSPs within this framework, simplifying previous approaches.

Findings

Edge colorings and graph decompositions express NP's full power.

Certain CSPs are characterized by finitely many forbidden lifted substructures.

The approach links CSPs to the MMSNP class, unifying complexity analysis.

Abstract

We present a definition of the class NP in combinatorial context as the set of languages of structures defined by finitely many forbidden lifted substructures. We apply this to special syntactically defined subclasses and show how they correspond to naturally defined (and intensively studied) combinatorial problems. We show that some types of combinatorial problems like edge colorings and graph decompositions express the full computational power of the class NP. We then characterize Constraint Satisfaction Problems (i.e. H-coloring problems) which are expressible by finitely many forbidden lifted substructures. This greatly simplifies and generalizes the earlier attempts to characterize this problem. As a corollary of this approach we perhaps find a proper setting of Feder and Vardi analysis of CSP languages within the class MMSNP.