NP by means of lifts and shadows

TL;DR

This paper demonstrates that all NP problems can be reduced to a simple digraph membership problem characterized by shadows and forbidden subgraphs, linking complexity to graph homomorphisms and coloring problems.

Contribution

It introduces a new combinatorial characterization of NP problems using shadows and lifted subgraphs, connecting them to CSPs and homomorphism dualities.

Findings

NP problems are polynomially equivalent to digraph membership problems.

Classes defined by forbidden lifted subgraphs are CSPs if structures are homomorphically equivalent to trees.

Restrictions of shadow classes align with coloring (CSP) classes in bounded expansion graph classes.

Abstract

We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden colored (lifted) subgraphs. Our characterization is motivated by the analysis of syntactical subclasses with the full computational power of NP, which were first studied by Feder and Vardi. Our approach applies to many combinatorial problems and it induces the characterization of coloring problems (CSP) defined by means of shadows. This turns out to be related to homomorphism dualities. We prove that a class of digraphs (relational structures) defined by finitely many forbidden colored subgraphs (i.e. lifted substructures) is a CSP class if and only if all the the forbidden structures are homomorphically equivalent to trees. We show a surprising richness of coloring problems when restricted to most frequent graph classes. Using results of Ne\v{s}et\v{r}il and Ossona de Mendez for bounded expansion classes (which include bounded degree and proper minor closed classes) we prove that the restriction of every class defined as the shadow of finitely many colored subgraphs equals to the restriction of a coloring (CSP) class.