On Canonical Forms of Complete Problems via First-order Projections

TL;DR

This paper explores how complete problems across various complexity classes can be decomposed into fundamental components, extending known results from NP to broader classes using properties of first-order reductions.

Contribution

It generalizes the decomposition of complete problems into independent-set-like fragments across multiple complexity classes, not just NP, based on simple properties of first-order reductions.

Findings

Decomposition applies to a wide range of complexity classes including nice classes.

Any complete problem can be decomposed into two non-equivalent parts.

Results rely on well-known properties of first-order reductions.

Abstract

The class of problems complete for NP via first-order reductions is known to be characterized by existential second-order sentences of a fixed form. All such sentences are built around the so-called generalized IS-form of the sentence that defines Independent-Set. This result can also be understood as that every sentence that defines a NP-complete problem P can be decomposed in two disjuncts such that the first one characterizes a fragment of P as hard as Independent-Set and the second the rest of P. That is, a decomposition that divides every such sentence into a quotient and residue modulo Independent-Set. In this paper, we show that this result can be generalized over a wide collection of complexity classes, including the so-called nice classes. Moreover, we show that such decomposition can be done for any complete problem with respect to the given class, and that two such decompositions are non-equivalent in general. Interestingly, our results are based on simple and well-known properties of first-order reductions.ow that this result can be generalized over a wide collection of complexity classes, including the so-called nice classes. Moreover, we show that such decomposition can be done for any complete problem with respect to the given class, and that two such decompositions are non-equivalent in general. Interestingly, our results are based on simple and well-known properties of first-order reductions.