Universal Structures and the logic of Forbidden Patterns

TL;DR

This paper demonstrates that Forbidden Patterns Problems (FPPs) can be reduced to Constraint Satisfaction Problems (CSPs) for certain classes of structures with low complexity, extending the understanding of their logical expressiveness.

Contribution

It proves that FPPs over structures with low tree-depth decomposition are equivalent to CSPs, generalizing previous results and broadening the scope to arbitrary relational structures.

Findings

FPPs reduce to CSPs for classes with low tree-depth decomposition

Results extend to arbitrary relational structures

Enhances understanding of MMSNP logic expressiveness

Abstract

Forbidden Patterns Problems (FPPs) are a proper generalisation of Constraint Satisfaction Problems (CSPs). However, we show that when the input is connected and belongs to a class which has low tree-depth decomposition (e.g. structure of bounded degree, proper minor closed class and more generally class of bounded expansion) any FPP becomes a CSP. This result can also be rephrased in terms of expressiveness of the logic MMSNP, introduced by Feder and Vardi in relation with CSPs. Our proof generalises that of a recent paper by Nesetril and Ossona de Mendez. Note that our result holds in the general setting of problems over arbitrary relational structures (not just for graphs).