Limits of CSP Problems and Efficient Parameter Testing

TL;DR

This paper develops a unified framework for understanding the limits of CSPs and efficient parameter testing using array exchangeability and cut decomposition, avoiding traditional regularity partitions.

Contribution

It introduces a representation theorem for hypergraph limits and applies it to CSP limits, advancing the theoretical understanding of testability and sample complexity.

Findings

Hypergraph limits are representable via a new theorem.

Generalized ground state energies are shown to be efficiently testable.

Sample complexity bounds are established for testable hypergraph parameters.

Abstract

We present a unified framework on the limits of constraint satisfaction problems (CSPs) and efficient parameter testing which depends only on array exchangeability and the method of cut decomposition without recourse to the weakly regular partitions. In particular, we formulate and prove a representation theorem for compact colored $r$-uniform directed hypergraph ($r$-graph) limits, and apply this to $r$CSP limits. We investigate the sample complexity of testable $r$-graph parameters, we discuss the generalized ground state energies and demonstrate that they are efficiently testable.