Universal Factor Graphs

TL;DR

This paper introduces the concept of universal factor graphs in constraint satisfaction problems, exploring their properties and implications for solving and approximating problems like max-kSAT, with connections to PCP and coding theory.

Contribution

It systematically studies universal factor graphs, presenting initial results for max-kSAT and establishing connections with PCP, preprocessing, and coding theory.

Findings

Defined universal factor graphs and their families.

Presented initial results for max-kSAT.

Connected universal factor graphs to PCP and coding theory.

Abstract

The factor graph of an instance of a symmetric constraint satisfaction problem on n Boolean variables and m constraints (CSPs such as k-SAT, k-AND, k-LIN) is a bipartite graph describing which variables appear in which constraints. The factor graph describes the instance up to the polarity of the variables, and hence there are up to 2km instances of the CSP that share the same factor graph. It is well known that factor graphs with certain structural properties make the underlying CSP easier to either solve exactly (e.g., for tree structures) or approximately (e.g., for planar structures). We are interested in the following question: is there a factor graph for which if one can solve every instance of the CSP with this particular factor graph, then one can solve every instance of the CSP regardless of the factor graph (and similarly, for approximation)? We call such a factor graph universal. As one needs different factor graphs for different values of n and m, this gives rise to the notion of a family of universal factor graphs. We initiate a systematic study of universal factor graphs, and present some results for max-kSAT. Our work has connections with the notion of preprocessing as previously studied for closest codeword and closest lattice-vector problems, with proofs for the PCP theorem, and with tests for the long code. Many questions remain open.