Balanced incomplete block designs and exact satisfiability

TL;DR

This paper establishes a connection between balanced incomplete block designs and linear CNF formulas, showing how properties in one domain translate to the other, including NP-completeness results for certain design problems.

Contribution

It introduces a novel correspondence between BIBDs and Boolean formulas, enabling transfer of results and proving NP-completeness of finding parallel classes in BIBDs.

Findings

Parallel classes in BIBDs correspond to XSAT solutions.

Finding a parallel class in a BIBD is NP-complete.

Results transfer between combinatorial design theory and satisfiability problems.

Abstract

The paper explores the correspondence between balanced incomplete block designs (BIBD) and certain linear CNF formulas by identifying the points of a block design with the clauses of the Boolean formula and blocks with Boolean variables. Parallel classes in BIBDs correspond to XSAT solutions in the corresponding formula. This correspondence allows for transfers of results from one field to the other. As a new result we deduce from known satisfiability theorems that the problem of finding a parallel class in a partially balanced incomplete block design is NP-complete.