Graph-like Scheduling Problems and Property B

TL;DR

This paper studies graph-like scheduling problems, a class of Boolean formulas generalizing graph coloring, focusing on the existence of solutions with binary variables and establishing bounds for uniform cases using probabilistic and constructive methods.

Contribution

It introduces the concept of graph-like scheduling problems, defines $ ext{lambda}$-uniform variants, and provides bounds on minimal problem sizes without property B, highlighting open problems.

Findings

Bounds are established for the size of minimal $ ext{lambda}$-uniform problems without property B.

Both random and constructive methods are used to derive these bounds.

Finding tight bounds remains an open problem in this area.

Abstract

Breuer and Klivans defined a diverse class of scheduling problems in terms of Boolean formulas with atomic clauses that are inequalities. We consider what we call graph-like scheduling problems. These are Boolean formulas that are conjunctions of disjunctions of atomic clauses $(x_i \neq x_j)$. These problems generalize proper coloring in graphs and hypergraphs. We focus on the existence of a solution with all $x_i$ taking the value of $0$ or $1$ (i.e. problems analogous to the bipartite case). When a graph-like scheduling problem has such a solution, we say it has property B just as is done for $2$-colorable hypergraphs. We define the notion of a $\lambda$-uniform graph-like scheduling problem for any integer partition $\lambda$. Some bounds are attained for the size of the smallest $\lambda$-uniform graph-like scheduling problems without property B. We make use of both random and constructive methods to obtain bounds. Just as in the case of hypergraphs finding tight bounds remains an open problem.