Bounded degree QBF and positional games

TL;DR

This paper explores the complexity of bounded-degree QBF, revealing polynomial-time solvability for degree two formulas but PSPACE-completeness for degree three, impacting the complexity classification of related positional games.

Contribution

It extends the study of bounded-degree constraints from SAT to QBF, showing new complexity results and implications for positional games.

Findings

Polynomial-time decidability for degree two QBF formulas

PSPACE-completeness of 3-regular QBF formulas

PSPACE-completeness of various positional games on bounded-degree hypergraphs

Abstract

The study of SAT and its variants has provided numerous NP-complete problems, from which most NP-hardness results were derived. Due to the NP-hardness of SAT, adding constraints to either specify a more precise NP-complete problem or to obtain a tractable one helps better understand the complexity class of several problems. In 1984, Tovey proved that bounded-degree SAT is also NP-complete, thereby providing a tool for performing NP-hardness reductions even with bounded parameters, when the size of the reduction gadget is a function of the variable degree. In this work, we initiate a similar study for QBF, the quantified version of SAT. We prove that, like SAT, the truth value of a maximum degree two quantified formula is polynomial-time computable. However, surprisingly, while the truth value of a 3-regular 3-SAT formula can be decided in polynomial time, it is PSPACE-complete for a 3-regular QBF formula. A direct consequence of these results is that Avoider-Enforcer and Client-Waiter positional games are PSPACE-complete when restricted to bounded-degree hypergraphs. To complete the study, we also show that Maker-Breaker and Maker-Maker positional games are PSPACE-complete for bounded-degree hypergraphs.