Shortest reconfiguration paths in the solution space of Boolean formulas

TL;DR

This paper investigates the computational complexity of finding the shortest reconfiguration path between satisfying assignments of Boolean formulas, establishing a trichotomy of polynomial, NP-complete, or PSPACE-complete cases, and highlighting cases where shortest paths are efficiently computable.

Contribution

It provides a comprehensive complexity classification for shortest reconfiguration paths in Boolean formulas, including cases where shortest paths can be found efficiently despite differing from the symmetric difference.

Findings

Shortest reconfiguration path problem has a trichotomy: P, NP-complete, or PSPACE-complete.

Polynomial-time algorithms exist for certain cases where shortest paths differ from symmetric difference.

The work extends understanding of reconfiguration problems by identifying cases with efficiently computable shortest paths.

Abstract

Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment $s$ to another satisfying assignment $t$ of a Boolean formula. Earlier work characterized the complexity of finding any (not necessarily the shortest) sequence of flips from one satisfying assignment to another using Schaefer's framework for classification of Boolean formulas. We build on it to provide a trichotomy for the complexity of finding the shortest sequence of flips and show that it is either in P, NP-complete, or PSPACE-complete. Our result adds to the small set of complexity results known for shortest reconfiguration sequence problems by providing an example where the shortest sequence can be found in polynomial time even though its length is not equal to the symmetric difference of the values of the variables in $s$ and $t$. This is in contrast to all reconfiguration problems studied so far, where polynomial time algorithms for computing the shortest path were known only for cases where the path modified the symmetric difference only.