Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

Amer E. Mouawad, Naomi Nishimura, Vinayak Pathak, Venkatesh Raman

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 an input Boolean formula. Earlier work characterized the complexity of deciding the existence of a sequence of flips between two given satisfying assignments 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 growing 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 the sequence flips variables that have the same value in both $s$ and $t$. This is in contrast to most reconfiguration problems studied so far, where polynomial-time algorithms for computing the shortest path were known only for cases where the path modified no more than the symmetric difference of $s$ and $t$. Our proof uses Birkhoff's representation theorem on a set system that we show to be a distributive lattice. The technique provides insights and can perhaps be used for other reconfiguration problems as well.