Exploiting Treewidth for Projected Model Counting and its Limits

TL;DR

This paper presents a new fixed-parameter tractable algorithm for projected model counting that exploits small treewidth, providing tight bounds and analyzing its computational limits under ETH.

Contribution

The paper introduces a novel algorithm for PMC that leverages treewidth and establishes tight lower bounds based on ETH, advancing the understanding of PMC's complexity.

Findings

Algorithm runs in $O(2^{2^{k+4}} n^2)$ time for treewidth k

PMC is fixed-parameter tractable when parameterized by treewidth

Lower bounds match the algorithm's runtime under ETH

Abstract

In this paper, we introduce a novel algorithm to solve projected model counting (PMC). PMC asks to count solutions of a Boolean formula with respect to a given set of projected variables, where multiple solutions that are identical when restricted to the projected variables count as only one solution. Our algorithm exploits small treewidth of the primal graph of the input instance. It runs in time $O({2^{2^{k+4}} n^2})$ where k is the treewidth and n is the input size of the instance. In other words, we obtain that the problem PMC is fixed-parameter tractable when parameterized by treewidth. Further, we take the exponential time hypothesis (ETH) into consideration and establish lower bounds of bounded treewidth algorithms for PMC, yielding asymptotically tight runtime bounds of our algorithm.