# Treewidth and Counting Projected Answer Sets

## TL;DR

This paper presents new algorithms for counting projected answer sets in logic programming, exploiting small treewidth and establishing complexity bounds, advancing the efficiency and theoretical understanding of #PAs.

## Contribution

The paper introduces novel dynamic programming algorithms for projected answer set counting that leverage small treewidth and extends existing methods to various classes of logic programs.

## Key findings

- Algorithms run in polynomial time with double exponential dependence on treewidth for certain programs.
- Extended algorithms for tight, normal, and disjunctive programs improve counting efficiency.
- Lower bounds under ETH show no significant runtime improvements are possible.

## Abstract

In this paper, we introduce novel algorithms to solve projected answer set counting (#PAs). #PAs asks to count the number of answer sets with respect to a given set of projected atoms, where multiple answer sets that are identical when restricted to the projected atoms count as only one projected answer set. Our algorithms exploit small treewidth of the primal graph of the input instance by dynamic programming (DP). We establish a new algorithm for head-cycle-free (HCF) programs and lift very recent results from projected model counting to #PAs when the input is restricted to HCF programs. Further, we show how established DP algorithms for tight, normal, and disjunctive answer set programs can be extended to solve #PAs. Our algorithms run in polynomial time while requiring double exponential time in the treewidth for tight, normal, and HCF programs, and triple exponential time for disjunctive programs. Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for #PAs. Under ETH, one cannot significantly improve our obtained worst-case runtimes.