# Answer Set Solving exploiting Treewidth and its Limits

## TL;DR

This paper explores the use of treewidth-based parameterized algorithms to solve Answer Set Programming problems efficiently, introduces a framework for projected model counting, and establishes lower bounds on algorithm improvements.

## Contribution

It presents novel algorithms exploiting treewidth for ASP, extends to projected model counting, and introduces a methodology for proving lower bounds on runtime improvements.

## Key findings

- Algorithms for ASP leveraging treewidth are competitive in counting answer sets.
- Framework for projected model counting applies to ASP, argumentation, and higher polynomial hierarchy problems.
- Most worst-case runtimes of the proposed algorithms cannot be significantly improved under standard complexity assumptions.

## Abstract

Parameterized algorithms have been subject to extensive research of recent years and allow to solve hard problems by exploiting a parameter of the corresponding problem instances. There, one goal is to devise algorithms, where the runtime is exponential exclusively in this parameter. One particular well-studied structural parameter is treewidth. Typically, a parameterized algorithm utilizing treewidth takes or computes a tree decomposition, which is an arrangement of a graph into a tree, and evaluates the problem in parts by dynamic programming on the tree decomposition. In our research, we want to exploit treewidth in the context of Answer Set Programming (ASP), a declarative modeling and solving framework, which has been successfully applied in several application domains and industries for years. So far, we presented algorithms for ASP for the full ASP-Core-2 syntax, which is competitive especially when it comes to counting answer sets. Since dynamic programming on tree decomposition lands itself well to counting, we designed a framework for projected model counting, which applies to ASP, abstract argumentation and even to problems higher in the polynomial hierarchy. Given standard assumptions in computational complexity, we established a novel methodology for showing lower bounds, and we showed that most worst-case runtimes of our algorithms cannot be significantly improved.