Characterizing Structural Hardness of Logic Programs: What makes Cycles and Reachability Hard for Treewidth?

TL;DR

This paper investigates the structural complexity of logic programs in ASP, focusing on how cycles and reachability contribute to computational hardness, and introduces a reduction that leverages treewidth to characterize problem difficulty.

Contribution

It presents a novel reduction from SAT to ASP that exploits structural properties to sublinearly reduce treewidth, providing a fine-grained hardness characterization.

Findings

Establishes a dependency between cycle length and treewidth for ASP hardness.

Shows that ASP evaluation can be slightly harder than SAT based on structural parameters.

Provides a reduction method that leverages ASP's structural power to analyze complexity.

Abstract

Answer Set Programming (ASP) is a problem modeling and solving framework for several problems in KR with growing industrial applications. Also for studies of computational complexity and deeper insights into the hardness and its sources, ASP has been attracting researchers for many years. These studies resulted in fruitful characterizations in terms of complexity classes, fine-grained insights in form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Recently, this lead to a novel result establishing that for the measure treewidth, which captures structural density of a program, the evaluation of the well-known class of normal programs is expected to be slightly harder than deciding satisfiability (SAT). However, it is unclear how to utilize this structural power of ASP. This paper deals with a novel reduction from SAT to normal ASP that goes beyond well-known encodings: We explicitly utilize the structural power of ASP, whereby we sublinearly decrease the treewidth, which probably cannot be significantly improved. Then, compared to existing results, this characterizes hardness in a fine-grained way by establishing the required functional dependency of the dependency graph's cycle length (SCC size) on the treewidth.