Treewidth-Aware Cycle Breaking for Algebraic Answer Set Counting

Abstract

Probabilistic reasoning, parameter learning, and most probable explanation inference for answer set programming have recently received growing attention. They are only some of the problems that can be formulated as Algebraic Answer Set Counting (AASC) problems. The latter are however hard to solve, and efficient evaluation techniques are needed. Inspired by Vlasser et al.'s Tp-compilation (JAR, 2016), we introduce Tp-unfolding, which employs forward reasoning to break the cycles in the positive dependency graph of a program by unfolding them. Tp-unfolding is defined for any normal answer set program and unfolds programs with respect to unfolding sequences, which are akin to elimination orders in SAT-solving. Using "good" unfolding sequences, we can ensure that the increase of the treewidth of the unfolded program is small. Treewidth is a measure adhering to a program's tree-likeness, which gives performance guarantees for AASC. We give sufficient conditions for the existence of good unfolding sequences based on the novel notion of component-boosted backdoor size, which measures the cyclicity of the positive dependencies in a program. The experimental evaluation of a prototype implementation, the AASC solver aspmc, shows promising results.

Treewidth-aware reductions of normal ASP to SAT – Is normal ASP harder than SAT after all?

“…In both cyclic as well as acyclic scenarios, our reduction and adaptions thereof seems promising and one might consider treewidth-aware reductions as a preprocessing tool in a portfolio setting. Interestingly, the reduction developed in this work already lead to follow-up studies [39] and treewidth-aware cycle breaking [36].…”

Exploiting Database Management Systems and Treewidth for Counting

“…We require a one-to-one correspondence between the answer sets and the satisfying assignments for the translation. Unfortunately, existing answer set counters focus on extended functionality like probabilistic reasoning (Fichte et al, 2022c), algebraic semirings (Eiter et al, 2021), or are tailored towards approximate counting (Kabir et al, 2022) or certain structural restrictions of the instance (Fichte et al, 2017). Therefore, we omit tools listed in (a) from an evaluation.…”

“…Set (S3) includes a very small set of instances of combinatorial problems. The instances in sets (S1) and (S2) have been used in previous works on ASP and counting (Eiter et al, 2021;Besin et al, 2021;Hecher, 2022). Set (S1) encodes finding extensions of an argumentation framework (Fichte et al, 2022b;Dvořák et al, 2020;Gaggl et al, 2020).…”

On the Structural Hardness of Answer Set Programming: Can Structure Efficiently Confine the Power of Disjunctions?

“…Parameterized complexity (Cygan et al 2015;Niedermeier 2006;Downey and Fellows 2013;Flum and Grohe 2006), offers a framework, enabling to analyze a problem's hardness in terms of certain parameter(s), which has been extensively applied to ASP (Gottlob, Scarcello, and Sideri 2002;Gottlob, Pichler, and Wei 2010;Lackner and Pfandler 2012;Fichte, Kronegger, and Woltran 2019). For ASP there is growing research on the prominent structural parameter treewidth (Jakl, Pichler, and Woltran 2009;Calimeri et al 2016;Bichler, Morak, and Woltran 2020;Bliem et al 2020;Eiter, Hecher, and Kiesel 2021). Intuitively, the measure treewidth enables the solving of numerous combinatorially hard problems in parts.…”