Answer Set Solving exploiting Treewidth and its Limits

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.

[1]  Stefan Woltran,et al.  Answer Set Solving with Bounded Treewidth Revisited , 2017, LPNMR.

[2]  Stefan Rümmele,et al.  A New Tree-Decomposition Based Algorithm for Answer Set Programming , 2011, 2011 IEEE 23rd International Conference on Tools with Artificial Intelligence.

[3]  Stefan Woltran,et al.  Exploiting Treewidth for Projected Model Counting and its Limits , 2018, SAT.

[4]  Johannes Klaus Fichte,et al.  TE-ETH: Lower Bounds for QBFs of Bounded Treewidth , 2019, ArXiv.

[5]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[6]  Stefan Woltran,et al.  Weighted Model Counting on the GPU by Exploiting Small Treewidth , 2018, ESA.

[7]  Johannes Klaus Fichte,et al.  Exploiting Treewidth for Counting Projected Answer Sets , 2018, KR.

[8]  Johannes Klaus Fichte,et al.  Default Logic and Bounded Treewidth , 2017, LATA.

[9]  Stefan Woltran,et al.  Expansion-based QBF Solving on Tree Decompositions , 2017, RCRA@AI*IA.

[10]  Florian Lonsing,et al.  Evaluating QBF Solvers: Quantifier Alternations Matter , 2018, CP.

[11]  Arne Meier,et al.  Counting Complexity for Reasoning in Abstract Argumentation , 2018, AAAI.

[12]  Martin Gebser,et al.  Answer Set Solving in Practice , 2012, Answer Set Solving in Practice.

[13]  Dániel Marx,et al.  Slightly superexponential parameterized problems , 2011, SODA '11.

[14]  Stefan Woltran,et al.  Dynamic Programming on Tree Decompositions in Practice - Some Lessons Learned , 2015, 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC).

[15]  Stefan Szeider,et al.  An SMT Approach to Fractional Hypertree Width , 2018, CP.

[16]  Stefan Woltran,et al.  D-FLAT2: Subset Minimization in Dynamic Programming on Tree Decompositions Made Easy , 2016, Fundam. Informaticae.

[17]  B. Mohar,et al.  Graph Minors , 2009 .

[18]  Martin Gebser,et al.  ASP-Core-2 Input Language Format , 2019, Theory and Practice of Logic Programming.

[19]  Arie M. C. A. Koster,et al.  Combinatorial Optimization on Graphs of Bounded Treewidth , 2008, Comput. J..

[20]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[21]  Miroslaw Truszczynski,et al.  Answer set programming at a glance , 2011, Commun. ACM.