Johannes K. Fichte
ABSTRACT
The problem of deciding the validity (QSat) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a successful parameter. A well-known result by Chen [9] is that QSat when parameterized by the treewidth of the primal graph and the quantifier rank of the input formula is fixed-parameter tractable. More precisely, the runtime of such an algorithm is polynomial in the formula size and exponential in the treewidth, where the exponential function in the treewidth is a tower, whose height is the quantifier rank. A natural question is whether one can significantly improve these results and decrease the tower while assuming the Exponential Time Hypothesis (ETH). In the last years, there has been a growing interest in the quest of establishing lower bounds under ETH, showing mostly problem-specific lower bounds up to the third level of the polynomial hierarchy. Still, an important question is to settle this as general as possible and to cover the whole polynomial hierarchy. In this work, we show lower bounds based on the ETH for arbitrary QBFs parameterized by treewidth and quantifier rank. More formally, we establish lower bounds for QSat and treewidth, namely, that under ETH there cannot be an algorithm that solves QSat of quantifier rank i in runtime significantly better than i-fold exponential in the treewidth and polynomial in the input size. In doing so, we provide a reduction technique to compress treewidth that encodes dynamic programming on arbitrary tree decompositions. Further, we describe a general methodology for a more finegrained analysis of problems parameterized by treewidth that are at higher levels of the polynomial hierarchy. Finally, we illustrate the usefulness of our results by discussing various applications of our results to problems that are located higher on the polynomial hierarchy, in particular, various problems from the literature such as projected model counting problems.
- Albert Atserias and Sergi Oliva. 2014. Bounded-width QBF is PSPACE-complete. J. Comput. Syst. Sci. 80, 7 (2014), 1415--1429. https://doi.org/10.1016/j.jcss.2014.04.014Google Scholar
Cross Ref
- Rehan A. Aziz. 2015. Answer Set Programming: Founded Bounds and Model Counting. Ph.D. Dissertation. Department of Computing and Information Systems, The University of Melbourne.Google Scholar
- Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh (Eds.). 2009. Handbook of Satisfiability. FAIA, Vol. 185. IOS Press.Google Scholar
- Hans L. Bodlaender. 1988. Dynamic Programming on Graphs with Bounded Treewidth. In Proceedings of the 15th International Colloquium on Automata, Languages and Programming (ICALP'88) (LNCS), Vol. 317. Springer, 105--118. https://doi.org/10.1007/3-540-19488-6Google ScholarCross Ref
- Hans L. Bodlaender. 1996. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 6 (1996), 1305--1317. https://doi.org/10.1137/S0097539793251219Google ScholarDigital Library
- Hans L. Bodlaender and Arie M. Koster. 2008. Combinatorial Optimization on Graphs of Bounded Treewidth. Comput. J. 51, 3 (2008), 255--269. https://doi.org/10.1093/comjnl/bxm037Google ScholarDigital Library
- John A. Bondy and Uppaluri S. R. Murty. 2008. Graph theory. Graduate Texts in Mathematics, Vol. 244. Springer. 655 pages. https://doi.org/10.1007/978-1-84628-970-5Google Scholar
- Günther Charwat and Stefan Woltran. 2019. Expansion-based QBF Solving on Tree Decompositions. Fundam. Inform. 167, 1-2 (2019), 59--92. https://doi.org/10.3233/FI-2019-1810Google ScholarCross Ref
- Hubie Chen. 2004. Quantified Constraint Satisfaction and Bounded Treewidth. In Proceedings of the 16th European Conference on Artificial Intelligence (ECAI'04), Vol. IOS Press. 161--170.Google Scholar
- Markus Chimani, Petra Mutzel, and Bernd Zey. 2012. Improved Steiner tree algorithms for bounded treewidth. J. Discrete Algorithms 16 (2012), 67--78. https://doi.org/10.1016/j.jda.2012.04.016Google ScholarDigital Library
- Bruno Courcelle. 1990. Graph Rewriting: An Algebraic and Logic Approach. In Handbook of Theoretical Computer Science, Vol. B. Elsevier, 193--242. https://doi.org/10.1016/b978-0-444-88074-1.50010-xGoogle ScholarDigital Library
- Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. 2001. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discr. Appl. Math. 108, 1-2 (2001), 23--52. https://doi.org/10.1016/S0166-218X(00)00221-3Google ScholarDigital Library
- Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. 2015. Parameterized Algorithms. Springer. XVII, 613 pages. https://doi.org/10.1007/978-3-319-21275-3Google Scholar
- Rina Dechter. 2006. Tractable Structures for Constraint Satisfaction Problems. In Handbook of Constraint Programming. Vol. I. Elsevier, Chapter 7, 209--244. https://doi.org/10.1016/S1574-6526(06)80011-8Google Scholar
- Reinhard Diestel. 2012. Graph Theory, 4th Edition. Graduate Texts in Mathematics, Vol. 173. Springer. 410 pages.Google Scholar
- Rodney G. Downey and Michael R. Fellows. 2013. Fundamentals of Parameterized Complexity. Springer. https://doi.org/10.1007/978-1-4471-5559-1Google Scholar
- Arnaud Durand, Miki Hermann, and Phokion G. Kolaitis. 2005. Sub-tractive reductions and complete problems for counting complexity classes. Theor. Comput. Sci. 340, 3 (2005), 496--513. https://doi.org/10.1016/j.tcs.2005.03.012Google ScholarDigital Library
- Wolfgang Dvořák, Reinhard Pichler, and Stefan Woltran. 2012. Towards fixed-parameter tractable algorithms for abstract argumentation. Artif. Intell. 186 (2012), 1--37. https://doi.org/10.1016/j.artint.2012.03.005Google ScholarDigital Library
- Uwe Egly, Thomas Eiter, Hans Tompits, and Stefan Woltran. 2000. Solving Advanced Reasoning Tasks Using Quantified Boolean Formulas. In Proceedings of the Seventeenth National Conference on Artificial Intelligence and Twelfth Conference on on Innovative Applications of Artificial Intelligence (AAAI/IAAI 2000). AAAI Press / The MIT Press, 417--422.Google ScholarDigital Library
- Thomas Eiter and Georg Gottlob. 1995. The Complexity of Logic-Based Abduction. J. ACM 42, 1 (1995), 3--42. https://doi.org/10.1145/200836.200838Google ScholarDigital Library
- Thomas Eiter and Georg Gottlob. 1995. On the computational cost of disjunctive logic programming: Propositional case. Ann. Math. Artif. Intell. 15, 3-4 (1995), 289--323. https://doi.org/10.1007/BF01536399Google ScholarCross Ref
- Michael Elberfeld, Andreas Jakoby, and Till Tantau. 2010. Logspace versions of the theorems of Bodlaender and Courcelle. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS'10). IEEE, 143--152. https://doi.org/10.1109/FOCS.2010.21Google ScholarDigital Library
- Ronald Fagin. 1974. Generalized First-Order Spectra and Polynomial-Time Recognizable Sets. In Proceedings of the 7th Symposia in Applied Mathematics. AMS.Google Scholar
- Alex Ferguson and Barry O'Sullivan. 2007. Quantified Constraint Satisfaction Problems: From Relaxations to Explanations. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI 2007). The AAAI Press, 74--79.Google Scholar
- Johannes K. Fichte and Markus Hecher. 2019. Treewidth and Counting Projected Answer Sets. In Proceedings of the 15th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2019) (LNCS), Vol. 11481. Springer, 105--119. https://doi.org/10.1007/978-3-030-20528-7_9Google Scholar
- Johannes K. Fichte, Markus Hecher, and Arne Meier. 2019. Counting Complexity for Reasoning in Abstract Argumentation. In Proceedings of the 33rd AAAI Conference on Artificial Intelligence, (AAAI 2019). The AAAI Press, 2827--2834. https://doi.org/10.1609/aaai.v33i01.33012827Google ScholarCross Ref
- Johannes K. Fichte, Markus Hecher, Michael Morak, and Stefan Woltran. 2018. Exploiting Treewidth for Projected Model Counting and its Limits. In Proceedings of the 21th International Conference on Theory and Applications of Satisfiability Testing (SAT'18) (LNCS), Vol. 10929. Springer, 165--184. https://doi.org/10.1007/978-3-319-94144-8_11Google ScholarDigital Library
- Johannes K. Fichte, Markus Hecher, Patrick Thier, and Stefan Woltran. 2020. Exploiting Database Management Systems and Treewidth for Counting. In Proceedings of the 22nd International Symposium on Practical Aspects of Declarative Languages (PADL 2020) (LNCS), Vol. 12007. Springer, 151--167. https://doi.org/10.1007/978-3-030-39197-3_10Google ScholarCross Ref
- Johannes K. Fichte and Stefan Szeider. 2015. Backdoors to Tractable Answer-Set Programming. Artif. Intell. 220, C (2015), 64--103. https://doi.org/10.1016/j.artint.2014.12.001Google Scholar
- Jörg Flum and Martin Grohe. 2006. Parameterized Complexity Theory. Springer. 495 pages. https://doi.org/10.1007/3-540-29953-XGoogle Scholar
- Eugene C. Freuder. 1985. A sufficient condition for backtrack-bounded search. J. ACM 32, 4 (1985), 755--761. https://doi.org/10.1145/4221.4225Google ScholarDigital Library
- Michael R. Garey and David S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman.Google Scholar
- Georg Gottlob, Reinhard Pichler, and Fang Wei. 2010. Bounded treewidth as a key to tractability of knowledge representation and reasoning. Artif. Intell. 174, 1 (2010), 105--132. https://doi.org/10.1016/j.artint.2009.10.003Google ScholarDigital Library
- Martin Grohe. 2017. Descriptive Complexity, Canonisation, and Definable Graph Structure Theory. Vol. 47. Cambridge University Press.Google Scholar
- Markus Hecher, Michael Morak, and Stefan Woltran. 2020. Structural Decompositions of Epistemic Logic Programs. In Proceedings of the 34th AAAI Conference on Artificial Intelligence, (AAAI 2020). The AAAI Press. In press.Google ScholarCross Ref
- Neil Immerman. 1999. Descriptive complexity. Springer.Google Scholar
- Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. 2001. Which Problems Have Strongly Exponential Complexity? J. Comput. Syst. Sci. 63, 4 (2001), 512--530. https://doi.org/10.1006/jcss.2001.1774Google ScholarDigital Library
- Michael Jakl, Reinhard Pichler, and Stefan Woltran. 2009. Answer-Set Programming with Bounded Treewidth. In Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI 2009), Vol. 2. 816--822.Google Scholar
- Hans Kleine Büning and Theodor Lettman. 1999. Propositional Logic: Deduction and Algorithms. Cambridge University Press, New York, NY, USA. 420 pages.Google ScholarDigital Library
- Ton Kloks. 1994. Treewidth, Computations and Approximations. LNCS, Vol.842. Springer. https://doi.org/10.1007/BFb0045375Google ScholarCross Ref
- Michael Lampis, Stefan Mengel, and Valia Mitsou. 2018. QBF as an Alternative to Courcelle's Theorem. In Proceedings of the 21th International Conference on Theory and Applications of Satisfiability Testing (SAT'18). Springer, 235--252. https://doi.org/10.1007/978-3-319-94144-8_15Google ScholarDigital Library
- Michael Lampis and Valia Mitsou. 2017. Treewidth with a Quantifier Alternation Revisited. In Proceedings of the 12th International Symposium on Parameterized and Exact Computation (IPEC'17) (LIPIcs), Vol. 89. Dagstuhl Publishing, 26:1--26:12. https://doi.org/10.4230/LIPIcs.IPEC.2017.26Google Scholar
- Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. 2018. Slightly Superexponential Parameterized Problems. SIAM J. Comput. 47, 3 (2018), 675--702. https://doi.org/10.1137/16M1104834Google ScholarCross Ref
- Florian Lonsing and Uwe Egly. 2018. Evaluating QBF Solvers: Quantifier Alternations Matter. In Proceedings of the 24th International Conference on Principles and Practice of Constraint Programming (CP 2018) (LNCS), Vol. 11008. Springer, 276--294. https://doi.org/10.1007/978-3-319-98334-9_19Google ScholarCross Ref
- Dániel Marx and Valia Mitsou. 2016. Double-Exponential and Triple-Exponential Bounds for Choosability Problems Parameterized by Treewidth. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) (LIPIcs), Vol. 55. Dagstuhl Publishing, 28:1--28:15. https://doi.org/10.4230/LIPIcs.ICALP.2016.28Google Scholar
- Sebastian Ordyniak and Stefan Szeider. 2013. Parameterized Complexity Results for Exact Bayesian Network Structure Learning. J. Artif. Intell. Res. 46 (2013), 263--302. https://doi.org/10.1613/jair.3744Google ScholarCross Ref
- Guoqiang Pan and Moshe Y. Vardi. 2006. Fixed-Parameter Hierarchies inside PSPACE. In LICS. IEEE Computer Society, 27--36. https://doi.org/10.1109/LICS.2006.25Google Scholar
- Christos H. Papadimitriou. 1994. Computational Complexity. Addison-Wesley. 523 pages.Google Scholar
- Reinhard Pichler, Stefan Rümmele, and Stefan Woltran. 2010. Counting and Enumeration Problems with Bounded Treewidth. In Proceedings of the 16th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR'10) (LNCS), Vol. 6355. Springer, 387--404. https://doi.org/10.1007/978-3-642-17511-4_22Google ScholarCross Ref
- Neil Robertson and Paul D. Seymour. 1983. Graph Minors. I. Excluding a Forest. J. Comb. Theory, Ser. B 35, 1 (1983), 39--61. https://doi.org/10.1016/0095-8956(83)90079-5Google ScholarCross Ref
- Neil Robertson and Paul D. Seymour. 1984. Graph Minors. III. Planar Tree-Width. J. Comb. Theory, Ser. B 36, 1 (1984), 49--64. https://doi.org/10.1016/0095-8956(84)90013-3Google ScholarCross Ref
- Neil Robertson and Paul D. Seymour. 1985. Graph Minors -- a Survey. In Surveys in Combinatorics 1985: Invited Papers for the 10th British Combinatorial Conference (London Mathematical Society Lecture Note Series). Cambridge University Press, 153--171. https://doi.org/10.1017/CBO9781107325678.009Google Scholar
- Neil Robertson and Paul D. Seymour. 1986. Graph Minors. II. Algorithmic Aspects of Tree-Width. J. Algorithms 7, 3 (1986), 309--322. https://doi.org/10.1016/0196-6774(86)90023-4Google ScholarCross Ref
- Neil Robertson and Paul D. Seymour. 1991. Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52, 2 (1991), 153--190.Google ScholarDigital Library
- Marko Samer and Stefan Szeider. 2010. Algorithms for propositional model counting. J. Discrete Algorithms 8, 1 (2010), 50--64. https://doi.org/10.1016/j.jda.2009.06.002Google ScholarDigital Library
- Yi-Dong Shen and Thomas Eiter. 2017. Evaluating Epistemic Negation in Answer Set Programming (Extended Abstract). In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI 2017). ijcai.org, 5060--5064. https://doi.org/10.24963/ijcai.2017/722Google ScholarCross Ref
- Larry J. Stockmeyer and Albert R. Meyer. 1973. Word problems requiring exponential time. In Proceedings of the 5th Annual ACM Symposium on Theory of Computing (STOC'73). ACM, 1--9. https://doi.org/10.1145/800125.804029Google Scholar
Index Terms
- Lower Bounds for QBFs of Bounded Treewidth
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