A Faster Algorithm for Propositional Model Counting Parameterized by Incidence Treewidth

The propositional model counting problem (#SAT) is known to be fixed-parameter-tractable (FPT) when parameterized by the width k of a given tree decomposition of the incidence graph. The running time of the fastest known FPT algorithm contains the exponential factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4^k$$\end{document}. We improve this factor to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^k$$\end{document} by utilizing fast algorithms for computing the zeta transform and covering product of functions representing partial model counts, thereby achieving the same running time as FPT algorithms that are parameterized by the less general treewidth of the primal graph. Our new algorithm is asymptotically optimal unless the Strong Exponential Time Hypothesis (SETH) fails.

[1]  Gerhard J. Woeginger,et al.  Exact Algorithms for NP-Hard Problems: A Survey , 2001, Combinatorial Optimization.

[2]  Toniann Pitassi,et al.  Algorithms and complexity results for #SAT and Bayesian inference , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[3]  Marko Samer,et al.  Algorithms for propositional model counting , 2007, J. Discrete Algorithms.

[4]  Harvey,et al.  Integer multiplication in time O(n log n) , 2021, Annals of Mathematics.

[5]  Hans L. Bodlaender,et al.  Discovering Treewidth , 2005, SOFSEM.

[6]  Dániel Marx,et al.  Lower bounds based on the Exponential Time Hypothesis , 2011, Bull. EATCS.

[7]  Marko Samer,et al.  Constraint satisfaction with bounded treewidth revisited , 2006, J. Comput. Syst. Sci..

[8]  Russell Impagliazzo,et al.  Which problems have strongly exponential complexity? , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[9]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[10]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

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

[12]  Ton Kloks.  Treewidth, Computations and Approximations , 1994, Lecture Notes in Computer Science.

[13]  Fedor V. Fomin,et al.  Exact exponential algorithms , 2013, CACM.

[14]  Robert Kennes,et al.  Computational aspects of the Mobius transformation of graphs , 1992, IEEE Trans. Syst. Man Cybern..

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

[16]  Michal Pilipczuk,et al.  A ck n 5-Approximation Algorithm for Treewidth , 2016, SIAM J. Comput..

[17]  Arie M. C. A. Koster,et al.  Treewidth: Computational Experiments , 2001, Electron. Notes Discret. Math..

[18]  Andreas Björklund,et al.  Fourier meets möbius: fast subset convolution , 2006, STOC '07.

[19]  Dan Roth,et al.  On the Hardness of Approximate Reasoning , 1993, IJCAI.