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New methods for 3SAT decision and worstcase analysis
 THEORETICAL COMPUTER SCIENCE
, 1999
"... We prove the worstcase upper bound 1:5045 n for the time complexity of 3SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2 and 3clauses, called "blocked clauses", generali ..."
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Cited by 74 (14 self)
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We prove the worstcase upper bound 1:5045 n for the time complexity of 3SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2 and 3clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3clausesets and can be applied also to arbitrary trees. Keywords: 3SAT, worstcase upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3SAT decision and prove the worstcase upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results...
Deterministic algorithms for kSAT based on covering codes and local search
 Proceedings of the 27th International Colloquium on Automata, Languages and Programming, ICALP'2000, volume 1853 of Lecture Notes in Computer Science
, 2000
"... Abstract. We show that satisfiability of formulas in kCNF can be decided deterministically in time close to (2k/(k + 1)) n, where n is the number of variables in the input formula. This is the best known worstcase upper bound for deterministic kSAT algorithms. Our algorithm can be viewed as a dera ..."
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Cited by 21 (10 self)
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Abstract. We show that satisfiability of formulas in kCNF can be decided deterministically in time close to (2k/(k + 1)) n, where n is the number of variables in the input formula. This is the best known worstcase upper bound for deterministic kSAT algorithms. Our algorithm can be viewed as a derandomized version of Schöning’s probabilistic algorithm presented in [15]. The key point of our algorithm is the use of covering codes together with local search. Compared to other “weakly exponential ” algorithms, our algorithm is technically quite simple. We also show how to improve the bound above by moderate technical effort. For 3SAT the improved bound is 1.481 n. 1
Algorithms, Measures and Upper Bounds for Satisfiability and Related Problems
 Department of Computer
, 2007
"... The topic of exact, exponentialtime algorithms for NPhard problems has received a lot of attention, particularly with the focus of producing algorithms with stronger theoretical guarantees, e.g. upper bounds on the running time on the form O (c n) for some c. Better methods of analysis may have an ..."
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Cited by 20 (1 self)
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The topic of exact, exponentialtime algorithms for NPhard problems has received a lot of attention, particularly with the focus of producing algorithms with stronger theoretical guarantees, e.g. upper bounds on the running time on the form O (c n) for some c. Better methods of analysis may have an impact not only on these bounds, but on the nature of the algorithms as well. The most classic method of analysis of the running time of dpllstyle (“branching ” or “backtracking”) recursive algorithms consists of counting the number of variables that the algorithm removes at every step. Notable improvements include Kullmann’s work on complexity measures, and Eppstein’s work on solving multivariate recurrences through quasiconvex analysis. Still, one limitation that remains in Eppstein’s framework is that it is difficult to introduce (nontrivial) restrictions on the applicability of a possible recursion.
Algorithms for counting 2SAT solutions and colorings with applications
 TR05033, Electronic Colloquium on Computational Complexity
, 2005
"... An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2Cnf formula. The worst case running time of ..."
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Cited by 19 (2 self)
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An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2Cnf formula. The worst case running time of
New upper bounds for MaxSat
 Charles University, Praha, Faculty of Mathematics and Physics
, 1998
"... We describe exact algorithms that provide new upper bounds for the Maximum Satisfiability problem (MaxSat). We prove that MaxSat can be solved in time O(F  · 1.3972 K), where F  is the length of a formula F in conjunctive normal form and K is the number of clauses in F. We also prove the time b ..."
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Cited by 14 (5 self)
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We describe exact algorithms that provide new upper bounds for the Maximum Satisfiability problem (MaxSat). We prove that MaxSat can be solved in time O(F  · 1.3972 K), where F  is the length of a formula F in conjunctive normal form and K is the number of clauses in F. We also prove the time bounds O(F  · 1.3995 k), where k is the maximum number of satisfiable clauses, and O((1.1279) F  ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K). An exponential time approximation algorithm by Dantsin et al. uses an exact algorithm for MaxSat as a building block and is therefore also improved.
A Tighter Bound for Counting MaxWeight Solutions to 2SAT Instances
"... We give an algorithm for counting the number of maxweight solutions to a 2SAT formula, and improve the bound on its running time to O (1.2377 n). The main source of the improvement is a refinement of the method of analysis, where we extend the concept of compound (piecewise linear) measures to mult ..."
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Cited by 8 (0 self)
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We give an algorithm for counting the number of maxweight solutions to a 2SAT formula, and improve the bound on its running time to O (1.2377 n). The main source of the improvement is a refinement of the method of analysis, where we extend the concept of compound (piecewise linear) measures to multivariate measures, also allowing the optimal parameters for the measure to be found automatically. This method extension should be of independent interest.
New upper bound for the #3SAT problem
"... We present a new deterministic algorithm for the #3SAT problem, based on the DPLL strategy. It uses a new approach for counting models of instances with low density. This allows us to assume the adding of more 2clauses than in previous algorithms. The algorithm achieves a running time of O(1.6423 ..."
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We present a new deterministic algorithm for the #3SAT problem, based on the DPLL strategy. It uses a new approach for counting models of instances with low density. This allows us to assume the adding of more 2clauses than in previous algorithms. The algorithm achieves a running time of O(1.6423 n) in the worst case which improves the current best bound of O(1.6737 n) by Dahllöf et al. 1
Determining the number of solutions to binary CSP instances
"... Counting the number of solutions to CSP instances has applications in several areas, ranging from statistical physics to artificial intelligence. We give an algorithm for counting the number of solutions to binary CSPs, which works by transforming the problem into a number of 2sat instances, where ..."
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Counting the number of solutions to CSP instances has applications in several areas, ranging from statistical physics to artificial intelligence. We give an algorithm for counting the number of solutions to binary CSPs, which works by transforming the problem into a number of 2sat instances, where the total number of solutions to these instances is the same as those of the original problem. The algorithm consists of two main cases, depending on whether the domain size d is even, in which case the algorithm runs in O(1.3247 n · (d/2) n) time, or odd, in which case it runs in O(1.3247 n · ((d 2 + d + 2)/4) n/2) if d = 4 · k + 1, and O(1.3247 n · ((d 2 + d)/4) n/2) if d = 4 · k + 3. We also give an algorithm for counting the number of possible 3colourings of a given graph, which runs in O(1.8171 n), an improvement over our general algorithm gained by using problem specific knowledge.
Computing #2SAT of Grids, GridCylinders and GridTori Boolean Formulas
"... We present an adaptation of transfer matrix method for signed grids, gridcylinders and gridtori. We use this adaptation to count the number of satisfying assignments of Boolean Formulas in 2CNF whose corresponding associated graph has such grid topologies. 1 ..."
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We present an adaptation of transfer matrix method for signed grids, gridcylinders and gridtori. We use this adaptation to count the number of satisfying assignments of Boolean Formulas in 2CNF whose corresponding associated graph has such grid topologies. 1
A Hierarchy of Tractable Subclasses for SAT and Counting SAT Problems
"... Finding subclasses of formulæ for which the SAT problem can be solved in polynomial time has been an important problem in computer science. We present a new hierarchy of propositional formulæ subclasses for which the SAT and counting SAT problems can be solved in polynomial time. Our tractable subcl ..."
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Finding subclasses of formulæ for which the SAT problem can be solved in polynomial time has been an important problem in computer science. We present a new hierarchy of propositional formulæ subclasses for which the SAT and counting SAT problems can be solved in polynomial time. Our tractable subclasses are those propositional formulæ in conjunctive normal form where any set of k +1clauses are related, i.e., there exists at least one literal in one clause that appears negated in another clause of the considered set of k +1clauses. We say this subclass of formulæ is of rank k and it is different from previously known subclasses that are solvable in polynomial time. This is an improvement over the SAT Dichotomy Theorem and the counting SAT Dichotomy Theorem, since our subclass can be moved out from the NPcomplete class to the P class. The membership problem for this new subclass can be solved in O(n · l k+1), where n, l and k are the number of variables, clauses and the rank (1 ≤ k ≤ l − 1), respectively. We give an efficient algorithm to approximate the number of assignments for any arbitrary conjunctive normal form propositional formula by an upper bound. 1