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Algorithms, measures and upper bounds for satisfiability and related problems
Linköping University, Department of Computer and Information Science, TCSLAB - Theoretical Computer Science Laboratory. Linköping University, The Institute of Technology.
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The topic of exact, exponential-time algorithms for NP-hard 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 DPLL-style ("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 (non-trivial) restrictions on the applicability of a possible recursion.

We introduce two new kinds of complexity measures, representing two ways to add such restrictions on applicability to the analysis. In the first measure, the execution of the algorithm is viewed as moving between a finite set of states (such as the presence or absence of certain structures or properties), where the current state decides which branchings are applicable, and each branch of a branching contains information about the resultant state. In the second measure, it is instead the relative sizes of the modelled attributes (such as the average degree or other concepts of density) that controls the applicability of branchings.

We adapt both measures to Eppstein's framework, and use these tools to provide algorithms with stronger bounds for a number of problems. The problems we treat are satisfiability for sparse formulae, exact 3-satisfiability, 3-hitting set, and counting models for 2- and 3-satisfiability formulae, and in every case the bound we prove is stronger than previously known bounds.

Place, publisher, year, edition, pages
Linköping, Sweden: Department of Computer and Information Science, Linköpings universitet , 2007. , 234 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1079
Keyword [en]
Exact algorithms, upper bounds, algorithm analysis, satisfiability
National Category
Computer Science
Identifiers
URN: urn:nbn:se:liu:diva-8714ISBN: 978-91-85715-55-8 (print)OAI: oai:DiVA.org:liu-8714DiVA: diva2:23420
Public defence
2007-04-27, Visionen, B-huset, Linköpings universitet, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2014-04-24
List of papers
1. Counting Satisfying Assignments in 2-SAT and 3-SAT
Open this publication in new window or tab >>Counting Satisfying Assignments in 2-SAT and 3-SAT
2003 (English)Conference paper, Published paper (Other academic)
Abstract [en]

We present an O(1.3247n) algorithm for counting the number of satisfying assignments for instances of 2-SAT and an O(1.6894n) algorithm for instances of 3-SAT. This is an improvement compared to the previously best known algorithms running in O(1.381n) and O(1.739n) time, respectively.

Series
Lecture Notes in Computer Science, ISSN 0302-9743 (Print) 1611-3349 (Online) ; 2387
National Category
Computer Science
Identifiers
urn:nbn:se:liu:diva-14394 (URN)10.1007/3-540-45655-4_57 (DOI)978-3-540-43996-7 (ISBN)
Conference
8th Annual International Conference, COCOON 2002, Singapore, August 15-17, 2002
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2017-02-23
2. Exact algorithms for finding minimum transversals in rank-3 hypergraphs
Open this publication in new window or tab >>Exact algorithms for finding minimum transversals in rank-3 hypergraphs
2004 (English)In: Journal of Algorithms, ISSN 0196-6774, Vol. 51, no 2, 107-121 p.Article in journal (Refereed) Published
Abstract [en]

We present two algorithms for the problem of finding a minimum transversal in a hypergraph of rank 3, also known as the 3-Hitting Set problem. This problem is a natural extension of the vertex cover problem for ordinary graphs. The first algorithm runs in time O(1.6538n) for a hypergraph with n vertices, and needs polynomial space. The second algorithm uses exponential space and runs in time O(1.6316n).

Keyword
Minimum transversal, Hypergraph, 3-Hitting Set, Exact algorithm
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-14395 (URN)10.1016/j.jalgor.2004.01.001 (DOI)
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2009-06-08
3. Counting models for 2sat and 3sat formulae
Open this publication in new window or tab >>Counting models for 2sat and 3sat formulae
2005 (English)In: Theoretical Computer Science, ISSN 0304-3975, Vol. 332, no 1-3, 265-291 p.Article in journal (Refereed) Published
Abstract [en]

We here present algorithms for counting models and max-weight models for 2SAT and 3SAT formulae. They use polynomial space and run in O(1.2561n) and O(1.6737n) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting non-weighted models for 2SAT and 3SAT, which run in O(1.3247n) and O(1.6894n) time, respectively. In order to prove these time bounds, we develop new measures of formula complexity, allowing us to conveniently analyze the effects of certain factors with a large impact on the total running time. We also provide an algorithm for the restricted case of separable 2SAT formulae, with fast running times for well-studied input classes. For all three algorithms we present interesting applications, such as computing the permanent of sparse 0/1 matrices.

Keyword
Counting models; Satisfiability; Exponential-time algorithms; Exact algorithms; Upper bounds
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-14396 (URN)10.1016/j.tcs.2004.10.037 (DOI)
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2017-02-23
4. Faster exact solving of SAT formulae with a low number of occurrences per variable
Open this publication in new window or tab >>Faster exact solving of SAT formulae with a low number of occurrences per variable
2005 (English)In: Theory and Applications of Satisfiability Testing, 8th International Conference, SAT 2005, 2005, 309-323 p.Conference paper, Published paper (Other academic)
Abstract [en]

We present an algorithm that decides the satisfiability of a formula F on CNF form in time O(1.1279(d – 2)n), if F has at most d occurrences per variable or if F has an average of d occurrences per variable and no variable occurs only once. For d 4, this is better than previous results. This is the first published algorithm that is explicitly constructed to be efficient for cases with a low number of occurrences per variable. Previous algorithms that are applicable to this case exist, but as these are designed for other (more general, or simply different) cases, their performance guarantees for this case are weaker.

Series
Lecture Notes in Computer Science, ISSN 0302-9743 (Print) 1611-3349 (Online) ; 3569
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-14397 (URN)10.1007/11499107_23 (DOI)978-3-540-26276-3 (ISBN)
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2009-06-08
5. An algorithm for the sat problem for formulae of linear length
Open this publication in new window or tab >>An algorithm for the sat problem for formulae of linear length
2005 (English)In: Algorithms – ESA 2005: 13th Annual European Symposium, Palma de Mallorca, Spain, October 3-6, 2005., 2005, 107 -118 p.Conference paper, Published paper (Other academic)
Abstract [en]

We present an algorithm that decides the satisfiability of a CNF formula where every variable occurs at most k times in time for some c (that is, O(αN) with for every fixed k). For k ≤ 4, the results coincide with an earlier paper where we achieved running times of O(20.1736N) for k = 3 and O(20.3472N) for k = 4 (for k ≤ 2, the problem is solvable in polynomial time). For k>4, these results are the best yet, with running times of O(20.4629N) for k = 5 and O(20.5408N) for k = 6. As a consequence of this, the same algorithm is shown to run in time O(20.0926L) for a formula of length (total number of literals) L. The previously best bound in terms of L is O(20.1030L). Our bound is also the best upper bound for an exact algorithm for a 3sat formula with up to six occurrences per variable, and a 4sat formula with up to eight occurrences per variable.

Series
Lecture Notes in Computer Science, ISSN 0302-9743 (Print) 1611-3349 (Online) ; 3669
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-14398 (URN)10.1007/11561071 (DOI)978-3-540-29118-3 (ISBN)
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2009-06-08

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