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Exact algorithms for finding minimum transversals in rank-3 hypergraphs
Linköping University, Department of Computer and Information Science, TCSLAB - Theoretical Computer Science Laboratory. Linköping University, The Institute of Technology.
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).

Place, publisher, year, edition, pages
2004. Vol. 51, no 2, 107-121 p.
Keyword [en]
Minimum transversal, Hypergraph, 3-Hitting Set, Exact algorithm
National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-14395DOI: 10.1016/j.jalgor.2004.01.001OAI: diva2:23416
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2009-06-08
In thesis
1. Algorithms, measures and upper bounds for satisfiability and related problems
Open this publication in new window or tab >>Algorithms, measures and upper bounds for satisfiability and related problems
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.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1079
Exact algorithms, upper bounds, algorithm analysis, satisfiability
National Category
Computer Science
urn:nbn:se:liu:diva-8714 (URN)978-91-85715-55-8 (ISBN)
Public defence
2007-04-27, Visionen, B-huset, Linköpings universitet, Linköping, 13:15 (English)
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2014-04-24

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