Inclusion/Exclusion Meets Measure and Conquer

Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/e...

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Veröffentlicht in:	Algorithmica Jg. 69; H. 3; S. 685 - 740
Hauptverfasser:	Nederlof, Jesper, van Rooij, Johan M. M., van Dijk, Thomas C.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Boston Springer US 01.07.2014 Springer
Schlagworte:	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Data Structures and Information Theory Exact sciences and technology Information retrieval. Graph Mathematics of Computing Theoretical computing Theory of Computation Branching Exact exponential algorithm NP-hard Dominating set Inclusion/Exclusion Expert system Algorithmics Graph theory Computational complexity Polynomial time Upper bound System reduction Domination number NP hard problem Exponential time Time complexity Satisfiability
ISSN:	0178-4617, 1432-0541
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Abstract	Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times. In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number . This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set , and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k - Set Splitting and its generalisation k - Not-All-Equal Satisfiability .
AbstractList	Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times. In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number . This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set , and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k - Set Splitting and its generalisation k - Not-All-Equal Satisfiability .
Author	van Dijk, Thomas C. van Rooij, Johan M. M. Nederlof, Jesper
Author_xml	– sequence: 1 givenname: Jesper surname: Nederlof fullname: Nederlof, Jesper email: j.nederlof@uu.nl organization: Department of Information and Computing Sciences, Utrecht University – sequence: 2 givenname: Johan M. M. surname: van Rooij fullname: van Rooij, Johan M. M. organization: Consultants in Quantitative Methods – sequence: 3 givenname: Thomas C. surname: van Dijk fullname: van Dijk, Thomas C. organization: Lehrstuhl für Informatik I, Universität Würzburg
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Keywords	Branching Exact exponential algorithm NP-hard Dominating set Inclusion/Exclusion Expert system Algorithmics Graph theory Computational complexity Polynomial time Upper bound System reduction Domination number NP hard problem Exponential time Time complexity Satisfiability
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Snippet	Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Data Structures and Information Theory Exact sciences and technology Information retrieval. Graph Mathematics of Computing Theoretical computing Theory of Computation
Title	Inclusion/Exclusion Meets Measure and Conquer
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