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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Vydáno v:Algorithmica Ročník 69; číslo 3; s. 685 - 740
Hlavní autoři: Nederlof, Jesper, van Rooij, Johan M. M., van Dijk, Thomas C.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.07.2014
Springer
Témata:
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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Popis
Shrnutí: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 .
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-013-9759-2