On Two Techniques of Combining Branching and Treewidth

Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations of...

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Bibliographic Details
Published in:Algorithmica Vol. 54; no. 2; pp. 181 - 207
Main Authors: Fomin, Fedor V., Gaspers, Serge, Saurabh, Saket, Stepanov, Alexey A.
Format: Journal Article Conference Proceeding
Language:English
Published: New York Springer-Verlag 01.06.2009
Springer
Springer Verlag
Subjects:
Algorithm Analysis and Problem Complexity
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Combinatorics
Combinatorics. Ordered structures
Computer Science
Computer science; control theory; systems
Computer Systems Organization and Communication Networks
Data Structures and Algorithms
Data Structures and Information Theory
Discrete Mathematics
Exact sciences and technology
Graph theory
Information retrieval. Graph
Mathematics
Mathematics of Computing
Sciences and techniques of general use
Theoretical computing
Theory of Computation
Exact exponential time algorithms
3-Coloring
Minimum maximal matching
Weighted vertex cover
Parameterized algorithms
NP hard problems
Minimum dominating set
Treewidth
Branching
Dominating set
Graph theory
k-Weighted vertex cover
Maximal Matching
NP hard problem
Graph degree
Graph colouring
Dynamic programming
Graph covering
Fast algorithm
Counting
Graph matching
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations of these techniques. The idea behind these algorithms is very natural: If a parameter like the treewidth of a graph is small, algorithms based on dynamic programming perform well. On the other side, if the treewidth is large, then there must be vertices of high degree in the graph, which is good for branching algorithms. We give several examples of possible combinations of branching and programming which provide the fastest known algorithms for a number of NP hard problems. All our algorithms require non-trivial balancing of these two techniques. In the first approach the algorithm either performs fast branching, or if there is an obstacle for fast branching, this obstacle is used for the construction of a path decomposition of small width for the original graph. Using this approach we give the fastest known algorithms for Minimum Maximal Matching and for counting all 3-colorings of a graph. In the second approach the branching occurs until the algorithm reaches a subproblem with a small number of edges (and here the right choice of the size of subproblems is crucial) and then dynamic programming is applied on these subproblems of small width. We exemplify this approach by giving the fastest known algorithm to count all minimum weighted dominating sets of a graph. We also discuss how similar techniques can be used to design faster parameterized algorithms.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-007-9133-3