A Fixed-Parameter Perspective on #BIS

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class # RH Π 1 . It is believed that #BIS does not have an efficient approximation algorithm but also that it is not...

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Veröffentlicht in:Algorithmica Jg. 81; H. 10; S. 3844 - 3864
Hauptverfasser: Curticapean, Radu, Dell, Holger, Fomin, Fedor, Goldberg, Leslie Ann, Lapinskas, John
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.10.2019
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Approximation
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Mathematical analysis
Mathematics of Computing
Parameterization
Parameters
Polynomials
Special Issue: Parameterized and Exact Computation
Theory of Computation
Parameterised complexity
Independent sets
Approximate counting
ISSN:0178-4617, 1432-0541
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Zusammenfassung:The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class # RH Π 1 . It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in # RH Π 1 , and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.
Bibliographie:ObjectType-Article-1
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00606-4