Computing Many Maximal Independent Sets for Hypergraphs in Parallel

A hypergraph is called uniformly δ-sparse if for every nonempty subset X ⊆ V of vertices, the average degree of the sub-hypergraph of induced by X is at most δ. We show that there is a deterministic algorithm that, given a uniformly δ-sparse hypergraph , and a positive integer k, outputs k or all minimal transversals for in O(δlog(1 + k)polylog(δ|V|))-time using |V|O(logδ)kO(δ) processors. Equivalently, the algorithm can be used to compute in parallel k or all maximal independent sets for .

[1]  Pierre Kelsen,et al.  A Simple NC-Algorithm for a Maximal Independent set in a Hypergraph of Poly-Log Arboricity , 1996, Inf. Process. Lett..

[2]  Leonid Khachiyan,et al.  On the Complexity of Dualization of Monotone Disjunctive Normal Forms , 1996, J. Algorithms.

[3]  Leonard Pitt,et al.  Efficient Read-Restricted Monotone CNF/DNF Dualization by Learning with Membership Queries , 1999, Machine Learning.

[4]  Vladimir Gurvich,et al.  A global parallel algorithm for the hypergraph transversal problem , 2007, Inf. Process. Lett..

[5]  Eugene L. Lawler,et al.  Generating all Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms , 1980, SIAM J. Comput..

[6]  R. Read.  Every one a Winner or how to Avoid Isomorphism Search when Cataloguing Combinatorial Configurations , 1978 .

[7]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[8]  Marek Karpinski,et al.  An Efficient Parallel Algorithm for Computing a Maximal Independent Set in a Hypergraph of Dimension 3 , 1991, Inf. Process. Lett..

[9]  Mark K. Goldberg,et al.  A New Parallel Algorithm for the Maximal Independent Set Problem , 1989, SIAM J. Comput..

[10]  Mark K. Goldberg,et al.  Constructing a Maximal Independent Set in Parallel , 1989, SIAM J. Discret. Math..

[11]  Eli Upfal,et al.  The Complexity of Parallel Search , 1988, J. Comput. Syst. Sci..

[12]  Georg Gottlob,et al.  Identifying the Minimal Transversals of a Hypergraph and Related Problems , 1995, SIAM J. Comput..

[13]  Tomasz Luczak,et al.  A Parallel Randomized Algorithm for Finding a Maximal Independent Set in a Linear Hypergraph , 1997, J. Algorithms.

[14]  P. Hammer,et al.  Dual subimplicants of positive Boolean functions , 1998 .

[15]  Piotr Indyk,et al.  A small approximately min-wise independent family of hash functions , 1999, SODA '99.

[16]  Mihalis Yannakakis,et al.  On Generating All Maximal Independent Sets , 1988, Inf. Process. Lett..

[17]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1985, JACM.

[18]  Vladimir Gurvich,et al.  An Efficient Incremental Algorithm for Generating All Maximal Independent Sets in Hypergraphs of Bounded Dimension , 2000, Parallel Process. Lett..

[19]  Vladimir Gurvich,et al.  Dual-Bounded Generating Problems: All Minimal Integer Solutions for a Monotone System of Linear Inequalities , 2002, SIAM J. Comput..