On parallelizing a greedy heuristic for finding small dominant sets

We analyse a greedy heuristic for finding small dominating sets in graphs: bounds on the size of the dominating set so produced had previously been derived in terms of the size of a smallest dominating set and the number of vertices and edges in the graph, respectively, We show that computing the resulting small dominating set isP-hard and so cannot be done efficiently in parallel (in the context of the PRAM model of parallel computation). We also consider a related non-deterministic greedy heuristic.

[1]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[2]  Dorit S. Hochbaum,et al.  Approximation Algorithms for the Set Covering and Vertex Cover Problems , 1982, SIAM J. Comput..

[3]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[4]  Raymond Greenlaw.  Ordered Vertex Removal and Subgraph Problems , 1989, J. Comput. Syst. Sci..

[5]  John H. Reif,et al.  Depth-First Search is Inherently Sequential , 1985, Inf. Process. Lett..

[6]  Abhay Parekh,et al.  Analysis of a Greedy Heuristic for Finding Small Dominating Sets in Graphs , 1991, Inf. Process. Lett..

[7]  Richard J. Anderson,et al.  Parallelism and the Maximal Path Problem , 1987, Inf. Process. Lett..

[8]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[9]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.