Reliable assignments of processors to tasks and factoring on matroids

In the simple assignment problem, there are n processors, m tasks, and a relation between the processors and tasks; this relation indicates the ability of the processor to perform the task. When the processors fail independently with known probabilities, two performance issues arise. First, with what probability can the operating processors all be kept busy? Second, with what probability can the operating processors perform the same number of tasks that all processors could? We formulate these questions on the underlying transversal matroid. We rst prove that counting minimum cardinality circuits in this matroid is #P-complete, and hence that both questions are also #P-complete. Secondly, we devise a factoring algorithm with series and parallel reductions to compute exact solutions of the above problems. We then outline some efcient strategies for bounding the probabilities.

[1]  James G. Oxley,et al.  Matroid theory , 1992 .

[2]  J. S. Provan,et al.  Bounds on the reliability polynomial for shellable independence systems , 1982 .

[3]  Charles J. Colbourn,et al.  The Combinatorics of Network Reliability , 1987 .

[4]  Mark K. Chang,et al.  Network reliability and the factoring theorem , 1983, Networks.

[5]  J. Edmonds.  Minimum partition of a matroid into independent subsets , 1965 .

[6]  Colin McDiarmid,et al.  Strict Gammoids and Rank Functions , 1972 .

[7]  Charles J. Colbourn,et al.  The complexity of completing partial Latin squares , 1984, Discret. Appl. Math..

[8]  Tom Brylawski,et al.  A combinatorial model for series-parallel networks , 1971 .

[9]  James Oxley.  Theory of Matroids: Graphs and Series-Parallel Networks , 1986 .

[10]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[11]  Charles J. Colbourn,et al.  Counting almost minimum cutsets with reliability applications , 1987, Math. Program..

[12]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[13]  A. Ingleton,et al.  Gammoids and transversal matroids , 1973 .

[14]  Alexander Schrijver,et al.  Matroids and linking systems , 1979, J. Comb. Theory B.

[15]  R. Kevin Wood.  A factoring algorithm using polygon-to-chain reductions for computing K-terminal network reliability , 1985, Networks.

[16]  A. Satyanarayana,et al.  An O(|E|) Time Algorithm for Computing the Reliability of a Class of Directed Networks , 1984, Oper. Res..

[17]  Ian Holyer,et al.  The NP-Completeness of Some Edge-Partition Problems , 1981, SIAM J. Comput..

[18]  Charles J. Colbourn,et al.  Edge-packing of graphs and network reliability , 1988, Discret. Math..

[19]  E. BixbyR..  The Minimum Number of Edges and Vertices in a Graph with Edge Connectivity n and m n-Bonds , 1975 .

[20]  Avinash Agrawal,et al.  A Survey of Network Reliability and Domination Theory , 1984, Oper. Res..

[21]  Charles J. Colbourn,et al.  Matroid Steiner problems, the Tutte polynomial and network reliability , 1989, J. Comb. Theory, Ser. B.