Improved Inclusion-Exclusion Identities and Bonferroni Inequalities with Applications to Reliability Analysis of Coherent Systems

[1]  Richard M. Karp,et al.  Dynamic programming meets the principle of inclusion and exclusion , 1982, Oper. Res. Lett..

[2]  L. Weisner,et al.  Abstract theory of inversion of finite series , 1935 .

[3]  Herbert Edelsbrunner,et al.  Inclusion-exclusion complexes for pseudodisk collections , 1997, Discret. Comput. Geom..

[4]  J. Galambos,et al.  Bonferroni-type inequalities with applications , 1996 .

[5]  K. Borsuk On the imbedding of systems of compacta in simplicial complexes , 1948 .

[6]  Hiroshi Narushima,et al.  Principle of inclusion-exclusion on partially ordered sets , 1978, Discret. Math..

[7]  Felix Lazebnik,et al.  New upper bounds for the greatest number of proper colorings of a (V, E)-graph , 1990, J. Graph Theory.

[8]  D. Naiman,et al.  INCLUSION-EXCLUSION-BONFERRONI IDENTITIES AND INEQUALITIES FOR DISCRETE TUBE-LIKE PROBLEMS VIA EULER CHARACTERISTICS , 1992 .

[9]  Lajos Takács,et al.  On the Method of Inclusion and Exclusion , 1967 .

[10]  S. L. Édel'man Closure operators on a lattice , 1980 .

[11]  Shun-Chen Niu,et al.  Reliability of Consecutive-k-out-of-n:F System , 1981, IEEE Transactions on Reliability.

[12]  Klaus Dohmen Inclusion-exclusion: Which terms cancel? , 2000 .

[13]  András Prékopa,et al.  Probability Bounds with Cherry Trees , 2001, Math. Oper. Res..

[14]  A. Satyanarayana,et al.  A Unified Formula for Analysis of Some Network Reliability Problems , 1982, IEEE Transactions on Reliability.

[15]  R. E. Barlow,et al.  Computing k-out-of-n System Reliability , 1984, IEEE Transactions on Reliability.

[16]  Eckhard Manthei Domination Theory and Network Reliability Analysis , 1991, J. Inf. Process. Cybern..

[17]  David A. Grable Hypergraphs and sharpened sieve inequalities , 1994, Discret. Math..

[18]  Herbert Edelsbrunner,et al.  The union of balls and its dual shape , 1993, SCG '93.

[19]  William T. Tutte A Ring in Graph Theory , 1947 .

[20]  Richard P. Stanley Acyclic orientations of graphs , 1973, Discret. Math..

[21]  Paul H. Edelman,et al.  The theory of convex geometries , 1985 .

[22]  Klaus Dohmen Bounds to the Chromatic Polynomial of a Graph , 1998 .

[23]  Günter M. Ziegler,et al.  Broken circuit complexes: Factorizations and generalizations , 1991, J. Comb. Theory B.

[24]  J. D. Esary,et al.  Coherent Structures of Non-Identical Components , 1963 .

[25]  H. Crapo A higher invariant for matroids , 1967 .

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

[27]  Peter Tittmann,et al.  A new two-variable generalization of the chromatic polynomial , 2003, Discret. Math. Theor. Comput. Sci..

[28]  Andreas Blass,et al.  Mobius functions of lattices , 1997 .

[29]  John A. Buzacott,et al.  Node partition formula for directed graph reliability , 1987, Networks.

[30]  Bruce E. Sagan,et al.  A Generalization of Rota′s NBC Theorem , 1995 .

[31]  Klaus Dohmen,et al.  A note on Möbius inversion over power set lattices , 1997 .

[32]  Owen D. Byer Some new bounds for the maximum number of vertex colorings of a (v, e)-graph , 1998, J. Graph Theory.

[33]  Cut-Set Intersections and Node Partitions , 1984, IEEE Transactions on Reliability.

[34]  A. Satyanarayana,et al.  A New Algorithm for the Reliability Analysis of Multi-Terminal Networks , 1981, IEEE Transactions on Reliability.

[35]  Gary Gordon,et al.  Convexity and the Beta Invariant , 1999, Discret. Comput. Geom..

[36]  R. Prim Shortest connection networks and some generalizations , 1957 .

[37]  G. Boole An Investigation of the Laws of Thought: On which are founded the mathematical theories of logic and probabilities , 2007 .

[38]  Some inequalities of Bonferroni—Galambos type , 1989 .

[39]  Pavel Zorin-Kranich,et al.  Habilitationsschrift , 1970 .

[40]  David A. Grable,et al.  Sharpened Bonferroni Inequalities , 1993, J. Comb. Theory, Ser. B.

[41]  J. Scott Provan,et al.  Computing Network Reliability in Time Polynomial in the Number of Cuts , 1984, Oper. Res..

[42]  W. Preuss,et al.  Reliability of consecutive-k-out-of-n:F systems with nonidentical component reliabilities , 1989 .

[43]  Ioan Tomescu,et al.  Chromatic Coefficients of Linear Uniform Hypergraphs , 1998, J. Comb. Theory, Ser. B.

[44]  Yuan Xu,et al.  A new method for generating Bonferroni-type inequalities by iteration , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[45]  A. Prabhakar,et al.  New Topological Formula and Rapid Algorithm for Reliability Analysis of Complex Networks , 1978, IEEE Transactions on Reliability.

[46]  Klaus Dohmen A Contribution to The Chromatic Theory of Uniform Hypergraphs , 1995 .

[47]  Klaus Dohmen On Partial Sums of Chromatic Polynomials , 1999, Ars Comb..

[48]  James G. Oxley,et al.  The Broken-circuit Complex: Its Structure and Factorizations , 1981, Eur. J. Comb..

[49]  M. Farber,et al.  Convexity in graphs and hypergraphs , 1986 .

[50]  G. Rota On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[51]  Edward Howorka A characterization of ptolemaic graphs , 1981, J. Graph Theory.

[52]  Robert E. Tarjan,et al.  Bounds on Backtrack Algorithms for Listing Cycles, Paths, and Spanning Trees , 1975, Networks.

[53]  Shuji Tsukiyama,et al.  An Algorithm to Enumerate All Cutsets of a Graph in Linear Time per Cutset , 1980, J. ACM.

[54]  Klaus Dohmen,et al.  An improvement of the inclusion-exclusion principle , 1999 .

[55]  Klaus D. Heidtmann,et al.  Improved Method of Inclusion-Exclusion Applied to k-out-of-n Systems , 1982, IEEE Transactions on Reliability.

[56]  Ioan Tomescu,et al.  Hypertrees and Bonferroni inequalities , 1986, J. Comb. Theory, Ser. B.

[57]  K. B. Misra,et al.  Reliability calculation of redundant systems with non-identical units , 1976 .

[58]  Andreas Blass,et al.  Bijective proofs of two broken circuit theorems , 1986, J. Graph Theory.

[59]  K. Worsley An improved Bonferroni inequality and applications , 1982 .

[60]  Klaus Dohmen,et al.  A Broken-Circuits-Theorem for hypergraphs , 1995 .

[61]  Klaus Dohmen,et al.  Improved Bonferroni Inequalities via Union-Closed Set Systems , 2000, J. Comb. Theory, Ser. A.

[62]  D R Shier Algebraic Aspects of Computing Network Reliability. , 1986 .

[63]  J. Galambos The role of graph theory in some sieve arguments of probability theory , 1972 .

[64]  Hiroshi Narushima,et al.  Principle of Inclusion-Exclusion on Semilattices , 1974, J. Comb. Theory, Ser. A.

[65]  Peng Huang,et al.  Stochastic Models in Reliability , 1999, Technometrics.

[66]  Eliezer L. Lozinskii Counting Propositional Models , 1992, Inf. Process. Lett..

[67]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[68]  G. Birkhoff A Determinant Formula for the Number of Ways of Coloring a Map , 1912 .

[69]  A INVARIANT FOR GREEDOIDS AND ANTIMATROIDS , 1997 .

[70]  J. Kruskal On the shortest spanning subtree of a graph and the traveling salesman problem , 1956 .

[71]  J. Galambos Methods for Proving Bonferroni Type Inequalities , 1975 .

[72]  Seymour M. Kwerel Most Stringent Bounds on Aggregated Probabilities of Partially Specified Dependent Probability Systems , 1975 .

[73]  K. Reidemeister Einführung in die kombinatorische Topologie , 1951 .

[74]  Douglas R. Shier,et al.  Inclusion-exclusion, cancellation, and consecutive sets , 1991, Australas. J Comb..

[75]  Herbert S. Wilf,et al.  Algorithms and Complexity , 1994, Lecture Notes in Computer Science.

[76]  Klaus Dohmen On graph invariants satisfying the deletion-contraction formula , 1996, J. Graph Theory.

[77]  J.M. Kontoleon,et al.  Reliability Determination of a r-Successive-out-of-n:F System , 1980, IEEE Transactions on Reliability.

[78]  Tamás F. Móri,et al.  A note on the background of several Bonferroni−Galambos-type inequalities , 1985 .

[79]  H. Whitney A logical expansion in mathematics , 1932 .

[80]  Eugene Seneta DEGREE, ITERATION AND PERMUTATION IN IMPROVING BONFERRONI-TYPE BOUNDS , 1988 .

[81]  D. Shier Network Reliability and Algebraic Structures , 1991 .

[82]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[83]  Edward A. Bender,et al.  A Theoretical Analysis of Backtracking in the Graph Coloring Problem , 1985, J. Algorithms.

[84]  Victor Reiner,et al.  Counting the Interior Points of a Point Configuration , 2000, Discret. Comput. Geom..

[85]  Klaus Dohmen,et al.  On Sums Over Partially Ordered Sets , 1999, Electron. J. Comb..

[86]  W. T. Tutte,et al.  A Contribution to the Theory of Chromatic Polynomials , 1954, Canadian Journal of Mathematics.

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

[88]  Michael O. Ball,et al.  Computational Complexity of Network Reliability Analysis: An Overview , 1986, IEEE Transactions on Reliability.

[89]  Klaus Dohmen,et al.  Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes , 1999 .

[90]  Daniel Q. Naiman,et al.  Abstract tubes, improved inclusion-exclusion identities and inequalities and importance sampling , 1997 .

[91]  J. George Shanthikumar,et al.  Recursive Algorithm to Evaluate the Reliability of a Consecutive-k-out-of-n:F System , 1982, IEEE Transactions on Reliability.

[92]  J. S. Provan,et al.  Reliability covering problems , 1991, Networks.

[93]  Z. Birnbaum,et al.  Multi-Component Systems and Structures and Their Reliability , 1961 .

[94]  David J. Hunter An upper bound for the probability of a union , 1976, Journal of Applied Probability.

[95]  Robert E. Jamison-Waldner PARTITION NUMBERS FOR TREES AND ORDERED SETS , 1981 .

[96]  G. Grätzer General Lattice Theory , 1978 .

[97]  Doron Zeilberger Garsia and Milne's bijective proof of the inclusion-exclusion principle , 1984, Discret. Math..

[98]  Klaus Dohmen A Note on Narushima's Principle of Inclusion–Exclusion on Partition Lattices , 2001, Graphs Comb..

[99]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[100]  Klaus Dohmen,et al.  Lower bounds and upper bounds for chromatic polynomials , 1993, J. Graph Theory.

[101]  F. Hwang,et al.  Fast Solutions for Consecutive-k-out-of-n: F System , 1982, IEEE Transactions on Reliability.

[102]  Klaus D. Heidtmann Zur Approximation der Verfügbarkeit technischer Systeme , 1981 .

[103]  Klaus Dohmen Some remarks on the sieve formula, the Tutte polynomial and Crapo's beta invariant , 2000 .

[104]  Klaus Dohmen,et al.  Improved inclusion-exclusion identities via closure operators , 2000, Discret. Math. Theor. Comput. Sci..

[105]  Seymour M. Kwerel Bounds on the probability of the union and intersection of m events , 1975, Advances in Applied Probability.