Electronic Journal of Probability

Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes

Klaus Dohmen

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Recently, Naiman and Wynn introduced the concept of an abstract tube in order to obtain improved inclusion-exclusion identities and inequalities that involve much fewer terms than their classical counterparts. In this paper, we introduce a particular class of abstract tubes which plays an important role with respect to chromatic polynomials and network reliability. The inclusion-exclusion identities and inequalities associated with this class simultaneously generalize several well-known results such as Whitney's broken circuit theorem, Shier's expression for the reliability of a network as an alternating sum over chains in a semilattice and Narushima's inclusion-exclusion identity for posets. Moreover, we show that under some restrictive assumptions a polynomial time inclusion-exclusion algorithm can be devised, which generalizes an important result of Provan and Ball on network reliability.

Article information

Electron. J. Probab., Volume 4 (1999), paper no. 5, 12 pp.

Accepted: 26 March 1999
First available in Project Euclid: 4 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05A19: Combinatorial identities, bijective combinatorics
Secondary: 05A20: Combinatorial inequalities 05C15: Coloring of graphs and hypergraphs 60C05: Combinatorial probability 68M15: Reliability, testing and fault tolerance [See also 94C12] 90B12 90B25: Reliability, availability, maintenance, inspection [See also 60K10, 62N05]

Inclusion-exclusion Bonferroni inequalities sieve formula abstract tube abstract simplicial complex partial order chain dynamic programming graph coloring chromatic polynomial broken circuit complex network reliability

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Dohmen, Klaus. Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes. Electron. J. Probab. 4 (1999), paper no. 5, 12 pp. doi:10.1214/EJP.v4-42. https://projecteuclid.org/euclid.ejp/1457125514

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