Electronic Journal of Probability

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

Klaus Dohmen

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457125514

Digital Object Identifier
doi:10.1214/EJP.v4-42

Mathematical Reviews number (MathSciNet)
MR1684161

Zentralblatt MATH identifier
0920.05008

Subjects
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]

Keywords
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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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