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.
Highlights
Inclusion-exclusion identities and inequalities play an important role in many areas of mathematics
There is no real restriction in using indicator functions rather than measures, since the above inequalities can be integrated with respect to any measure on the σ-algebra generated by {Av}v∈V
Naiman and Wynn [10] introduced the notion of an abstract tube in order to improve and generalize the classical inclusion-exclusion inequalities
Summary
Inclusion-exclusion identities and inequalities play an important role in many areas of mathematics. For any finite collection of sets {Av}v∈V and any n ∈ N0 = N ∪ {0}, the classical inclusion-exclusion inequalities ( known as Bonferroni inequalities) state that χ. Note that for n ≥ |V | the equals sign holds and we have the classical inclusion-exclusion identity ( known as the sieve formula). Naiman and Wynn [10] introduced the notion of an abstract tube in order to improve and generalize the classical inclusion-exclusion inequalities. In the subsection on chromatic polynomials, a link is established between the theory of abstract tubes and the theory of broken circuit complexes, which was initiated by Wilf [17]
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