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<p><it>Abstract</it>—A coterie, which is used to realize mutual exclusion in a distributed system, is a family <it>C</it> of incomparable subsets such that every pair of subsets in <it>C</it> has at least one element in common. Associate with a family of subsets <it>C</it> a positive (i.e., monotone) Boolean function <it>f</it><sub><it>C</it></sub> such that <it>f</it><sub><it>C</it></sub>(<it>x</it>) = 1 if the Boolean vector <it>x</it> is equal to or greater than the characteristic vector of some subset in <it>C</it>, and 0 otherwise. It is known that <it>C</it> is a coterie if and only if <it>f</it><sub><it>C</it></sub> is dual-minor, and is a nondominated (ND) coterie if and only if <it>f</it><sub><it>C</it></sub> is self-dual.</p><p>In this paper, we introduce an operator ρ, which transforms a positive self-dual function into another positive self-dual function, and the concept of almost-self-duality, which is a close approximation to self-duality and can be checked in polynomial time (the complexity of checking positive self-duality is currently unknown). After proving several interesting properties of them, we propose a simple algorithm to check whether a given positive function is self-dual or not. Although this is not a polynomial algorithm, it is practically efficient in most cases. Finally, we present an incrementally polynomial algorithm that generates all positive self-dual functions (ND coteries) by repeatedly applying ρ operations. Based on this algorithm, all ND coteries of up to seven variables are computed.</p>
Almost-self-dual functions, coteries, dualization, monotone Boolean functions, mutual-exclusion, nondominated coteries, positive Boolean functions, self-dual functions.
Toshihide Ibaraki, Jan C. Bioch, "Generating and Approximating Nondominated Coteries", IEEE Transactions on Parallel & Distributed Systems, vol. 6, no. , pp. 905-914, September 1995, doi:10.1109/71.466629
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