Generating and approximating nondominated coteries

Abstract

A coterie, which is used to realize mutual exclusion in a distributed system is a family C of incomparable subsets such that every pair of subsets in C has at least one element in common. Associate with a family of subsets C a positive (i.e., monotone) Boolean function f/sub c/ such that f/sub c/(x)=1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f/sub c/ is dual-minor, and is a nondominated (ND) coterie if and only if f/sub c/ is self-dual. We introduce an operator /spl rho/, 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 p operations. Based on this algorithm, all ND coteries of up to seven variables are computed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>