An algorithm for generating all maximal independent subsets of posets

Abstract

This paper gives an algorithm for determining the setU0(Θ) of all maximal independent subsets of a finite poset Θ=(A, O), based on a (complex) Θ-induced arrangement ofU0(Θ) as a rooted tree. By this procedure, all essential data manipulations can be restricted to certain bipartite graphs, thereby leading to a computation complexity not exceedingO(|A|·|\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} \)) per maximal independent subset obtained (where\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} \) denotes the immediate ordering relations). In fact, an intensive study of examples (with |A| ranging from 10–200) even showed almost total independence of external parameters such as |A|, |O| or |\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} \)| and at the same time led to considerably low computation times. Thus, in particular, the application of standard graph-theoretical methods to the associated comparability graph of Θ would no longer seem a reasonable approach to the considered problem.