An approximation algorithm for random generation of capacities

TL;DR

This paper introduces a 2-layer approximation algorithm for efficiently generating capacities on finite sets, reducing computational time while maintaining accuracy compared to existing Markov chain methods.

Contribution

The paper presents a novel 2-layer approximation method that efficiently generates a subset of linear extensions for capacities, improving speed over traditional Markov chain approaches.

Findings

The 2-layer method is significantly faster than Markov chain algorithms.

The approximation maintains similar accuracy to existing methods.

The approach effectively filters out low-probability linear extensions.

Abstract

Capacities on a finite set are sets functions vanishing on the empty set and being monotonic w.r.t. inclusion. Since the set of capacities is an order polytope, the problem of randomly generating capacities amounts to generating all linear extensions of the Boolean lattice. This problem is known to be intractable even as soon as $n>5$, therefore approximate methods have been proposed, most notably one based on Markov chains. Although quite accurate, this method is time consuming. In this paper, we propose the 2-layer approximation method, which generates a subset of linear extensions, eliminating those with very low probability. We show that our method has similar performance compared to the Markov chain but is much less time consuming.