Witnessing Subsystems for Probabilistic Systems with Low Tree Width
Abstract
A standard way of justifying that a certain probabilistic property holds in a system is to provide a witnessing subsystem (also called critical subsystem) for the property. Computing minimal witnessing subsystems is NPhard already for acyclic Markov chains, but can be done in polynomial time for Markov chains whose underlying graph is a tree. This paper considers the problem for probabilistic systems that are similar to trees or paths. It introduces the parameters directed treepartition width (dtpw) and directed pathpartition width (dppw) and shows that computing minimal witnesses remains NPhard for Markov chains with bounded dppw (and hence also for Markov chains with bounded dtpw). By observing that graphs of bounded dtpw have bounded width with respect to all known tree similarity measures for directed graphs, the hardness result carries over to these other tree similarity measures. Technically, the reduction proceeds via the conceptually simpler matrixpair chain problem, which is introduced and shown to be NPcomplete for nonnegative matrices of fixed dimension. Furthermore, an algorithm which aims to utilise a given directed tree partition of the system to compute a minimal witnessing subsystem is described. It enumerates partial subsystems for the blocks of the partition along the tree order, and keeps only necessary ones. A preliminary experimental analysis shows that it outperforms other approaches on certain benchmarks which have directed tree partitions of small width.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.08326
 Bibcode:
 2021arXiv210908326J
 Keywords:

 Computer Science  Logic in Computer Science
 EPrint:
 In Proceedings GandALF 2021, arXiv:2109.07798. A full version of this paper, containing all proofs, appears at arXiv:2108.08070