Witnessing Subsystems for Probabilistic Systems with Low Tree Width

TL;DR

This paper investigates the complexity of finding minimal witnessing subsystems in probabilistic systems with tree-like structures, proving NP-hardness for certain parameters and proposing an algorithm with promising preliminary results.

Contribution

It introduces new parameters for measuring directed tree-like structures and proves NP-hardness of minimal witness computation for these classes, along with a partial algorithm and experimental insights.

Findings

NP-hardness for bounded directed path-partition width

Algorithm leveraging directed tree partitions shows promising results

Bounded dtpw graphs relate to all known tree similarity measures

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 NP-hard 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 tree-partition width (dtpw) and directed path-partition width (dppw) and shows that computing minimal witnesses remains NP-hard 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 matrix-pair chain problem, which is introduced and shown to be NP-complete 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.