Witnessing subsystems for probabilistic systems with low tree width
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by
Simon Jantsch, Jakob Piribauer, Christel Baier
2021
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.
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