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.

中文翻译：

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低树宽概率系统的见证子系统

证明系统中某个概率属性成立的标准方法是为该属性提供见证子系统（也称为临界子系统）。对于非循环马尔可夫链，计算最小见证子系统已经是 NP-hard 问题，但对于底层图是树的马尔可夫链，可以在多项式时间内完成。本文考虑了类似于树或路径的概率系统的问题。它引入了参数有向树分区宽度 (dtpw) 和有向路径分区宽度 (dppw)，并表明计算最小见证对于具有有界 dppw 的马尔可夫链（因此对于具有有界 dtpw 的马尔可夫链）仍然是 NP 难的。通过观察有界 dtpw 的图对于有向图的所有已知树相似性度量具有有界宽度，硬度结果延续到这些其他树的相似性度量。从技术上讲，减少通过概念上更简单的矩阵对链问题进行，该问题被引入并显示为固定维度的非负矩阵的 NP 完全问题。此外，还描述了一种旨在利用系统的给定有向树分区来计算最小见证子系统的算法。它沿树顺序枚举分区块的部分子系统，并仅保留必要的子系统。初步实验分析表明，它在某些具有小宽度定向树分区的基准上优于其他方法。对于固定维数的非负矩阵，它被引入并显示为 NP 完全的。此外，还描述了一种旨在利用系统的给定有向树分区来计算最小见证子系统的算法。它沿树顺序枚举分区块的部分子系统，并仅保留必要的子系统。初步实验分析表明，它在某些具有小宽度定向树分区的基准上优于其他方法。对于固定维数的非负矩阵，它被引入并显示为 NP 完全的。此外，还描述了一种旨在利用系统的给定有向树分区来计算最小见证子系统的算法。它沿树顺序枚举分区块的部分子系统，并仅保留必要的子系统。初步实验分析表明，它在某些具有小宽度定向树分区的基准上优于其他方法。