Characterization of joint probability distribution for large networks of random variables remains a challenging task in data science. Probabilistic graph approximation with simple topologies has practically been resorted to; typically the tree topology makes joint probability computation much simpler and can be effective for statistical inference on insufficient data. However, to characterize network components where multiple variables cooperate closely to influence others, model topologies beyond a tree are needed, which unfortunately are infeasible to acquire. In particular, our previous work has related optimal approximation of Markov networks of tree-width k >=2 closely to the graph-theoretic problem of finding maximum spanning k-tree (MSkT), which is a provably intractable task. This paper investigates optimal approximation of Markov networks with k-tree topology that retains some designated underlying subgraph. Such a subgraph may encode certain background information that arises in scientific applications, for example, about a known significant pathway in gene networks or the indispensable backbone connectivity in the residue interaction graphs for a biomolecule 3D structure. In particular, it is proved that the \beta-retaining MSkT problem, for a number of classes \beta of graphs, admit O(n^{k+1})-time algorithms for every fixed k>= 1. These \beta-retaining MSkT algorithms offer efficient solutions for approximation of Markov networks with k-tree topology in the situation where certain persistent information needs to be retained.

翻译：对大规模随机变量网络联合概率分布的刻画仍然是数据科学中的一项挑战性任务。实践中常采用具有简单拓扑的概率图近似方法；通常树拓扑使联合概率计算更为简化，且能在数据不足时有效支持统计推断。然而，当需要刻画多变量紧密协作影响其他变量的网络组件时，需要超越树结构的模型拓扑，但这类拓扑往往难以获取。特别地，我们前期的研究将树宽k≥2的马尔可夫网络最优逼近问题，与图论中寻找最大生成k-树（MSkT）这一已被证明难以处理的问题紧密关联。本文研究保留特定指定底层子图的k-树拓扑马尔可夫网络的最优逼近问题。此类子图可编码科学应用中出现的背景信息，例如基因网络中已知的重要通路，或生物分子三维结构中残基相互作用图不可或缺的主干连接。特别地，对于图类β，我们证明了β-保留MSkT问题对每个固定k≥1均存在O(n^{k+1})时间复杂度的算法。这些β-保留MSkT算法为需要保留特定持久信息的场景下，基于k-树拓扑的马尔可夫网络逼近提供了高效解决方案。