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.
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