On Finding Optimal Polytrees

TL;DR

This paper investigates the computational complexity of finding optimal polytrees, showing polynomial-time solutions under certain conditions and exploring fixed-parameter tractability based on the number of deletions.

Contribution

It introduces a matroid intersection approach for bounded arc deletions and analyzes fixed-parameter tractability for node and arc deletion scenarios.

Findings

Polynomial-time algorithm for bounded arc deletions.

Fixed-parameter tractability for certain restricted problems.

Intractability results for node deletion parameterization.

Abstract

Inferring probabilistic networks from data is a notoriously difficult task. Under various goodness-of-fit measures, finding an optimal network is NP-hard, even if restricted to polytrees of bounded in-degree. Polynomial-time algorithms are known only for rare special cases, perhaps most notably for branchings, that is, polytrees in which the in-degree of every node is at most one. Here, we study the complexity of finding an optimal polytree that can be turned into a branching by deleting some number of arcs or nodes, treated as a parameter. We show that the problem can be solved via a matroid intersection formulation in polynomial time if the number of deleted arcs is bounded by a constant. The order of the polynomial time bound depends on this constant, hence the algorithm does not establish fixed-parameter tractability when parameterized by the number of deleted arcs. We show that a restricted version of the problem allows fixed-parameter tractability and hence scales well with the parameter. We contrast this positive result by showing that if we parameterize by the number of deleted nodes, a somewhat more powerful parameter, the problem is not fixed-parameter tractable, subject to a complexity-theoretic assumption.