On finding minimal w-cutset

TL;DR

This paper investigates the problem of finding minimal w-cutsets in graphs, relating it to set multi-cover, proving NP-completeness, and proposing a greedy algorithm with empirical evaluation.

Contribution

It introduces a formal approach to find minimal w-cutsets, relates the problem to set multi-cover, and provides a greedy algorithm with empirical results.

Findings

The problem is NP-complete.

A greedy algorithm can find approximate solutions.

Empirical results demonstrate the algorithm's effectiveness.

Abstract

The complexity of a reasoning task over a graphical model is tied to the induced width of the underlying graph. It is well-known that the conditioning (assigning values) on a subset of variables yields a subproblem of the reduced complexity where instantiated variables are removed. If the assigned variables constitute a cycle-cutset, the rest of the network is singly-connected and therefore can be solved by linear propagation algorithms. A w-cutset is a generalization of a cycle-cutset defined as a subset of nodes such that the subgraph with cutset nodes removed has induced-width of w or less. In this paper we address the problem of finding a minimal w-cutset in a graph. We relate the problem to that of finding the minimal w-cutset of a treedecomposition. The latter can be mapped to the well-known set multi-cover problem. This relationship yields a proof of NP-completeness on one hand and a greedy algorithm for finding a w-cutset of a tree decomposition on the other. Empirical evaluation of the algorithms is presented.