Randomized Algorithms for the Loop Cutset Problem

Abstract

We show h o w to nd a minimum weight loop cutset in a Bayesian network with high probability. Finding such a loop cutset is the rst step in the method of conditioning for inference. Our randomized algorithm for nding a loop cutset outputs a minimum loop cutset after O(c 6 k kn) steps with probability at least 1 ; (1 ; 1 6 k ) c6 k , where c > 1 i s a constant speci ed by the user, k is the minimal size of a minimum weight loop cutset, and n is the number of vertices. We also show empirically that a variant of this algorithm often nds a loop cutset that is closer to the minimum weight loop cutset than the ones found by the best deterministic algorithms known.

Improved algorithms for feedback vertex set problems

“…By conditions (2) and (3), the vertex w 1 has a unique neighbor in V 2 . Also by conditions (2) and (3), the vertex w 1 must have a parent w in the tree T in G[V 1 ].…”

“…The best known randomized parameterized algorithms for the FVS problems are due to Becker et al. [2], who developed a randomized algorithm of running time O(4 k kn 2 ) for the FVS problem on unweighted graphs, and a randomized algorithm of running time O(6 k kn 2 ) for the FVS problem on weighted graphs. To our knowledge, no deterministic algorithm of running time f (k)n O (1) for any function f was known prior to our results for the weighted FVS problem.…”

Planar Feedback Vertex Set and Face Cover: Combinatorial Bounds and Subexponential Algorithms

“…A fixed parameter algorithm (or simply FPT-algorithm) is one that solves the problem in O(f (k) · |I | O (1) ) steps where f is a function depending exclusively on the parameter k (for more on parameterized complexity and algorithms, see [17,28]). …”

“…Many FPT-algorithms were proposed for FEEDBACK VERTEX SET. The best current results in this direction are the O(4 k kn) step probabilistic algorithm in [1] and the O(5 k kn 2 ) step algorithm in [6] (throughout the paper and for both problems, we denote by n the number of vertices of the input graph).…”

“…algorithms running in O (2 o(k) ·n O (1) ) steps. The first results of this kind were given by Kloks et al in [26] for both problems.…”