Analysis of Markov chain algorithms on spanning trees, rooted forests, and connected subgraphs

Johannes Fehrenbach ¹   ; Ludger Rüschendorf ¹

¹ Department of Mathematics University of Freiburg Eckerstr. 1 79104 Freiburg, Germany

Applicationes Mathematicae, Tome 32 (2005) no. 3, pp. 341-365

Résumé

We analyse a natural edge exchange Markov chain on the set of spanning trees of an undirected graph by the method of multicommodity flows. The analysis is then refined to obtain a canonical path analysis. The construction of the flow and of the canonical paths is based on related path constructions in a paper of Cordovil and Moreira (1993) on block matroids. The estimates of the congestion measure imply a polynomial bound on the mixing time. The canonical paths for spanning trees also yield polynomial time mixing rates for some related Markov chains on the set of forests with roots and on the set of connected spanning subgraphs. We obtain a parametric class of stationary distributions from which we can efficiently sample. For rooted forests this includes the uniform distribution. For connected spanning subgraphs the uniform distribution is not covered.

Détail

DOI : 10.4064/am32-3-7

Keywords: analyse natural edge exchange markov chain set spanning trees undirected graph method multicommodity flows analysis refined obtain canonical path analysis construction flow canonical paths based related path constructions paper cordovil moreira block matroids estimates congestion measure imply polynomial bound mixing time canonical paths spanning trees yield polynomial time mixing rates related markov chains set forests roots set connected spanning subgraphs obtain parametric class stationary distributions which efficiently sample rooted forests includes uniform distribution connected spanning subgraphs uniform distribution covered