eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-12-14
246
260
10.4230/LIPIcs.FSTTCS.2015.246
article
Counting Euler Tours in Undirected Bounded Treewidth Graphs
Balaji, Nikhil
Datta, Samir
Ganesan, Venkatesh
We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a GapL upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic programming on bounded clique-width graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize #SAC^1) is relatively easy, establishing a uniform #SAC^1 bound needs a careful use of polynomial interpolation.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol045-fsttcs2015/LIPIcs.FSTTCS.2015.246/LIPIcs.FSTTCS.2015.246.pdf
Euler Tours
Bounded Treewidth
Bounded clique-width
Hamiltonian cycles
Parallel algorithms