Parallel algorithms with optimal speedup for bounded treewidth

Abstract

We describe the first parallel algorithm with optimal speedup for constructing minimum-width tree decompositions of graphs of bounded treewidth. On n-vertex input graphs, the algorithm works in O((logn)2) time using O(n) operations on the EREW PRAM. We also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a vanety of problems on graphs of bounded treewidth, including all decision problems expressible in monadic second-order logic. On n-vertex input graphs, the algorithms use O(n) operations together with O(lognlog*n) time on the EREW PRAM, or 0 (log n) time on the CRCW PRAM.

Graph Operations Characterizing Rank-Width and Balanced Graph Expressions

“…Without loss of generality we assume that V G is always linearly ordered. This order will be used to represent edg G by a square matrix over GF (2).…”

“…Another practical use of balanced terms is the design of parallel algorithms. This is considered for example by Bodlaender to design parallel algorithms to construct minimum-width tree-decompositions of graphs or to solve some NP-complete problems [1,2].…”

Embedding Graphs with Bounded Treewidth into Their Optimal Hypercubes

“…On the other hand, for some special classes of graphs, e.g., permutation graphs, cotriangulated graphs, convex graphs, chordal bipartite graphs, and circle graphs, there exist polynomial time algorithms to determine the treewidth (see, e.g., [24]). For graphs with constant treewidth an algorithm for constructing a minimal tree-decomposition is described in [6]. This algorithm works in time O log 2 n using O n operations on a EREW PRAM.…”

“…Recall that in general it is -hard to decide the treewidth of a given graph. As shown in [6], a minimal tree-decomposition for a graph with constant treewidth can be computed on a EREW PRAM in time O log 2 n . Using sorting, this algorithm can be immediately transformed into a hypercube algorithm with running time O log 3 n log log 2 n .…”

“…It will be shown, that this embedding can be computed in time O max T LR n dt 2 log 2 n , where T LR n denotes the time complexity for list ranking on the hypercube. If the given graph has constant treewidth, a minimal tree-decomposition for this graph can be computed on the optimal hypercube in time O log 3 n log log 2 n , using a result in [6].…”