Parallel Algorithms with Optimal Speedup for Bounded Treewidth

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 $O((\log n)^2)$ 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 variety 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 $O(\log n\Tlogstar n)$ time on the EREW PRAM, or $O(\log n)$ time on the CRCW PRAM