Log-space Algorithms for Paths and Matchings in k-trees

TL;DR

This paper presents new log-space algorithms for reachability, shortest paths, and perfect matchings in directed k-trees, improving the understanding of their computational complexity within bounded tree-width graphs.

Contribution

It introduces the first log-space algorithms for reachability and path problems in directed k-trees, and proves L-completeness for perfect matching problems in k-trees.

Findings

Log-space algorithms for reachability in directed k-trees.

Log-space algorithms for shortest and longest paths in directed acyclic k-trees.

L-completeness of perfect matching decision and search problems in k-trees.

Abstract

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect macthing (decision version), and if so, finding a perfect match- ing (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [DKR08]. Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from [JT07] and [LMR07].