Paths Problems in Symmetric Logarithmic Space

Abstract

This paper studies space complexity of some basic problems for undirected graphs. In particular, the two disjoint paths problem (TPP) is investigated that asks whether for a given undirected graph G and nodes s ₁, s ₂, t ₁, t ₂, G admits two node disjoint paths connecting s ₁ with t ₁ and s ₂ with t ₂. The solving of this problem belongs to the fundamental tasks of routing algorithms and VLSI design, where two pairs of nodes have to be connected via disjoint paths in a given network. One of the most important results of this paper says that TPP can be solved by a symmetric nondeterministic Turing machine that works in logarithmic space. It is well known that switching from undirected to directed graphs, TPP becomes intractable.

Furthermore, the space complexity of minor detections is discussed. We show that testing for K ₄ -minor can be reduced (via log-space many-one reduction) to planarity testing and that detecting K ₅-minor is hard for $\mathcal{S}\mathcal{L}$ . As a corollary we obtain that series-parallel graphs can be recognised in $\mathcal{S}\mathcal{L}$ . Finally, the problem to determine the number of self avoiding walks in undirected series-parallel graphs is considered. It is proved that this problem can be solved in $\mathcal{F}\mathcal{L}^{\left\langle {\mathcal{S}\mathcal{L}} \right\rangle }$ .