Parameterized Graph Connectivity and Polynomial-Time Sub-Linear-Space Short Reductions

Abstract

We are focused on the solvability/insolvability of the directed s-t connectivity problem (DSTCON) parameterized by suitable size parameters m(x) on multi-tape deterministic Turing machines working on instances x to DSTCON by consuming simultaneously polynomial time and sub-linear space, where the informal term “sub-linear” refers to a function of the form $m(x)^{\varepsilon } \ell (|x|)$ on instances x for a certain absolute constant $\varepsilon \in (0,1)$ and a certain polylogarithmic function $\ell (n)$. As natural size parameters, we take the numbers $m_{ver}(x)$ of vertices and of edges $m_{edg}(x)$ of a graph cited in x. Parameterized problems solvable simultaneously in polynomial time using sub-linear space form a complexity class $\mathrm {PsubLIN}$ and it is unknown whether $\mathrm {DSTCON}$ parameterized by $m_{ver}$ belongs to $\mathrm {PsubLIN}$. Toward this open question, we wish to investigate the relative complexity of $\mathrm {DSTCON}$ and its natural variants and classify them according to a restricted form of many-one and Turing reductions, known as “short reductions,” which preserve the polynomial-time sub-linear-space complexity. As variants of $\mathrm {DSTCON}$, we consider the breadth-first search problem, the minimal path problem, and the topological sorting problem. Certain restricted forms of them fall into $\mathrm {PsubLIN}$. We also consider a stronger version of “sub-linear,” called “hypo-linear.” Additionally, we refer to a relationship to a practical working hypothesis known as the linear space hypothesis.

This work was done at the University of Toronto between August 2016 and March 2017 and was supported by the Natural Sciences and Engineering Council of Canada.