# 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.

## Keywords

Sub-linear space Hypo-linear space Directed s-t-connectivity NL search NL optimization Short reduction Linear space hypothesis## References

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