Parameterized-NL Completeness of Combinatorial Problems by Short Logarithmic-Space Reductions and Immediate Consequences of the Linear Space Hypothesis

TL;DR

This paper introduces new NL-complete problems based on natural parameterizations of classic NP-complete problems, demonstrating their complexity equivalences and implications under the linear space hypothesis.

Contribution

It expands the list of NL-complete problems with natural parameterizations and analyzes their complexity under space constraints and the linear space hypothesis.

Findings

Proposes new NL-complete problems from NP-complete problems with natural parameters.

Shows equivalence of these problems to known problems via short logarithmic-space reductions.

Under the linear space hypothesis, these problems cannot be solved in sub-linear space polynomial time.

Abstract

The concept of space-bounded computability has become significantly important in handling vast data sets on memory-limited computing devices. To replenish the existing short list of NL-complete problems whose instance sizes are dictated by log-space size parameters, we propose new additions obtained directly from natural parameterizations of three typical NP-complete problems -- the vertex cover problem, the exact cover by 3-sets problem, and the 3-dimensional matching problem. With appropriate restrictions imposed on their instances, the proposed decision problems parameterized by appropriate size parameters are proven to be equivalent in computational complexity to either the parameterized $3$-bounded 2CNF Boolean formula satisfiability problem or the parameterized degree-$3$ directed $s$-$t$ connectivity problem by ``short'' logarithmic-space reductions. Under the assumption of the linear space hypothesis, furthermore, none of the proposed problems can be solved in polynomial time if the memory usage is limited to sub-linear space.