Kernelizing Problems on Planar Graphs in Sublinear Space and Polynomial Time

TL;DR

This paper presents a new scheme for kernelizing problems on planar graphs using sublinear space and polynomial time, enabling efficient algorithms for problems like Dominating Set and Vertex Cover.

Contribution

It introduces a novel kernelization scheme exploiting planarity for sublinear space and polynomial time, with applications to Dominating Set and Vertex Cover.

Findings

Kernelization scheme operates in O((sqrt(n)+k) log n) space.

Dominating Set kernelization algorithm runs in polynomial time and O(k log n) space.

Restricted-memory procedures for region decompositions are developed.

Abstract

In this paper, we devise a scheme for kernelizing, in sublinear space and polynomial time, various problems on planar graphs. The scheme exploits planarity to ensure that the resulting algorithms run in polynomial time and use O((sqrt(n) + k) log n) bits of space, where n is the number of vertices in the input instance and k is the intended solution size. As examples, we apply the scheme to Dominating Set and Vertex Cover. For Dominating Set, we also show that a well-known kernelization algorithm due to Alber et al. (JACM 2004) can be carried out in polynomial time and space O(k log n). Along the way, we devise restricted-memory procedures for computing region decompositions and approximating the aforementioned problems, which might be of independent interest.