Approximation Schemes for Covering and Packing in the Streaming Model

TL;DR

This paper adapts the shifting strategy for approximation schemes to the streaming model, enabling efficient algorithms for graph properties and the novel unit disc cover problem in streaming settings.

Contribution

It introduces a streaming-friendly shifting strategy and combines it with coresets to develop new approximation algorithms for unit disc graphs and the UDC problem.

Findings

Streaming algorithms for graph properties of unit disc graphs.

New approximation algorithms for the unit disc cover problem.

Lower bounds for the UDC problem in streaming.

Abstract

The shifting strategy, introduced by Hochbaum and Maass, and independently by Baker, is a unified framework for devising polynomial approximation schemes to NP-Hard problems. This strategy has been used to great success within the computational geometry community in a plethora of different applications; most notably covering, packing, and clustering problems. In this paper, we revisit the shifting strategy in the context of the streaming model and develop a streaming-friendly shifting strategy. When combined with the shifting coreset method introduced by Fonseca et al., we obtain streaming algorithms for various graph properties of unit disc graphs. As a further application, we present novel approximation algorithms and lower bounds for the unit disc cover (UDC) problem in the streaming model, for which currently no algorithms are known.