Improved Approximation Algorithms for Geometric Set Cover

TL;DR

This paper develops improved approximation algorithms for geometric set cover problems, leveraging a new decomposition technique and extending shallow cuttings to achieve constant-factor guarantees in various geometric scenarios.

Contribution

It introduces a general condition for approximation algorithms based on geometric decompositions, extending shallow cuttings, and provides new constant-factor algorithms for specific geometric set cover cases.

Findings

Constant-factor approximation for unit cubes in R^3

Improved guarantees for fat triangles of arbitrary size

Effective algorithms for guarding terrains and covering fat objects

Abstract

Given a collection S of subsets of some set U, and M a subset of U, the set cover problem is to find the smallest subcollection C of S such that M is a subset of the union of the sets in C. While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually U = R^d. Extending prior results, we show that approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R of S and function f(), there is a decomposition of the portion of U not covered by R into an expected f(|R|) regions, each region of a particular simple form. We show that under this condition, a cover of size O(f(|C|)) can be found. Our proof involves the generalization of shallow cuttings to more general geometric situations. We obtain constant-factor approximation algorithms for covering by unit cubes in R^3, for guarding a one-dimensional terrain, and for covering by similar-sized fat triangles in R^2. We also obtain improved approximation guarantees for fat triangles, of arbitrary size, and for a class of fat objects.