Geometric Packing under Non-uniform Constraints

TL;DR

This paper introduces a framework for approximating solutions to geometric packing problems with capacity constraints, providing constant-factor approximations for specific cases like fat triangles, and establishing limits on approximation quality.

Contribution

The authors develop a general approximation framework for geometric packing with capacities and demonstrate its effectiveness on fat triangles, including hardness results.

Findings

Achieved an O(1)-approximation for fat triangles of similar size.

Proved that no PTAS exists for this problem under certain conditions.

Provided a dual framework for point selection with capacity constraints.

Abstract

We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity. We provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings. Using this framework we get a flotilla of results on this problem (and also on its dual, where one wants to pick a maximum weight subset of the points when the regions have capacities). For example, for the case of fat triangles of similar size, we show an O(1)-approximation and prove that no \PTAS is possible.