A better lower bound for Lower-Left Anchored Rectangle Packing

TL;DR

This paper improves the lower bound on the total area covered by axis-aligned rectangles anchored at points in a unit square, advancing understanding of Freedman's conjecture and analyzing greedy algorithms for the problem.

Contribution

It provides a new lower bound of 0.1039 for the area covered, using a novel analysis of a greedy algorithm, and establishes an upper bound of 3/4 for a class of greedy algorithms.

Findings

Lower bound on covered area increased from 0.09121 to 0.1039.

Greedy algorithm achieves at least 0.1039 approximation ratio.

Upper bound of 3/4 on approximation ratio for certain greedy algorithms.

Abstract

Given any set of points $S$ in the unit square that contains the origin, does a set of axis aligned rectangles, one for each point in $S$, exist, such that each of them has a point in $S$ as its lower-left corner, they are pairwise interior disjoint, and the total area that they cover is at least 1/2? This question is also known as Freedman's conjecture (conjecturing that such a set of rectangles does exist) and has been open since Allen Freedman posed it in 1969. In this paper, we improve the best known lower bound on the total area that can be covered from 0.09121 to 0.1039. Although this step is small, we introduce new insights that push the limits of this analysis. Our lower bound uses a greedy algorithm with a particular order of the points in $S$. Therefore, it also implies that this greedy algorithm achieves an approximation ratio of 0.1039. We complement the result with an upper bound of 3/4 on the approximation ratio for a natural class of greedy algorithms that includes the one that achieves the lower bound.