Tight Hardness Results for Maximum Weight Rectangles
Abstract
Given $n$ weighted points (positive or negative) in $d$ dimensions, what is the axisaligned box which maximizes the total weight of the points it contains? The best known algorithm for this problem is based on a reduction to a related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in time $O(n^d)$. It was conjectured [Barbay et al., CCCG'13] that this runtime is tight up to subpolynomial factors. We answer this conjecture affirmatively by providing a matching conditional lower bound. We also provide conditional lower bounds for the special case when points are arranged in a grid (a well studied problem known as Maximum Subarray problem) as well as for other related problems. All our lower bounds are based on assumptions that the best known algorithms for the AllPairs Shortest Paths problem (APSP) and for the MaxWeight kClique problem in edgeweighted graphs are essentially optimal.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 arXiv:
 arXiv:1602.05837
 Bibcode:
 2016arXiv160205837B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Computer Science  Computational Geometry