Tight Hardness Results for Maximum Weight Rectangles
Artūrs Bačkursa, Nishanth Dikkalaa, Christos Tzamosa
- aMassachusetts Institute of Technology, Cambridge, United States
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Abstract
Given $n$ weighted points (positive or negative) in $d$ dimensions, what is the axis-aligned 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 All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique problem in edge-weighted graphs are essentially optimal.
TL;DRThis paper answers the conjecture that this runtime of the Weighted Depth problem is tight up to subpolynomial factors affirmatively by providing a matching conditional lower bound and provides conditional lower bounds for the special case when points are arranged in a grid. (Semantic Scholar)