Tight Hardness Results for Maximum Weight Rectangles

  title={Tight Hardness Results for Maximum Weight Rectangles},
  author={Arturs Backurs and Nishanth Dikkala and Christos Tzamos},
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… 

Tables from this paper

New hardness results for planar graph problems in p and an algorithm for sparsest cut
The lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P and is the first poly(n) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter.
Smallest k-Enclosing Rectangle Revisited
The first near quadratic time algorithm for this problem is presented, improving over the previous near- O(n^{5/2}) O ( n 5 / 2 ) -time algorithm by Kaplan et al.
Improving Viterbi is Hard: Better Runtimes Imply Faster Clique Algorithms
It is shown that the Viterbi algorithm runtime is optimal up to subpolynomial factors even when the number of distinct observations is small, 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 tight.
Polynomial formulations as a barrier for reduction-based hardness proofs
Evidence is provided that such fine-grained reductions will be difficult to derive Set Cover Conjecture from SETH, and that proving c n lower bound for any constant c > 1 (not just c = 2) under SETH for any of the problems above would imply new circuit lower bounds.
Average-Case Fine-Grained Hardness Marshall Ball
Based on the average-case hardness and structural properties of the functions, the construction of a Proof of Work scheme is outlined and possible approaches to constructing finegrained One-Way Functions are discussed.
Fine-grained Complexity Meets IP = PSPACE
The main contribution is showing reductions from exact to approximate solution for a host of such problems, and leveraging the techniques to show new barriers for deterministic approximation algorithms for LCS.
Towards Hardness of Approximation for Polynomial Time Problems
A framework that exhibits barriers for truly subquadratic and deterministic algorithms with good approximation guarantees is introduced and highlights a novel connection between deterministic approximation algorithms for natural problems in P and circuit lower bounds.
Better Approximations for Tree Sparsity in Nearly-Linear Time
This work designs (1+e)-approximation algorithms for the Tree Sparsity problem that run in nearly-linear time, and shows that if the exact version of the TreeSparsity problem can be solved in strongly subquadratic time, then the (min, +) convolution problem can been solved in strong subquadraatic time as well.
More consequences of falsifying SETH and the orthogonal vectors conjecture
It is shown that if the OV-conjecture fails, then two problems for which the authors are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms.
Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n5/3) Time
This work presents an explicit and efficient construction of additively weighted Voronoi diagrams on planar graphs, and uses this construction to compute the diameter of a directed planar graph with real arc lengths in $\tilde{O}(n^{5/3})$ time.


D S ] 3 M ar 2 01 6 Tight Hardness Results for Maximum Weight Rectangles
This paper provides conditional lower bounds for the special case when points are arranged in a grid (a well studied problem known as M AXIMUM SUBARRAY problem) as well as for other related problems, based on assumptions that the best own algorithms for the ALL -PAIRS SHORTEST PATHS problem (APSP) and for the M AX -WEIGHT k-CLIQUE problem in edge-weighted graphs are essentially optimal.
Matching Triangles and Basing Hardness on an Extremely Popular Conjecture
Novel reductions from 3-SUM, APSP, and CNF-SAT are designed, and interesting consequences of this very plausible conjecture are derived, including tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs and new conditional lower bound for the Single-Source-Max-Flow problem.
On a class of O(n2) problems in computational geometry
Klee's Measure Problem Made Easy
  • Timothy M. Chan
  • Computer Science
    2013 IEEE 54th Annual Symposium on Foundations of Computer Science
  • 2013
We present a new algorithm for a classic problem in computational geometry, Klee's measure problem: given a set of n axis-parallel boxes in d-dimensional space, compute the volume of the union of the
Faster all-pairs shortest paths via circuit complexity
A new randomized method for computing the min-plus product of two n × n matrices is presented, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense n-node directed graphs with arbitrary edge weights.
Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems
It is proved that sufficient progress would imply a breakthrough on one of five major open problems in the theory of algorithms, including dynamic versions of bipartite perfect matching, bipartites maximum weight matching, single source reachability, single sources shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems.
On Dynamic Shortest Paths Problems
Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest- Paths problem.