On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic

Abstract

Given a set of n points in R d , the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the p -metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d = ω(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18], ).In this paper, we show that for every p ∈ R ≥1 ∪ {0}, under the Strong Exponential Time Hypothesis (SETH), for every ε > 0, the following holds:No algorithm running in time O(n 2−ε ) can solve the Closest Pair problem in d = (log n) Ωε(1) dimensions in the p -metric.ThereIn particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to n ε factor in the running time) for d = (log n) Ωε(1) dimensions.Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of n points in n o(1) -dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product.At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on n vertices with n 2−ε edges whose vertices can be realized as points in a (log n) Ωε(1) -dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].

SETH-Hardness of Coding Problems

“…(Our gadget differs slightly from [DMS03] in that we require all vectors to be at the same distance. The same idea is used in [AS18] in the context of lattices and in [KM19] in a very different context. )…”

Inapproximability of Clustering in Lp Metrics

“…Also, there was no techniques developed in previous works to address all p -metrics for the clustering problems. We make an interesting connection to contact dimension of a graph, motivated by recent advances in hardness of approximation in fine-grained complexity [30], [31]. Elaborating, from the vertex coverage instance G = (V, E) we create the bipartite graph on partite sets V and E where we have an edge (i, {j, j }) ∈ V × E if and only if i = j or i = j .…”

“…We remark here that the above definition is a variant of the notion gap contact dimension introduced in [31] in the sense that the authors in [31] required that for all distinct…”

On Hardness of Approximation of Parameterized Set Cover and Label Cover: Threshold Graphs from Error Correcting Codes

“…More concretely, one can ask if it is possible to recover using the DPCPF all the inapproximability results that are currently only obtained using the TGC technique and vice-versa? In [KM19] the authors made the connection that if one could construct certain high dimensional extremal combinatorial objects then it is possible to prove the inapproximability of k-One-Sided-Biclique through DPCPF (specifically by using the result of [KLM19] on k-MaxCover). However, the construction of the desired combinatorial objects seem far from reach using current techniques.…”