We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to w...
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Tverberg's theorem states that a set of n points in Rd can be partitioned into ⌈n/(d + 1)⌉ sets whose convex hulls all intersect. A point in the intersection (aka Tverberg point) is a centerpoint, or high-dimensio...
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The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we first consider these problems in the asymmetric str...
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We present an (eO(p)log /log log )-approximation algorithm for socially fair clustering with the p-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of ...
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In this article, we consider the Euclidean dispersion problems. Let P = {p1, p2, . . ., pn} be a set of n points in R2. For each point p ∈ P and S ⊆ P, we define costγ(p, S) as the sum of Euclidean distance from p t...
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In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum nu...
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The Lasserre Hierarchy, [18, 19], is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a hi...
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Graph parameters such as the diameter, radius, and vertex eccentricities are not defined in a useful way in Directed Acyclic Graphs (DAGs) using the standard measure of distance, since for any two nodes, there is no p...
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Graph parameters such as the diameter, radius, and vertex eccentricities are not defined in a useful way in Directed Acyclic Graphs (DAGs) using the standard measure of distance, since for any two nodes, there is no path between them in one of the two directions. So it is natural to consider the distance between two nodes as the length of the shortest path in the direction in which this path exists, motivating the definition of the min-distance. The min-distance between two nodes u and v is the minimum of the shortest path distances from u to v and from v to u. As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in O͂(mn) time. So it is natural to resort to approximation algorithms in O͂(mn1−ϵ) time for some positive ϵ. Abboud, Vassilevska W., and Wang [SODA 2016] first studied min-distance problems achieving constant factor approximation algorithms on DAGs, and Dalirrooyfard et al [ICALP 2019] gave the first constant factor approximation algorithms on general graphs for min-diameter, min-radius and min-eccentricities. Abboud et al obtained a 3-approximation algorithm for min-radius on DAGs which works in O͂(m√n) time, and showed that any (2 − δ)-approximation requires n2−o(1) time for any δ > 0, under the Hitting Set Conjecture. We close the gap, obtaining a 2-approximation algorithm which runs in O͂(m√n) time. As the lower bound of Abboud et al only works for sparse DAGs, we further show that our algorithm is conditionally tight for dense DAGs using a reduction from Boolean matrix multiplication. Moreover, Abboud et al obtained a linear time 2-approximation algorithm for min-diameter along with a lower bound stating that any (3/2 − δ)-approximation algorithm for sparse DAGs requires n2−o(1) time under SETH. We close this gap for dense DAGs up to an addit
Given two sets S and T of points in the plane, of total size n, a many-to-many matching between S and T is a set of pairs (p, q) such that p ∈ S, q ∈ T and for each r ∈ S ∪ T, r appears in at least one such pair. ...
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The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) prob...
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