In edge-covering problem the goal is finding the minimum number of guards to cover the edges of a simple polygon. This problem is NP-hard, and to our knowledge there is just one approximation algorithm for a restricte...
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We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a vector cost ce ∈ R≥0. For a ...
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We consider the problem of aligning sequences related by a given evolutionary tree: given a fixed tree with its leaves labeled with sequences, find ancestral sequences to label the internal nodes so as to minimize the...
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We define and study a combinatorial problem called WEIGHTED DIAGNOSTIC COVER (WDC) that models the use of a laboratory technique called genotyping in the diagnosis of a important class of chromosomal aberrations. An o...
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We consider the problem of covering arbitrary polygons, without any acute interior angles, using a preferably minimum number of squares. The squares must lie entirely within the polygon. Let P be an arbitrary input po...
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In network activation problems we are given a directed or undirected graph G = (V,E) with a family {fuv (xu, xv) : (u,v) ∈ E} of monotone non-decreasing activation functions from D2 to {0,1}, where D is a constant-si...
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Graph routing problem (GRP) and its generalizations have been extensively studied because of their broad applications in the real world. In this paper, we study a variant of GRP called the general cluster routing prob...
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We consider the problem of aligning of k sequences of length n. The cost function is sum of pairs, and satisfies triangle inequality. Earlier results on finding approximation algorithms for this problem are due to Gus...
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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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ISBN:
(纸本)9783959771955
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 Õ(mn) time. So it is natural to resort to approximation algorithms in Õ(mn1-∈) time for some positive ∈. 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 Õ(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 Õ(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 by obtaining a 3/
Graph Burning models information spreading in a given graph as a process such that in each step one node is infected (informed) and also the infection spreads to all neighbors of previously infected nodes. Formally, g...
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