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QPTAS for geometric setcover problems via optimal separators
, 2014
"... Weighted geometric setcover problems arise naturally in several geometric and nongeometric settings (e.g. the breakthrough of BansalPruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric setcover). More than two decades of research has succeeded in settling ..."
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Weighted geometric setcover problems arise naturally in several geometric and nongeometric settings (e.g. the breakthrough of BansalPruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric setcover). More than two decades of research has succeeded in settling the (1+)approximability status for most geometric setcover problems, except for four basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasisampling technique, which together with improvements by Chan et al. (SODA 2012), yielded a O(1)approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes disks, unitheight rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces inR3, for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of AdamaszekWiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APXhard, assuming NP * DTIME(2polylog(n)). Together with the recent work of ChanGrant (CGTA 2014), this settles the APXhardness status for all natural geometric setcover problems.
Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation
, 2014
"... We revisit two NPhard geometric partitioning problems – convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasipolynomial time algorithms for these problems with improved approximation guarantees. ..."
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We revisit two NPhard geometric partitioning problems – convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasipolynomial time algorithms for these problems with improved approximation guarantees.
Approximation Algorithms for PolynomialExpansion and LowDensity Graphs
, 2015
"... We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion. We present efficient (1 + ε)approximation algorith ..."
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We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion. We present efficient (1 + ε)approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTAS’s for these problems are known for subclasses of this graph family. These results have immediate interesting applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow). We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth.
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"... Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams? ..."
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Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams?
A QPTAS for the Base of the Number of Triangulations of a Planar Point Set
"... The number of triangulations of a planar n point set is known to be cn, where the base c lies between 2.43 and 30. The fastest known algorithm for counting triangulations of a planar n point set runs in O∗(2n) time. The fastest known arbitrarily close approximation algorithm for the base of the numb ..."
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The number of triangulations of a planar n point set is known to be cn, where the base c lies between 2.43 and 30. The fastest known algorithm for counting triangulations of a planar n point set runs in O∗(2n) time. The fastest known arbitrarily close approximation algorithm for the base of the number of triangulations of a planar n point set runs in time subexponential in n. We present the first quasipolynomial approximation scheme for the base of the number of triangulations of a planar point set. 1
A Variant of the Maximum Weight Independent Set Problem
, 2014
"... We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph G = (V,E), a weight function w: V → R+, a budget function b: V → Z+, and a positive integer B. The weight (resp. budget) of a sub ..."
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We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph G = (V,E), a weight function w: V → R+, a budget function b: V → Z+, and a positive integer B. The weight (resp. budget) of a subset of vertices is the sum of weights (resp. budgets) of the vertices in the subset. A kbudgeted independent set in G is a subset of vertices, such that no pair of vertices in that subset are adjacent, and the budget of the subset is at most k. The goal is to find a Bbudgeted independent set in G such that its weight is maximum among all the Bbudgeted independent sets in G. We refer to this problem as MWBIS. Being a generalization of MWIS, MWBIS also has several applications in Scheduling, Wireless networks and so on. Due to the hardness results implied from MWIS, we study the MWBIS problem in several special classes of graphs. We design exact algorithms for trees, forests, cycle graphs, and interval graphs. In unweighted case we design an approximation algorithm for d+ 1claw free graphs whose approximation ratio (d) is competitive with the approximation ratio (d2) of MWIS (unweighted). Furthermore, we extend Baker’s technique [6] to get a PTAS for MWBIS in planar graphs. 1
A Separator Theorem for Intersecting Objects in the Plane
"... Separators in graphs are instrumental in the design of algorithms, having proven to be the key technical tool in approximation algorithms for many optimization problems. In the geometric setting, this naturally translates into the study of separators in the intersection graphs of geometric objects. ..."
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Separators in graphs are instrumental in the design of algorithms, having proven to be the key technical tool in approximation algorithms for many optimization problems. In the geometric setting, this naturally translates into the study of separators in the intersection graphs of geometric objects. Recently a number of new separator theorems have been proven for the case of geometric objects in the plane. In this paper we present a new separator theorem that unifies and generalizes some earlier results. 1