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24
Approximation Algorithms for Maximum Independent Set of PseudoDisks
, 2008
"... We present approximation algorithms for maximum independent set of pseudodisks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation ..."
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Cited by 22 (4 self)
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We present approximation algorithms for maximum independent set of pseudodisks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation of the problem, that leads to a constantfactor approximation. Most previous algorithms for maximum independent set (in geometric settings) relied on packing arguments that are not applicable in this case. As such, the analysis of both algorithms requires some new combinatorial ideas, which we believe to be of independent interest.
A constant factor approximation algorithm for unsplittable flow on paths
 In Proceedings of the 52th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011
, 2011
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A QPTAS for maximum weight independent set of polygons with polylogarithmically many vertices
 In Proceedings of the TwentyFifth Annual ACMSIAM Symposium on Discrete Algorithms, SODA 2014
"... The Maximum Weight Independent Set of Polygons (MWISP) problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the twodimensional plane, the goal is to find a set of pairwise nonoverlapping polygons with maximum total weight. Due to its wide range of applica ..."
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Cited by 8 (0 self)
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The Maximum Weight Independent Set of Polygons (MWISP) problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the twodimensional plane, the goal is to find a set of pairwise nonoverlapping polygons with maximum total weight. Due to its wide range of applications and connections to other problems, the MWISP problem and its special cases have been extensively studied both in the approximation algorithms and the computational geometry community. Despite a lot of research, its general case is not wellunderstood yet. Currently the best known polynomial time algorithm achieves an approximation ratio of n [Fox and Pach, SODA 2011], and it is not even clear whether the problem is APXhard. We present a (1+)approximation algorithm, assuming that each polygon in the input has at most a polylogarithmic number of vertices. Our algorithm has quasipolynomial running time, i.e., it runs in time 2poly(logn,1/). In particular, our result implies that for this setting the problem is not APXhard, unless NP ⊆ DTIME(2poly(logn)). We use a recently introduced framework for approximating maximum weight independent set in geometric intersection graphs. The framework has been used to construct a QPTAS in the much simpler case of axisparallel rectangles. We extend it in two ways, to adapt it to our much more general setting. First, we show that its technical core can be reduced to the case when all input polygons are triangles. Secondly, we replace its key technical ingredient which is a method to partition the plane using only few edges such that the objects stemming from the optimal solution are evenly distributed among the resulting faces and each object is intersected only a few times. Our new procedure for this task is not even more complex than the original one and, importantly, it can handle the arising difficulties due to the arbitrary angles of the input polygons. Note that already this obstacle makes the known analysis for the above framework fail. Also, in general it is not well understood how to handle this difficulty by efficient approximation algorithms.
Quasipolynomial time approximation scheme for sparse subsets of polygons
 In Proc. 30th Annu. Sympos. Comput. Geom. (SoCG
, 2014
"... We describe how to approximate, in quasipolynomial time, the largest independent set of polygons, in a given set of polygons. Our algorithm works by extending the result of Adamaszek and Wiese [AW13, AW14] to polygons of arbitrary complexity. Surprisingly, the algorithm also works for computing the ..."
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Cited by 6 (1 self)
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We describe how to approximate, in quasipolynomial time, the largest independent set of polygons, in a given set of polygons. Our algorithm works by extending the result of Adamaszek and Wiese [AW13, AW14] to polygons of arbitrary complexity. Surprisingly, the algorithm also works for computing the largest subset of the given set of polygons that has some sparsity condition. For example, we show that one can approximate the largest subset of polygons, such that the intersection graph of the subset does not contain a cycle of length 4 (i.e., K2,2). 1.
On isolating points using disks
 In Proceedings of the 19th European Conference on Algorithms, ESA’11
, 2011
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Wimax/ofdma burst scheduling algorithm to maximize scheduled data
 IEEE Transactions on Mobile Computing
, 2011
"... Abstract—OFDMA resource allocation algorithms manage the distribution and assignment of shared OFDMA resources among the users serviced by the basestation. The OFDMA resource allocation algorithms determine which users to schedule, how to allocate subcarriers to them, and how to determine the approp ..."
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Cited by 4 (1 self)
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Abstract—OFDMA resource allocation algorithms manage the distribution and assignment of shared OFDMA resources among the users serviced by the basestation. The OFDMA resource allocation algorithms determine which users to schedule, how to allocate subcarriers to them, and how to determine the appropriate power levels for each user on each subcarrier. In WiMAX, the downlink (DL) TDD OFDMA subframe structure is a rectangular area of N subchannels K time slots. Users are assigned rectangular bursts in the downlink subframe. The size of burst, in terms of number of subchannels and number of time slots, varies based on the user’s channel quality and data to be transmitted for the assigned user. In this paper, we study the problem of assigning users to bursts in WiMAX TDD OFDMA system with the objective of maximizing downlink system throughput for the Partially Used subcarrier (PUSC) subchannalization permutation mode. Our main contributions in this paper are: 1) we propose different methods to assign bursts to users, 2) we prove that our Best Channel burst assignment method achieves throughput within a constant factor of the optimal, 3) through extensive simulations with real system parameters, we study the performance of the Best Channel burst assignment method. To the best of our knowledge, we are the first to study the problem of DL Burst Assignment in the downlink OFDMA subframe for PUSC subchannalization permutation mode taking user’s channel quality into consideration in the assignment process. Index Terms—WiMAX, OFDMA, wireless scheduling, burst scheduling, throughput maximization Ç 1
Truthful Mechanisms for Exhibitions
"... We consider the following combinatorial auction: Given a range space (U, R), and m bidders interested in buying only ranges in R, each bidder j declares her bid bj: R → R+. We give a deterministic truthful mechanism, when the valuations are singleminded: when R is a collection of fat objects (respe ..."
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Cited by 4 (1 self)
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We consider the following combinatorial auction: Given a range space (U, R), and m bidders interested in buying only ranges in R, each bidder j declares her bid bj: R → R+. We give a deterministic truthful mechanism, when the valuations are singleminded: when R is a collection of fat objects (respectively, axisaligned rectangles) in the plane, there is a truthful mechanism with a 1 + ɛ (respectively, ⌈log n⌉)approximation of the social welfare (where n is an upper bound on the maximum integral coordinate of each rectangle). We also consider the nonsingleminded case, and design a randomized truthfulinexpectation mechanism with approximation guarantee O(1) (respectively, O(log m)).
Constant Integrality Gap LP Formulations of Unsplittable Flow on a Path
, 2013
"... The Unsplittable Flow Problem on a Path (UFPP) isacore problem in many important settings such as network flows, bandwidth allocation, resource constraint scheduling, and interval packing. We are given a path with capacities on the edges and a set of tasks, each task having a demand, a profit, a sou ..."
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Cited by 4 (4 self)
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The Unsplittable Flow Problem on a Path (UFPP) isacore problem in many important settings such as network flows, bandwidth allocation, resource constraint scheduling, and interval packing. We are given a path with capacities on the edges and a set of tasks, each task having a demand, a profit, a source and a destination vertex on the path. The goal is to compute a subset of tasks of maximum profit that does not violate the edge capacities. In practical applications generic approaches such as integer programming (IP) methods are desirable. Unfortunately, no IPformulation is known for the problem whose LPrelaxation has an integrality gap that is provably constant. For the unweighted case, we show that adding a few constraints to the standard LP of the problem is sufficient to make the integrality gap drop from Ω(n) to O(1). This positively answers an open question in [Chekuri et al., APPROX 2009]. For the general (weighted) case, we present an extended formulation with integrality gap bounded by 7+ε. This matches the best known approximation factor for the problem [Bonsma et al., FOCS 2011]. This result exploits crucially a technique for embedding dynamic programs into linear programs. We believe that this method could be useful to strengthen LPformulations for other problems as well and might eventually speed up computations due to stronger problem formulations.
Independent and hitting sets of rectangles intersecting a diagonal line
 In LATIN 2014: Theoretical Informatics Lecture Notes in Computer Science 8392
, 2014
"... Abstract. Finding a maximum independent set (MIS) of a given family of axisparallel rectangles is a basic problem in computational geometry and combinatorics. This problem has attracted significant attention since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the ..."
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Abstract. Finding a maximum independent set (MIS) of a given family of axisparallel rectangles is a basic problem in computational geometry and combinatorics. This problem has attracted significant attention since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the maximum possible ratio between the maximum independent set and the minimum hitting set (MHS), is bounded by a universal constant. An interesting special case, that may prove useful to tackling the general problem, is the diagonalintersecting case, in which the given family of rectangles is intersected by a diagonal. Indeed, Chepoi and Felsner recently gave a factor 6 approximation algorithm for MHS in this setting, and showed that the duality gap is between 3/2 and 6. In this paper we improve upon these results. First we show that MIS in diagonalintersecting families is NPcomplete, providing one smallest subclass for which MIS is provably hard. Then, we derive an O(n2)time algorithm for the maximum weight independent set when, in addition the rectangles intersect below the diagonal. This improves and extends a classic result of Lubiw, and amounts to obtain a 2approximation algorithm for the maximum weight independent set of rectangles intersecting a diagonal. Finally, we prove that for diagonalintersecting families the duality gap is between 2 and 4. The upper bound, which implies an approximation algorithm of the same factor, follows from a simple combinatorial argument, while the lower bound represents the best known lower bound on the duality gap, even in the general case. An extended abstract of a preliminary version of this work appears in the proceedings