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21
Dependent rounding and its applications to approximation algorithms
 JOURNAL OF THE ACM
, 2006
"... We develop a new randomized rounding approach for fractional vectors defined on the edgesets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include: ffl low congestion multipath rout ..."
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Cited by 60 (7 self)
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We develop a new randomized rounding approach for fractional vectors defined on the edgesets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include: ffl low congestion multipath routing; ffl richer randomgraph models for graphs with a given degreesequence; ffl improved approximation algorithms for: (i) throughputmaximization in broadcast scheduling, (ii) delayminimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and
Parameterized complexity of generalized vertex cover problems
 In Proc. 9th WADS, volume 3608 of LNCS
, 2005
"... Abstract. Important generalizations of the Vertex Cover problem ..."
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Cited by 27 (3 self)
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Abstract. Important generalizations of the Vertex Cover problem
Parameterized Complexity of Vertex Cover Variants
, 2006
"... Important variants of the Vertex Cover problem (among others, Connected Vertex Cover, Capacitated Vertex Cover, and Maximum ..."
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Cited by 23 (5 self)
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Important variants of the Vertex Cover problem (among others, Connected Vertex Cover, Capacitated Vertex Cover, and Maximum
Capacitated domination and covering: A parameterized perspective
 Proceedings 3rd International Workshop on Parameterized and Exact Computation, IWPEC 2008
"... Capacitated versions of Dominating Set and Vertex Cover have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for t ..."
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Cited by 16 (9 self)
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Capacitated versions of Dominating Set and Vertex Cover have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for the capacitated versions. Here we make an attempt to understand the behavior of the problems Capacitated Dominating Set and Capacitated Vertex Cover from the perspective of parameterized complexity. The original versions of these problems, Vertex Cover and Dominating Set, are known to be fixed parameter tractable when parameterized by a structure of the graph called the treewidth (tw). In this paper we show that the capacitated versions of these problems behave differently. Our results are: • Capacitated Dominating Set is W[1]hard when parameterized by treewidth. In fact, Capacitated Dominating Set is W[1]hard when parameterized by both treewidth and solution size k of the capacitated dominating set. • Capacitated Vertex Cover is W[1]hard when parameterized by treewidth. • Capacitated Vertex Cover can be solved in time 2O(tw log k) nO(1) where tw is the treewidth of the input graph and k is the solution size. As a corollary, we show that the weighted version of Capacitated Vertex Cover in general graphs can be solved in time 2O(k log k) nO(1). This improves the earlier algorithm of Guo et al. [15] running in time O(1.2k2 + n2). We would also like to point out that our W[1]hardness result for Capacitated Vertex Cover, when parameterized by treewidth, makes it (to the best of our knowledge) the first known “subset problem ” which has turned out to be fixed parameter tractable when parameterized by solution size but W[1]hard when parameterized by treewidth. 1
Algorithms for capacitated rectangle stabbing and lotsizing with joint setup costs
, 2007
"... In the rectangle stabbing problem we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this paper we study the capacitated version of this problem in which the input inc ..."
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Cited by 15 (2 self)
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In the rectangle stabbing problem we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this paper we study the capacitated version of this problem in which the input includes an integral capacity for each line. The capacity of a line bounds the number of rectangles that the line can cover. We consider two versions of this problem. In the first, one is allowed to use only a single copy of each line (hard capacities), and in the second, one is allowed to use multiple copies of every line, but the multiplicities are counted in the size (or weight) of the solution (soft capacities). We present an exact polynomialtime algorithm for the weighted one dimensional case with hard capacities that can be extended to the one dimensional weighted case with soft capacities. This algorithm is also extended to solve a certain capacitated multiitem lot sizing inventory problem with joint setup costs. For the case of ddimensional rectangle stabbing with soft capacities, we present a 3dapproximation algorithm for the unweighted case. For ddimensional rectangle stabbing problem with hard capacities, we present a bicriteria algorithm that computes 4dapproximate solutions that use at most two copies of every line. Finally, we present hardness results for rectangle stabbing when the dimension is part of the input and for a twodimensional weighted version with hard capacities.
A primaldual approximation algorithm for partial vertex cover: Making educated guesses
 In 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, volume 3624 of LNCS
, 2005
"... We study the partial vertex cover problem. Given a graph G = (V,E), a weight function w: V → R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NPhard as it generalizes the wellknown vertex cover problem. We provide ..."
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Cited by 13 (2 self)
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We study the partial vertex cover problem. Given a graph G = (V,E), a weight function w: V → R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NPhard as it generalizes the wellknown vertex cover problem. We provide a primaldual 2approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity ku. A solution consists of a function x: V → N0 and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most xuku. Our objective is to find a cover that minimizes � v∈V xvwv. This is the first 2approximation for the problem and also runs in O(nlog n + m) time.
A PrimalDual Bicriteria Distributed Algorithm for Capacitated Vertex Cover
, 2008
"... In this paper we consider the capacitated vertex cover problem which is the variant of vertex cover where each node is allowed to cover a limited number of edges. We present an efficient, deterministic, distributed approximation algorithm for the problem. Our algorithm computes a (2 + ǫ)approximate ..."
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Cited by 5 (0 self)
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In this paper we consider the capacitated vertex cover problem which is the variant of vertex cover where each node is allowed to cover a limited number of edges. We present an efficient, deterministic, distributed approximation algorithm for the problem. Our algorithm computes a (2 + ǫ)approximate solution which violates the capacity constraints by a factor of (4 + ǫ) in a polylogarithmic number of communication rounds. On the other hand, we also show that every efficient distributed approximation algorithm for this problem must violate the capacity constraints. Our result is achieved in two steps. We first develop a 2approximate, sequential primaldual algorithm that violates the capacity constraints by a factor of 2. Subsequently, we present a distributed version of this algorithm. We demonstrate that the sequential algorithm has an inherent need for synchronization which forces any naive distributed implementation to use a linear number of communication rounds. The challenge in this step is therefore to achieve a reduction of the communication complexity to a polylogarithmic number of rounds without worsening the approximation guarantee too much.
Packing to Angles and Sectors
 IN: GIBBONS, P. B., SCHEIDELER, C. (EDS.) SPAA
, 2007
"... Motivated by the widespread proliferation of wireless networks employing directional antennas, we study some capacitated covering problems arising in these networks. Geometrically, the area covered by a directional antenna with parameters α, ρ, ¯r is a set of points with polar coordinates (r, θ) suc ..."
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Cited by 4 (2 self)
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Motivated by the widespread proliferation of wireless networks employing directional antennas, we study some capacitated covering problems arising in these networks. Geometrically, the area covered by a directional antenna with parameters α, ρ, ¯r is a set of points with polar coordinates (r, θ) such that r ≤ ¯r and α ≤ θ ≤ α + ρ. Given a set of customers, their positions on the plane and their bandwidth demands, the capacitated covering problem considered here is to cover all the customers with the minimum number of directional antennas such that the demands of customers assigned to an antenna stays within a bound. We consider two settings of this capacitated cover problem arising in wireless networks. In the first setting where the antennas have variable angular range, we present an approximation algorithm with ratio 3. In the setting where the angular range of antennas is fixed, we improve this approximation ratio to 1.5. These results also apply for a related problem of bin packing with deadlines. In this problem we are are given a set of items, each with a weight, arrival time and deadline, and we want to pack each item into a bin after it arrives but before its deadline. The objective is to minimize the number of bins used. We present a 3approximation algorithm for this problem, and 1.5approximation algorithm for the special case when each difference between a deadline and the corresponding arrival time is the same.
Capacitated Domination Problem
"... We consider a generalization of the wellknown domination problem on graphs. The (soft) capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex ..."
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Cited by 3 (2 self)
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We consider a generalization of the wellknown domination problem on graphs. The (soft) capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex has a capacity that it can use to meet the demand of dominated vertices in its closed neighborhood, and the number of copies of each vertex allowed in D is unbounded. The demand constraint specifies that the demand of each vertex in V is met by the capacities of vertices in D dominating it. In this paper, we study the capacitated domination problem on trees from an algorithmic point of view. We present a linear time algorithm for the unsplittable demand model, and a pseudopolynomial time algorithm for the splittable demand model. In addition, we show that the capacitated domination problem on trees with splittable demand constraints is NPcomplete (even for its integer version) and provide a 3/2approximation algorithm. We also give a primaldual approximation algorithm for the weighted capacitated domination problem with splittable demand constraints on general graphs.
The multiradius cover problem
 In Proceedings of the 9th International Workshop on Algorithms and Data Structures (WADS’05
, 2005
"... Abstract. Let G = (V, E) be a graph with a nonnegative edge length lu,v for every (u, v) ∈ E. The vertices of G represent locations at which transmission stations are positioned, and each edge of G represents a continuum of demand points to which we should transmit. A station located at v is assoc ..."
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Cited by 2 (1 self)
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Abstract. Let G = (V, E) be a graph with a nonnegative edge length lu,v for every (u, v) ∈ E. The vertices of G represent locations at which transmission stations are positioned, and each edge of G represents a continuum of demand points to which we should transmit. A station located at v is associated with a set Rv of allowed transmission radii, where the cost of transmitting to radius r ∈ Rv is given by cv(r). The multiradius cover problem asks to determine for each station a transmission radius, such that for each edge (u, v) ∈ E the sum of the radii in u and v is at least lu,v, and such that the total cost is minimized. In this paper we present LProunding and primaldual approximation algorithms for discrete and continuous variants of multiradius cover. Our algorithms cope with the special structure of the problems we consider by utilizing greedy rounding techniques and a novel method for constructing primal and dual solutions.