Results 1  10
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28
Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs
, 1999
"... We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performa ..."
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Cited by 95 (6 self)
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We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performance ratio of 2  (1  o(1)) 2 ln ln \Delta ln \Delta for large \Delta, which improves the previously known ratio of 2 \Gamma log \Delta+O(1) \Delta obtained by Halldórsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For kuniform hypergraphs with n vertices, we achieve a ratio of k \Gamma (1 \Gamma o(1)) k ln ln n ln n for large n, and for kuniform hypergraphs with maximum degree at most \Delta, the algorithm achieves a ratio of k \Gamma (1 \Gamma o(1)) k(k\Gamma1) ln ln \Delta ln \Delta for large \Delta. These results considerably improve the previous best ratio of k(1\Gammac=\Delta 1 k\Gamma1 ) for bounded degree kuniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general kuniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldórsson.
Covering problems with hard capacities
 IN PROC OF. FOCS’02
, 2002
"... We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generali ..."
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Cited by 33 (0 self)
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We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generalization of the classical problems that also captures resource limitations in practical scenarios. We obtain the following results. For the unweighted vertex cover problem with hard capacities we give aapproximation algorithm which is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem. This is an interesting separation between the approximability of weighted and unweighted versions of a “natural ” graph problem. A logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of Wolsey [23] on submodular set cover. We provide in this paper a simple and intuitive proof for this bound.
Approximating the Weight of Shallow Steiner Trees
 DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science
, 1998
"... This paper deals with the problem of constructing Steiner trees of minimum weight with diameter bounded by d, spanning a given set of k vertices in a graph. Exact solutions or logarithmic ratio approximation algorithms were known before for the cases of d <= 5. Here we give a polynomial time approxi ..."
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Cited by 29 (3 self)
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This paper deals with the problem of constructing Steiner trees of minimum weight with diameter bounded by d, spanning a given set of k vertices in a graph. Exact solutions or logarithmic ratio approximation algorithms were known before for the cases of d <= 5. Here we give a polynomial time approximation algorithm of ratio O(log k) for constant d, which is asymptotically optimal unless P = NP , and an algorithm of ratio O( k^{\epsilon})), for any fixed 0 < \epsilon < 1, for general d. Keywords: NPhard problems, approximation algorithms, Steiner trees 1 Introduction 1.1 The problem This paper considers the problem of finding low diameter Steiner trees of minimum weight. Given an nvertex graph G(V
Approximating BuyatBulk and Shallowlight kSteiner trees
 In Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems
, 2006
"... Abstract We study two related network design problems with two cost functions. In the buyatbulk kSteiner tree problem we are given a graph G(V, E) with a set of terminals T ` V including aparticular vertex s called the root, and an integer k < = T . There are two cost functions on theedges of G ..."
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Cited by 16 (2 self)
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Abstract We study two related network design problems with two cost functions. In the buyatbulk kSteiner tree problem we are given a graph G(V, E) with a set of terminals T ` V including aparticular vertex s called the root, and an integer k < = T . There are two cost functions on theedges of G, a buy cost b: E! R+ and a distance cost r: E! R+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost Pe2H b(e)+Pt2Ts dist(t, s) is minimized, where dist(t, s) is the distance from t to s in H with respect to the r cost. We present an O(log4 n)approximation algorithm for the buyatbulk kSteiner tree problem. The second andclosely related one is bicriteria approximation algorithm for Shallowlight kSteiner trees. In theshallowlight kSteiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under bcost) kSteiner tree such thatthe diameter under rcost is at most some given bound D. We develop an (O(log n), O(log3 n))approximation algorithm for a relaxed version of Shallowlight kSteiner tree where the solutionhas at least k 8 terminals. Using this we obtain an (O(log 2 n), O(log4 n))approximation algorithm
Approximation khop minimumspanning trees
, 2004
"... Given a complete graph on n nodes with metric edge costs, the minimumcost k hop spanning tree (kHMST) problem asks for a spanning tree of minimum total cost such that the longest rootleafpath in the tree has at most k edges. We present an algorithm that computes such a tree of total expected co ..."
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Cited by 12 (0 self)
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Given a complete graph on n nodes with metric edge costs, the minimumcost k hop spanning tree (kHMST) problem asks for a spanning tree of minimum total cost such that the longest rootleafpath in the tree has at most k edges. We present an algorithm that computes such a tree of total expected cost O(log n) times that of a minimumcost khop spanningtree.
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 11 (1 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.
RandomTree Diameter and the DiameterConstrained MST
 MST,” Congressus Numerantium
, 2000
"... A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tre ..."
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Cited by 9 (1 self)
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A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The DiameterConstrained MST (DCMST) problem can be stated as follows: given an undirected, edgeweighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NPcomplete, for all values of k; 4 k #n  2). In this paper, we investigate the behavior of the diameter of MST in randomlyweighted complete graphs (in ErdsRnyi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is smallindependent of n, we present a onetimetreeconstruction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of t...
Computing A DiameterConstrained Minimum Spanning Tree
, 2001
"... In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path wi ..."
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Cited by 8 (0 self)
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In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NPcomplete for all values of k; 4 k (n  2), except when all edgeweights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree datastructure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomialtime algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter4 tree is also used for evaluating the quality of o...
Set Cover Revisited: Hypergraph Cover with Hard Capacities
"... In this paper, we consider generalizations of classical covering problems to handle hard capacities. In the hard capacitated set cover problem, additionally each set has a covering capacity which we are not allowed to exceed. In other words, after picking a set, we may cover at most a specified num ..."
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Cited by 3 (1 self)
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In this paper, we consider generalizations of classical covering problems to handle hard capacities. In the hard capacitated set cover problem, additionally each set has a covering capacity which we are not allowed to exceed. In other words, after picking a set, we may cover at most a specified number of elements. Based on the classical results by Wolsey, an O(logn) approximation follows for this problem. Chuzhoy and Naor [FOCS 2002], first studied the special case of unweighted vertex cover with hard capacities and developed an elegant 3 approximation for it based on rounding a natural LP relaxation. This was subsequently improved to a 2 approximation by Gandhi et al. [ICALP 2003]. These results are surprising in light of the fact that for weighted vertex cover with hard capacities, the problem is at least as hard as set cover to approximate. Hence this separates the unweighted problem from the weighted version. The set cover hardness precludes the possibility of a constant factor approximation for the hardcapacitated vertex cover problem on weighted graphs. However,
A note on two source location problems
"... We consider Source Location (SL) problems: given a capacitated network G = (V,E), cost c(v) and a demand d(v) for every v ∈ V, choose a mincost S ⊆ V so that λ(v,S) ≥ d(v) holds for every v ∈ V, where λ(v,S) is the maximum flow value from v to S. In the directed variant, we have demands d in (v) an ..."
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Cited by 2 (1 self)
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We consider Source Location (SL) problems: given a capacitated network G = (V,E), cost c(v) and a demand d(v) for every v ∈ V, choose a mincost S ⊆ V so that λ(v,S) ≥ d(v) holds for every v ∈ V, where λ(v,S) is the maximum flow value from v to S. In the directed variant, we have demands d in (v) and d out (v) and we require λ(S,v) ≥ d in (v) and λ(v,S) ≥ dout (v). Undirected SL is (weakly) NPhard on stars with r(v) = 0 for all v except the center. But, it is known to be polynomially solvable for uniform costs and uniform demands. For general instances, both directed an undirected SL admit a (ln D+1)approximation algorithms, where D is the sum of the demands; up to constant this is tight, unless P=NP. We give a pseudopolynomial algorithm for undirected SL on trees with running time O(V  ∆ 3), where ∆ = maxv∈V d(v). This algorithm is used to derive a linear time algorithm for undirected SL with ∆ ≤ 3. We also consider the Single Assignment Source Location (SASL) where every v ∈ V should be assigned to a single node s(v) ∈ S. While the undirected SASL is in P, we give a (ln V  + 1)approximation algorithm for the directed case, and show that this is tight, unless P=NP. 1