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55
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.
A constantfactor approximation algorithm for the multicommodity rentorbuy problem
 In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science
, 2002
"... We present the first constantfactor approximation algorithm for network design with multiple commodities and economies of scale. We consider the rentorbuy problem, a type of multicommodity buyatbulk network design in which there are two ways to install capacity on any given edge. Capacity can b ..."
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Cited by 83 (9 self)
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We present the first constantfactor approximation algorithm for network design with multiple commodities and economies of scale. We consider the rentorbuy problem, a type of multicommodity buyatbulk network design in which there are two ways to install capacity on any given edge. Capacity can be rented, with cost incurred on a perunit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of sourcesink pairs, we seek a minimumcost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. Recent work on buyatbulk network design has concentrated on the special case where all sinks are identical; existing constantfactor approximation algorithms for this special case make crucial use of the assumption that all commodities ship flow to the same sink vertex and do not obviously extend to the multicommodity rentorbuy problem. Prior to our work, the best heuristics for the multicommodity rentorbuy problem achieved only logarithmic performance guarantees and relied on the machinery of relaxed metrical task systems or of metric embeddings. By contrast, we solve the network design problem directly via a novel primaldual algorithm. 1
Facility Location with Nonuniform Hard Capacities
 Proceedings of the 42nd IEEE Symposium on the Foundations of Computer Science
, 2001
"... In this paper we give the first constant factor approximation algorithm for the Facility Location Problem with nonuniform, hard capacities. Facility location problems have received a great deal of attention in recent years. Approximation algorithms have been developed for many variants. Most of thes ..."
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Cited by 43 (0 self)
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In this paper we give the first constant factor approximation algorithm for the Facility Location Problem with nonuniform, hard capacities. Facility location problems have received a great deal of attention in recent years. Approximation algorithms have been developed for many variants. Most of these algorithms are based on linear programming, but the LP techniques developed thus far have been unsuccessful in dealing with hard capacities.
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.
On the Approximability of Some Network Design Problems
"... Consider the following classical network design problem: a set of terminals T: {t.i} wants to send traffic to a "root" r in an 'xnode graph G: (V, E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e ca ..."
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Cited by 28 (3 self)
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Consider the following classical network design problem: a set of terminals T: {t.i} wants to send traffic to a "root" r in an 'xnode graph G: (V, E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some base capacity ue and hence provisioning k x ue bandwidth on edge e incurs a cost of [k] times the cost of that edge. The objective is a minimumcost feasible solution. This is one of many network design problems widely studied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables to be purchased on them, or certain qualityofservice requirements may have to be met. In this work, we show that the above problem, and in fact, several basic problems in this general network design framework, cannot be approximated better than ~(log log n) unless NP c _ OTIME(,r~°(l°gl°gl°gn)). In particular, we show that this inapproximability threshold holds for (i) the PrioritySteiner Tree problem [7], (ii) the CostDistance problem [31], and the singlesink version of an even more fundamental problem, (iii) Fixed Charge Network Flow [33]. Our results provide a further breakthrough in the understanding of the level of complexity of network design problems. These are the first nonconstant hardness results known for all these problems.
On Network Design Problems: Fixed Cost Flows and the Covering Steiner Problem
, 2001
"... Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow. ..."
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Cited by 23 (3 self)
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Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow.
Tight Approximation Results for General Covering Integer Programs
 In Proc. of the FortySecond Annual Symposium on Foundations of Computer Science
, 2001
"... In this paper we study approximation algorithms for solving a general covering integer program. An nvector x of nonnegative integers is sought, which minimizes c T x; subject to Ax b; x d: The entries of A; b; c are nonnegative. Let m be the number of rows of A: Covering problems have been hea ..."
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Cited by 20 (3 self)
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In this paper we study approximation algorithms for solving a general covering integer program. An nvector x of nonnegative integers is sought, which minimizes c T x; subject to Ax b; x d: The entries of A; b; c are nonnegative. Let m be the number of rows of A: Covering problems have been heavily studied in combinatorial optimization. We focus on the effect of the multiplicity constraints, x d; on approximability. Two longstanding open questions remain for this general formulation with upper bounds on the variables. (i) The integrality gap of the standard LP relaxation is arbitrarily large. Existing approximation algorithms that achieve the wellknown O(log m)approximation with respect to the LP value do so at the expense of violating the upper bounds on the variables by the same O(log m) multiplicative factor. What is the smallest possible violation of the upper bounds that still achieves cost within O(log m) of the standard LP optimum ? (ii) The best known approximation ratio for the problem has been O(log(max j P i A ij )) since 1982. This bound can be as bad as polynomial in the input size. Is an O(log m)approximation, like the one known for the special case of Set Cover, possible? We settle these two open questions. To answer the first question we give an algorithm based on the relatively simple new idea of randomly rounding variables to smallerthaninteger units. To settle the second question we give a reduction from approximating the problem while respecting multiplicity constraints to approximating the problem with a bounded violation of the multiplicity constraints. 1 Research partially supported by NSERC Grant 22780900 and a CFI New Opportunities Award 1.
A Fast Approximation Scheme for Fractional Covering Problems with Box Constraints
, 2004
"... We present the first combinatorial approximation scheme that yields a pure approximation guarantee for linear programs that are either covering problems with upper bounds on variables, or their duals. Existing approximation schemes for mixed covering and packing problems do not simultaneously satis ..."
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Cited by 18 (2 self)
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We present the first combinatorial approximation scheme that yields a pure approximation guarantee for linear programs that are either covering problems with upper bounds on variables, or their duals. Existing approximation schemes for mixed covering and packing problems do not simultaneously satisfy packing and covering constraints exactly. We present the first combinatorial approximation scheme that returns solutions that simultaneously satisfy general positive covering constraints and upper bounds on variable values. For input parameter ffl? 0, the returned solution has positive linear objective function value at most 1 + ffl times the optimal value. The general algorithm requires O(ffl2m log(cTu)) iterations, where c is the objective cost vector, u is the vector of upper bound values, and m is the number of variables. Each iteration uses an oracle that finds an (approximately) most violated constraint. A natural set of problems that our work addresses are linear programs for various network design problems: generalized Steiner network, vertex connectivity, directed connectivity, capacitated network design, group Steiner forest. The integer versions of these problems are all NPhard. For each of them, there is an approximation algorithm that rounds the solution to the corresponding linear program relaxation. If the LP solution is not feasible, then the corresponding integer solution will also not be feasible. Solving the linear program is often the computational bottleneck in these problems, and thus a fast approximation scheme for the LP relaxation means faster approximation algorithms. For these applications, we introduce a new modification of the pushrelabel maximum flow algorithm that allows us to perform each iteration in amortized O(jEj+jV j log jV j) time, instead of one maximum flow per iteration that is implied by the straight forward adaptation of our general algorithm. In conjunction with an observation that reduces the number of iterations to jEj log jV j for f0; 1g constraint matrices, the modification allows us to obtain an algorithm that is faster than existing exact or approximate algorithms by a factor of at least O(jEj) and by a factor of O(jEj log jV j) if the number of demand pairs is \Omega (jV j).
LPbased approximation algorithms for capacitated facility location
 in Proc. of IPCO’04, 2004
"... In the capacitated facility location problem with hard capacities, we are given a set of facilities, F, and a set of clients D in a common metric space. Each facility i has a facility opening cost fi and capacity ui that specifies the maximum number of clients that may be assigned to this facility. ..."
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Cited by 15 (1 self)
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In the capacitated facility location problem with hard capacities, we are given a set of facilities, F, and a set of clients D in a common metric space. Each facility i has a facility opening cost fi and capacity ui that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set F and assign each client to an open facility so that at most ui clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance cij, and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NPhard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of singledemand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.