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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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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.
Dependent Rounding in Bipartite Graphs
"... We combine the pipage rounding technique of Ageev &Sviridenko with a recent rounding method developed by Srinivasan, to develop a new randomized rounding approachfor fractional vectors defined on the edgesets of bipartite graphs. We show various ways of combining this techniquewith other idea ..."
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Cited by 35 (5 self)
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We combine the pipage rounding technique of Ageev &Sviridenko with a recent rounding method developed by Srinivasan, to develop a new randomized rounding approachfor fractional vectors defined on the edgesets of bipartite graphs. We show various ways of combining this techniquewith other ideas, leading to the following applications: ffl richer randomgraph models for graphs with a givendegreesequence; ffl improved approximation algorithms for: (i) throughputmaximization in broadcast scheduling, (ii) delayminimization in broadcast scheduling, and (iii) capacitated vertex cover;
On the equivalence between the primaldual schema and the local ratio technique
 In 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX). Number 2129 in Lecture Notes in Computer Science
, 2001
"... Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transform ..."
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Cited by 32 (8 self)
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Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primaldual algorithm. This was done in the case of the 2approximation algorithms for the feedback vertex set problem and in the case of the first primaldual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447–478]. In this paper we answer this question by showing that the two paradigms are equivalent.
PrimalDual Based Distributed Algorithms for Vertex Cover with SemiHard Capacities
, 2005
"... In this paper we consider the weighted, capacitated vertex cover problem with hard capacities (capVC). Here, we are given an undirected graph G = (V, E), nonnegative vertex weightswtv for all vertices v ∈ V, and nodecapacities Bv ≥ 1 for all v ∈ V. A feasible solution to a givencapVC instance cons ..."
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Cited by 9 (1 self)
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In this paper we consider the weighted, capacitated vertex cover problem with hard capacities (capVC). Here, we are given an undirected graph G = (V, E), nonnegative vertex weightswtv for all vertices v ∈ V, and nodecapacities Bv ≥ 1 for all v ∈ V. A feasible solution to a givencapVC instance consists of a vertex cover C ⊆ V. Each edge e ∈ E is assigned to one of its endpoints in C and the number of edges assigned to any vertex v ∈ C is at most Bv. The goal is to minimize the total weight of C. For a parameter ɛ> 0 we give a deterministic, distributed algorithm for thecapVC problem that computes a vertex cover C of weight at most (2 + ɛ) ·opt whereopt is the weight of a minimumweight feasible solution to the given instance. The number of edges assigned to any node v ∈ C is at most (4 + ɛ) · Bv. The running time of our algorithm is O(log(nW)/ɛ), where n is the number of nodes in the network and W =wtmax/wtmin is the ratio of largest to smallest weight. This result is complemented by a lowerbound saying that any distributed algorithm for capVC which requires a polylogarithmic number of rounds is bound to violate the capacity constraints by a factor two. The main feature of the algorithm is that it is derived in a systematic fashion starting from a primaldual sequential algorithm.
A Model for Minimizing Active Processor Time
"... We introduce the following elementary scheduling problem. We are given a collection of n jobs, where each job Ji has an integer length ℓi as well as a set Ti of time intervals in which it can be feasibly scheduled. Given a parameter B, the processor can schedule up to B jobs at a timeslot t solongas ..."
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Cited by 4 (3 self)
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We introduce the following elementary scheduling problem. We are given a collection of n jobs, where each job Ji has an integer length ℓi as well as a set Ti of time intervals in which it can be feasibly scheduled. Given a parameter B, the processor can schedule up to B jobs at a timeslot t solongasit is “active”att. The goalis toschedule allthe jobs in the fewestnumber of active timeslots. The machine consumes a fixed amount of energy per active timeslot, regardless of the number of jobs scheduled in that slot (as long as the number of jobs is nonzero). In other words, subject to ℓi units of each job i being scheduled in its feasible region and at each slot at most B jobs being scheduled, we are interested in minimizing the total time during which the machine is active. We present a linear time algorithm for the case where jobs are unit length and each Ti is a single interval. For general Ti, we show that the problem is NPcomplete even for B = 3. However when B = 2, we show that it can be efficiently solved. In addition, we consider a version of the problem where jobs have arbitrary lengths and can be preempted at any point in time. For general B, the problem can be solved by linear programming. For B = 2, the problem amounts to finding a trianglefree 2matching on a special graph. We extend the algorithm of Babenko et. al. [5] to handle our variant, and also to handle nonunit length jobs. This yields an O ( √ Lm) time algorithm to solve the preemptive scheduling problem for B = 2, where L = ∑ iℓi. We alsoshow that for B = 2 and unit length jobs, the optimal nonpreemptive schedule has ≤ 4/3times the activetime of the optimal preemptive schedule; this bound extends to several versions of the problem when jobs have arbitrary length. 1
LP Rounding for kCenters with Nonuniform Hard Capacities
"... Abstract—In this paper we consider a generalization of the classical kcenter problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node ..."
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Abstract—In this paper we consider a generalization of the classical kcenter problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. This problem is NPhard, even when centers have no capacity restrictions and optimal factor 2 approximation algorithms are known. With capacities, when all centers have identical capacities, a 6 approximation is known with no better lower bounds than for the infinite capacity version. While many generalizations and variations of this problem have been studied extensively, no progress was made on the capacitated version for a general capacity function. We develop the first constant factor approximation algorithm for this problem. Our algorithm uses an LP rounding approach to solve this problem, and works for the case of nonuniform hard capacities, when multiple copies of a node may not be chosen and can be extended to the case when there is a hard bound on the number of copies of a node that may be selected. Finally, for nonuniform soft capacities we present a much simpler 11approximation algorithm, which we find as one more evidence that hard capacities are much harder to deal with. Keywordsapproximation algorithms; kcenter; nonuniform capacities; hard capacities; LP rounding; I.
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"... Efficient dispersal of information using a communication network is among the most ..."
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Efficient dispersal of information using a communication network is among the most
A MinEdge Cost Flow Framework for Capacitated Covering Problems
"... In this work, we introduce the CovMECF framework, a special case of minimumedge cost flow in which the input graph is bipartite. We observe that several important covering (and multicovering) problems are captured in this unifying model and introduce a new heuristic LPO for any problem in this fr ..."
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In this work, we introduce the CovMECF framework, a special case of minimumedge cost flow in which the input graph is bipartite. We observe that several important covering (and multicovering) problems are captured in this unifying model and introduce a new heuristic LPO for any problem in this framework. The essence of LPO harnesses as an oracle the fractional solution in deciding how to greedily modify the partial solution. We empirically establish that this heuristic returns solutions that are higher in quality than those of Wolsey’s algorithm. We also apply the analogs of Leskovec et. al.’s [25] optimization to LPO and introduce a further freezing optimization to both algorithms. We observe that the former optimization generally benefits LPO more than Wolsey’s algorithm, and that the additional freezing step often corrects suboptimalities while further reducing the number of subroutine calls. We tested these implementations on randomly generated testbeds, several instances from the Second DIMACS Implementation Challenge and a couple networks modeling realworld dynamics. 1
On the MultiRadius Cover Problem ∗
"... An instance of the multiradius cover problem consists of a graph G = (V, E) with edge lengths l: E → R +. Each vertex u ∈ V represents a transmission station for which a transmission radius ru must be picked. Edges represent a continuum of demand points to be satisfied, that is, for every edge (u, ..."
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An instance of the multiradius cover problem consists of a graph G = (V, E) with edge lengths l: E → R +. Each vertex u ∈ V represents a transmission station for which a transmission radius ru must be picked. Edges represent a continuum of demand points to be satisfied, that is, for every edge (u, v) ∈ E we ask that ru + rv ≥ luv. The cost of transmitting at radius r from vertex u is given by an arbitrary nondecreasing cost function cu(r). Our goal is to find a cover with minimum total cost P u cu(ru). The multiradius cover problem is NPhard as it generalizes the wellknown vertex cover problem. In this paper we present a 2approximation algorithm for it. 1