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CostDistance: Two Metric Network Design
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Abstract We present the CostDistance problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of sourcesink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for CostDistance, where k is the numbe ..."
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Abstract We present the CostDistance problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of sourcesink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for CostDistance, where k is the number of sources. We reduce many common network design problems to CostDistance, obtaining (in some cases) the first known logarithmic approximation for them. These problems include singlesink buyatbulk with variable pipe types between different sets of nodes, facility location with buyatbulk type costs on edges, and maybecast with combind cost and distance metrics. Our algorithm is also the algorithm of choice for several previous network design problems, due to its ease of implementation and fast running time. 1 Introduction Consider designing a network from the ground up. We are given a set of customers, and need to place various servers and network links in order to cheaply provide sufficient service. If we only need to place the servers, this becomes the facility location problem and constantapproximations are known. If a single server handles all customers, and we impose the additional constraint that the set of available network link types is the same for every pair of nodes (subject to constant scaling factors on cost) then this is the single sink buyatbulk problem. We give the first known approximation for the general version of this problem with both servers and network links. We reduce the network design problem to an elegant theoretical framework: the CostDistance problem. We are given a graph with a single distinguished sink node (server). Every edge in this graph can be measured along two metrics; the first will be called cost and the second will be length. Note that the two metrics are entirely independent, and that there may be any number of parallel edges in the graph. We are given a set of sources (customers). Our objective is to construct a Steiner tree connecting the sources to the sink while minimizing the combined sum of the cost of the edges in the tree and sum over sources of the weighted length from source to sink.
On the Approximability of Some Network Design Problems
, 2005
"... Consider the following classical network design problem: aset of terminals T = ftig wants to send traffic to a "root" r in an nnode graph G = (V; E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocatedon the edges to permit this. However, bandwidth on an ..."
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Cited by 33 (4 self)
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Consider the following classical network design problem: aset of terminals T = ftig wants to send traffic to a "root" r in an nnode graph G = (V; E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocatedon the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some basecapacity ue and hence provisioning k \Theta ue bandwidth onedge e incurs a cost of dke times the cost of that edge. Theobjective is a minimumcost feasible solution. This is one of many network design problems widelystudied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables tobe 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 designframework, cannot be approximated better than \Omega (log log n)unless NP ` DTIME \Gamma nO(log log log n) \Delta. In particular,
Multicommodity facility location
, 2004
"... Multicommodity facility location refers to the extension of facility location to allow for different clients having demand for different goods, from among a finite set of goods. This leads to several optimization problems, depending on the costs of opening facilities (now a function of the commoditi ..."
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Cited by 17 (2 self)
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Multicommodity facility location refers to the extension of facility location to allow for different clients having demand for different goods, from among a finite set of goods. This leads to several optimization problems, depending on the costs of opening facilities (now a function of the commodities it serves). In this paper, we introduce and study some variants of multicommodity facility location, and provide approximation algorithms and hardness results for them.
On Columnrestricted and Priority Covering Integer Programs ⋆
"... Abstract. In a columnrestricted covering integer program (CCIP), all the nonzero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the int ..."
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Abstract. In a columnrestricted covering integer program (CCIP), all the nonzero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the integrality gaps of the underlying 0,1CIP. If the underlying 0,1CIP has an integrality gap O(γ), and assuming that the integrality gap of the priority version of the 0,1CIP is O(ω), we give a factor O(γ + ω) approximation algorithm for the CCIP. Priority versions of 0,1CIPs (PCIPs) naturally capture quality of service type constraints in a covering problem. We investigate priority versions of the line (PLC) and the (rooted) tree cover (PTC) problems. Apart from being natural objects to study, these problems fall in a class of fundamental geometric covering problems. We bound the integrality of certain classes of this PCIP by a constant. Algorithmically, we give a polytime exact algorithm for PLC, show that the PTC problem is APXhard, and give a factor 2approximation algorithm for it. 1
Approximability of Capacitated Network Design
"... Abstract. In the capacitated survivable network design problem (CapSNDP), we are given an undirected multigraph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimumcut requirements. Unlike its classical specia ..."
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Abstract. In the capacitated survivable network design problem (CapSNDP), we are given an undirected multigraph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimumcut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of CapSNDP is not well understood; even in very restricted settings no known algorithm achieves a o(m) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of CapSNDP. We give an O(log n) approximation for a special case of CapSNDP where the global minimum cut is required to be at least R, byrounding the natural cutbased LP relaxation strengthened with valid knapsackcover inequalities. We then show that as we move away from global connectivity, the single pair case (that is, when only one pair (s, t) has positive connectivity requirement) captures much of the difficulty of CapSNDP: even strengthened with KC inequalities, the LP has an Ω(n) integrality gap. Furthermore, in directed graphs, we show that single pair CapSNDP is 2 log1−δ nhard to approximate for any fixed constant δ> 0. We also consider a variant of the CapSNDP in which multiple copies of an edge can be bought: we give an O(log k) approximationforthis case, where k is the number of vertex pairs with nonzero connectivity requirement. This improves upon the previously known O(min{k, log Rmax})approximation for this problem when the largest minimumcut requirement, namely Rmax, islarge.Ontheotherhand,weobserve
Swarm Intelligence Inspired Multicast Routing: an Ant Colony Optimization Approach ⋆
"... Abstract. The advancement of network induces great demands on a series of applications such as the multicast routing. This paper firstly makes a brief review on the algorithms in solving routing problems. Then it proposes a novel algorithm called the distance complete ant colony system (DCACS), whic ..."
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Abstract. The advancement of network induces great demands on a series of applications such as the multicast routing. This paper firstly makes a brief review on the algorithms in solving routing problems. Then it proposes a novel algorithm called the distance complete ant colony system (DCACS), which is aimed at solving the multicast routing problem by utilizing the ants to search for the best routes to send data packets from a source node to a group of destinations. The algorithm bases on the framework of the ant colony system (ACS) and adopts the Prim’s algorithm to probabilistically construct a tree. Both the pheromone and heuristics influence the selection of the nodes. The destination nodes in the multicast network are given priority in the selection by the heuristics and a proper reinforcement proportion to the destination nodes is studied in the case experiments. Three types of heuristics are tested, and the results show that a modest heuristic reinforcement to the destination nodes can accelerate the convergence of the algorithm and achieve better results. 1
Primaldual algorithms for QoS multimedia multicast
 IN PROCEEDINGS. OF IEEE GLOBECOM, 3631. 5505—CHAPTER 71—4/9/2006—01:53—SRIDHAR—14632—XML MODEL CRC12A
, 2003
"... The QoS Steiner Tree Problem asks for the most costefficient way to multicast multimedia to a heterogeneous collection of users with different consumption rates. We assume that the cost of using a link is not constant but rather depends on the maximum bandwidth routed through the link. Formally, gi ..."
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The QoS Steiner Tree Problem asks for the most costefficient way to multicast multimedia to a heterogeneous collection of users with different consumption rates. We assume that the cost of using a link is not constant but rather depends on the maximum bandwidth routed through the link. Formally, given a graph with costs on the edges, a source node and a set of terminal nodes, each one with a bandwidth requirement, the goal is to find a Steiner tree containing the source, and the cheapest assignment of bandwidth to each of its edges so that each sourcetoterminal path in the tree has bandwidth at least as large as the bandwidth required by the terminal. Our main contributions are: (1) new coveringtype integer linear program formulations for the problem; (2) two new heuristics based on the primaldual framework; (3) a primaldual constantfactor approximation algorithm; (4) an extensive experimental study of the new heuristics and of several previously proposed algorithms.
On capacitated set cover problems
 APPROX
"... Abstract. Recently, Chakrabarty et al. [5] initiated a systematic study of capacitated set cover problems, and considered the question of how their approximability relates to that of the uncapacitated problem on the same underlying set system. Here, we investigate this connection further and give se ..."
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Abstract. Recently, Chakrabarty et al. [5] initiated a systematic study of capacitated set cover problems, and considered the question of how their approximability relates to that of the uncapacitated problem on the same underlying set system. Here, we investigate this connection further and give several results, both positive and negative. In particular, we show that if the underlying set system satisfies a certain hereditary property, then the approximability of the capacitated problem is closely related to that of the uncapacitated version. We also give related lower bounds, and show that the hereditary property is necessary to obtain nontrivial results. Finally, we give some results for capacitated covering problems on set systems with low hereditary discrepancy and low VC dimension. 1
Improved approximation algorithms for MinMax Tree Cover, Bounded Tree Cover, ShallowLight and BuyatBulk kSteiner Tree, and (k, 2)Subgraph
, 2011
"... In this thesis we provide improved approximation algorithms for the MinMax kTree Cover, Bounded Tree Cover and ShallowLight kSteiner Tree, (k, 2)subgraph problems. In Chapter 2 we consider the MinMax kTree Cover (MMkTC). Given a graph G = (V, E) with weights w: E → Z +, a set T1, T2,..., Tk ..."
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In this thesis we provide improved approximation algorithms for the MinMax kTree Cover, Bounded Tree Cover and ShallowLight kSteiner Tree, (k, 2)subgraph problems. In Chapter 2 we consider the MinMax kTree Cover (MMkTC). Given a graph G = (V, E) with weights w: E → Z +, a set T1, T2,..., Tk of subtrees of G is called a tree cover of G if V = ⋃ k i=1 V (Ti). In the MMkTC problem we are given graph G and a positive integer