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Multicommodity maxflow mincut theorems and their use in designing approximation algorithms
 J. ACM
, 1999
"... In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound implied by ..."
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Cited by 357 (6 self)
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In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound implied by the mincut. The result (which is existentially optimal) establishes an important analogue of the famous 1commodity maxflow mincut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomialtime (polylog ntimesoptimal) approximation algorithms for wellknown NPhard optimization problems such as graph partitioning, mincut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.
DivideandConquer Approximation Algorithms via Spreading Metrics
, 1996
"... We present a novel divideandconquer paradigm for approximating NPhard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divideandconquer approach is applicable. Second, a fractional spreading metric is computable in polynomial tim ..."
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Cited by 110 (9 self)
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We present a novel divideandconquer paradigm for approximating NPhard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divideandconquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns rational lengths to either edges or vertices of the input graph, such that all subgraphs on which the optimization problem is nontrivial have large diameters. In addition, the spreading metric provides a lower bound, ø , on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is O (minflog ø log log ø; log k log log kg), where k denotes the number of "interesting" vertices in the problem instance, and is at most the number of vertices. We present seven problems that can be formulated to fit the paradigm. For all these problems our algorithm improves ...
Multiway Cuts in Directed and Node Weighted Graphs
 in Proc. 21st ICALP, Lecture Notes in Computer Science 820
, 1994
"... this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for t ..."
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Cited by 46 (4 self)
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this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for this. Define an isolating cut for terminal s i to be a cut that separates s i from the rest of the terminals. A minimum isolating cut for s i can be computed in polynomial time by identifying the remaining terminals, and finding a minimum cut separating them from s i . The algorithm in [2] finds such cuts for each terminal, discards the heaviest cut, and picks the union of the remaining. The approximation factor is proven by observing that on doubling each edge in the optimum multiway cut, we can partition these edges into k isolating cuts, one for each Department of Computer Science and Engg., Indian Institute of Technology, New Delhi, India
Improved Bounds on the MaxFlow MinCut Ratio for Multicommodity Flows
, 1993
"... In this paper we consider the worst case ratio between the capacity of mincuts and the value of maxflow for multicommodity flow problems. We improve the best known bounds for the mincut maxflow ratio for multicommodity flows in undirected graphs, by replacing the O(log D) in the bound by O(log ..."
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Cited by 22 (2 self)
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In this paper we consider the worst case ratio between the capacity of mincuts and the value of maxflow for multicommodity flow problems. We improve the best known bounds for the mincut maxflow ratio for multicommodity flows in undirected graphs, by replacing the O(log D) in the bound by O(log k), where D denotes the sum of all demands, and k denotes the number of commodities. In essence we prove that up to constant factors the worst mincut maxflow ratios appear in problems where demands are integral and polynomial in the number of commodities. Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the mincut maxflow ratio in general undirected graphs is bounded by O(logC log D), where C denotes the sum of all the capacities. Tragoudas has improved this bound to O(logn log D), where n is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this to O(log k log D). Klein, Plotkin and Ra...
Bounds on the Throughput Gain of Network Coding in Unicast and Multicast Wireless Networks
"... Abstract—Gupta and Kumar established that the per node throughput of ad hoc networks with multipair unicast traffic scales with an increasing number of nodes n as λ(n) = Θ(1 / √ n log n), thus indicating that performance does not scale well. However, Gupta and Kumar did not consider network coding ..."
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Cited by 10 (0 self)
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Abstract—Gupta and Kumar established that the per node throughput of ad hoc networks with multipair unicast traffic scales with an increasing number of nodes n as λ(n) = Θ(1 / √ n log n), thus indicating that performance does not scale well. However, Gupta and Kumar did not consider network coding and wireless broadcasting, which recent works suggest have the potential to significantly improve throughput. Here, we establish bounds on the improvement provided by such techniques. For random networks of any dimension under either the protocol or physical model that were introduced by Gupta and Kumar, we show that network coding and broadcasting lead to at most a constant factor improvement in per node throughput. For the protocol model, we provide bounds on this factor. We also establish bounds on the throughput benefit of network coding and broadcasting for multiple source multicast in random networks. Finally, for an arbitrary network deployment, we show that the coding benefit ratio is at most O(log n) for both the protocol and physical communication models. These results give guidance on the application space of network coding, and, more generally, indicate the difficulty in improving the scaling behavior of wireless networks without modification of the physical layer.
Multicommodity Flows and Approximation Algorithms
, 1994
"... This thesis is about multicommodity flows and their use in designing approximation algorithms for problems involving cuts in graphs. In a groundbreaking work Leighton and Rao [34] showed an approximate maxflow mincut theorem for uniform multicommodity flow and used this to obtain an approximation ..."
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Cited by 5 (0 self)
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This thesis is about multicommodity flows and their use in designing approximation algorithms for problems involving cuts in graphs. In a groundbreaking work Leighton and Rao [34] showed an approximate maxflow mincut theorem for uniform multicommodity flow and used this to obtain an approximation algorithm for the flux of a graph. We consider the multicommodity flow problem in which the object is to maximize the sum of the flows routed and prove the following approximate maxflow minmulticut theorem minmulticut O(log k) maxflow minmulticut where k is the number of commodities. Our proof is based on a rounding technique from [34]. Further, we show that this theorem is tight. For a multicommodity flow instance with specified demands, the ratio of the maximum concurrent flow to the sparsest cut was shown to be bounded by O(log 2 k) [30, 57, 17, 47]. We use ideas from our proof of the approximate maxflow minmulticut theorem and a geometric scaling technique from [1] to provi...
An informationtheoretic metatheorem on edgecut bounds
 IN PROCEEDINGS IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY (ISIT
, 2012
"... We consider the problem of multiple unicast in wireline networks. Edgecut based bounds which are simple bounds on the rates achievable by routing flow are not in general, fundamental, i.e. they are not outer bounds on the capacity region. It has been observed that when the problem has some kind of ..."
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Cited by 4 (2 self)
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We consider the problem of multiple unicast in wireline networks. Edgecut based bounds which are simple bounds on the rates achievable by routing flow are not in general, fundamental, i.e. they are not outer bounds on the capacity region. It has been observed that when the problem has some kind of symmetry involved, then flows and edgecut based bounds are ‘close’, i.e. within a constant or polylogarithmic factor of each other. In this paper, we make the observation that in these very cases, such edgecut based bounds are actually ‘close ’ to fundamental yielding an approximate characterization of the capacity region for these problems. We demonstrate this in the case of kunicast in undirected networks, kpair unicast in directed networks with symmetric demands i.e. for every source communicating to a destination at a certain rate, the destination communicates an independent message back to the source at the same rate, and sumrate of kgroupcast in directed networks, i.e. a group of nodes, each of which has an independent message for every other node in the group. We place our work in context of existing results to suggest a metatheorem: if there is inherent symmetry either in the network connectivity or in the traffic pattern, then edgecut bounds are nearfundamental and flows approximately achieve capacity.
Network Capacity under Traffic Symmetry: Wireline and Wireless Networks
"... The problem of designing near optimal strategies for multiple unicast traffic in wireline networks is wideopen; however, channel symmetry or traffic symmetry can be leveraged to show that routing can achieve with a polylogarithmic approximation factor of the edgecut bound. For the same problem, t ..."
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The problem of designing near optimal strategies for multiple unicast traffic in wireline networks is wideopen; however, channel symmetry or traffic symmetry can be leveraged to show that routing can achieve with a polylogarithmic approximation factor of the edgecut bound. For the same problem, the edgecut bound is known to only upper bound rates of routing flows and unlike the information theoretic cutset bound, it does not upper bound (capacityachieving) information rates with general strategies. In this paper, we demonstrate that under channel or traffic symmetry, the edgecut bound upperbounds general information rates, thus providing a capacity approximation result. The key technique is a combinatorial result relating edgecut bounds to generalized network sharing bounds. Finally, we generalize the results to wireless networks via an intermediary class of combinatorial graphs known as polymatroidal networks – our main result is that a natural architecture separating the physical and networking layers is near optimal when the traffic is symmetric among sourcedestination pairs, even when the channel is asymmetric (due to asymmetric power constraints, or prior frequency allocation like frequency division duplexing). This result is complementary to our earlier work proving a similar result under channel symmetry [1]. I.