Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Leighton, Tom; Rao, Satish

doi:10.1145/331524.331526

Cited by 693 publications

(677 citation statements)

References 59 publications

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“…Conductance is widely used to capture the intuition of a good community; it is a fundamental combinatorial quantity; and it has a very natural interpretation in terms of random walks on the interaction graph. Moreover, since there exist a rich suite of both theoretical and practical algorithms [87,149,107,108,17,95,96,162,54], we can for point (4) compare and contrast several methods to approximately optimize it. To illustrate conductance, note that of the three 5-node sets A, B, and C illustrated in the graph in Figure 1, B has the best (the lowest) conductance and is thus the most community-like.…”

Section: Overview Of Our Approachmentioning

confidence: 99%

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,494

1,082

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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions.Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a "real" communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the "best" possible community-according to the conductance measure-over a wide range of size scales. We study over 100 large real-world networks, ranging from traditional and on-line social networks, to technological and information networks and web graphs, and ranging in size from thousands up to tens of millions of nodes.Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. Our observations agree with previous work on small networks, but we show that large networks have a very different structure. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales (up to ≈ 100 nodes); and communities of size scale beyond ≈ 100 nodes gradually "blend into" the expanderlike core of the network and thus become less "community-like," with a roughly inverse relationship between community size and optimal community quality. This observation agrees well with the so-called Dunbar number which gives a limit to the size of a well-functioning community.However, this behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. The relatively gradual increase of the network community profile plot as a function of increasing community size depends in a subtle manner on the way in which local clustering information is propagated from smaller to larger size scales in the network. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe i...

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Section: Overview Of Our Approachmentioning

confidence: 99%

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,494

1,082

View full text Add to dashboard Cite

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“…In case of latter, we do not get to separate s from t. This simple example also highlights that the two-edge-disjoint-path distance measure behaves quite differently from the usual shortest path distance metric. It appears that the standard region growing algorithms [21,15] and embedding methods are difficulty to adapt to the 2-route setting. There is no simple cycle between s and t but an edge has to be removed to 2-separate s from t in the edge-disjoint case.…”

Section: Algorithmic Ideasmentioning

confidence: 99%

Algorithms for 2-Route Cut Problems

Chekuri

Khanna

2008

Automata, Languages and Programming

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In this paper we study approximation algorithms for multi-route cut problems in undirected graphs. In these problems the goal is to find a minimum cost set of edges to be removed from a given graph such that the edge-connectivity (or node-connectivity) between certain pairs of nodes is reduced below a given threshold K. In the usual cut problems the edge connectivity is required to be reduced below 1 (i.e. disconnected). We consider the case of K = 2 and obtain poly-logarithmic approximation algorithms for fundamental cut problems including single-source, multiway-cut, multicut, and sparsest cut. These cut problems are dual to multi-route flows that are of interest in fault-tolerant networks flows. Our results show that the flow-cut gap between 2-route cuts and 2-route flows is poly-logarithmic in undirected graphs with arbitrary capacities. 2-route cuts are also closely related to well-studied feedback problems and we obtain results on some new variants. Multi-route cuts pose interesting algorithmic challenges. The new techniques developed here are of independent technical interest, and may have applications to other cut and partitioning problems.

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“…Leighton and Rao [9] gave a pseudo-approximation for the weighted b-balanced cut problem. Let B denote the cost of an optimal b-balanced cut.…”

Section: Related Workmentioning

confidence: 99%

“…Let B denote the cost of an optimal b-balanced cut. A polynomial time algorithm described in [9] finds for every > 0 a (b − )-balanced cut of cost O( log n B). This is true in the undirected case.…”

Section: Related Workmentioning

confidence: 99%