## Approximate Max-Flow Min-(multi)cut Theorems and Their Applications (1993)

Venue: | SIAM Journal on Computing |

Citations: | 144 - 3 self |

### BibTeX

@ARTICLE{Garg93approximatemax-flow,

author = {Naveen Garg and Vijay V. Vazirani and Mihalis Yannakakis},

title = {Approximate Max-Flow Min-(multi)cut Theorems and Their Applications},

journal = {SIAM Journal on Computing},

year = {1993},

volume = {25},

pages = {698--707}

}

### Years of Citing Articles

### OpenURL

### Abstract

Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate max-flow min-multicut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us to find a multicut within O(log k) of the max flow (and hence also the optimal multicut). In addition, the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands, of Leighton-Rao and Klein et.al., and thereby obtain an improved bound for the latter problem. 1 Introduction Much of flow theory, and the theory of cuts in graphs, is built around a single theorem - the celebrated max-flow min-cut theorem of Ford and Fulkerson [FF], and Elias, Feinstein and Shannon [EFS]. The power of this theorem lies in that it relates two fundamental graph-theoretic entities via the potent mechanism of a min-max relation. The importance of this theor...

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Citation Context ...all the demands are unity. They proved the following approximate max-flow min-cut theorem: ff O(log n)sfsff; where n is the number of vertices in the graph. Subsequently, Klein, Agrawal, Ravi and Rao =-=[KARR]-=- managed to attack the arbitrary demands problem, and proved: ff O(log C log D)sfsff; where C is the sum of capacities of all edges and D is the sum of all demands. However, one restriction they impos... |

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Citation Context ...m 6.1 ff O(log k log D)sfsff Thus, for a multicommodity flow to be feasible it is necessary that the sparsest cut ratio, ff, be at least 1 and sufficient that it be O(log k log D). Plotkin and Tardos =-=[PT]-=- give a method of scaling demands so that the log D factor in Theorem 6.1 can be replaced by log k. Hence they improve the bound in Theorem 6.1 to O(log 2 k). For uniform multicommodity flow on bounde... |

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Citation Context ...ven set of source-sink pairs for the commodities consists of all pairs of vertices from a given subset S of terminals, the gap between min cut and max flow is much smaller, it is at most 2 \Gamma 2 k =-=[Cu]-=-. Of course, if S is the whole set of vertices (i.e., there is one commodity for every pair of vertices), the problem is trivial and there is no gap: max-flow = min-cut = total capacity of the graph. ... |

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Citation Context ...imate max-flow min-multicut theorem) FsMsF \Delta O(log k) Further this bound on the ratio of the minimum multicut and maxflow is tight as shown in Theorem 5.4. For planar graphs, Tardos and Vazirani =-=[TV]-=- obtain a constant factor approximation for the minimum multicut. Garg, Vazirani and Yannakakis [GVY] approximate the minimum multicut on trees to within twice the optimal. They also give a factor 1 2... |

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Citation Context ...fsff. Klein et.al. show that the "throughput" f is at least ff O(logC log D) , where C is the sum of all capacities and D is the total demand. This was later improved to ff O(logn log D) by =-=Tragoudas [Trag]-=-. We improve this result by providing a tighter bound on f . Theorem 6.1 ff O(log k log D)sfsff Thus, for a multicommodity flow to be feasible it is necessary that the sparsest cut ratio, ff, be at le... |

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Citation Context ...ed bound for the latter problem. 1 Introduction Much of flow theory, and the theory of cuts in graphs, is built around a single theorem - the celebrated max-flow min-cut theorem of Ford and Fulkerson =-=[FF]-=-, and Elias, Feinstein and Shannon [EFS]. The power of this theorem lies in that it relates two fundamental graph-theoretic entities via the potent mechanism of a min-max relation. The importance of t... |

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Citation Context ...he optimal integral solution to the dual is the minimum multicut. In general, the vertices of the dual polyhedron are not integral. However, for the case of a single commodity, they are integral (see =-=[GV]-=- for an exact characterization), and the max-flow min-cut theorem is simply a consequence of the LP duality theorem. For the multicommodity case, the LP duality theorem shows only that maximum flow is... |

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Citation Context ...tated by Hu in 1963 [Hu]. For k = 1, the problem coincides of course with the ordinary min cut problem. For k = 2, it can be also solved in polynomial time by two applications of a max flow algorithm =-=[YKCP]-=-. The problem was proven NP-hard and MAX SNP-hard for any ks3 by Dahlhaus, Johnson, Papadimitriou, Seymour and Yannakakis [DJPSY]. As a consequence of the MAX SNP-hardness, there is no polynomial time... |