Finding minimum-quotient cuts in planar graphs

Park, James K.; Phillips, Cynthia A.

doi:10.1145/167088.167284

Cited by 30 publications

(52 citation statements)

References 14 publications

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“…If the performance guarantee of this algorithm is improved (as it has been for certain classes of graphs such as planar graphs or graphs with excluded minors [Park and Phillips 1993;Rao 1992;Klein et al 1993] and dense graphs [Arora et al 1995]), then it will be possible to derive corresponding improvements in the performance guarantees in the other algorithms described in the section.…”

Section: Applications To Approximation Algorithmsmentioning

confidence: 99%

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

Leighton

Rao

1999

J. ACM

697

666

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Abstract. In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut 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 polynomial-time (polylog n-times-optimal) approximation algorithms for well-known NP-hard optimization problems such as graph partitioning, min-cut 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.

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Section: Applications To Approximation Algorithmsmentioning

confidence: 99%

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

Leighton

Rao

1999

J. ACM

697

666

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“…Park and Phillips [18] devise a pseudo-polynomial time algorithm for solving (exactly) the minimum quotient edgecut problem (improving over a constant factor approximation in [19,20]). Klein, Plotkin and Rao [11] give a constant factor approximation for the directed version of this problem.…”

Section: Related Workmentioning

confidence: 99%

“…The algorithms proceed by searching for simple cycles in the dual of the planar graph. In particular, Park and Phillips [18] define a cost and weight for each edge in the dual graph such that the cost of any simple cycle in the dual graph is equal to the cost of the corresponding cut in the input graph, and the weight of the cycle corresponds to the weight that the cut separates. Then, they can modify methods for finding minimum mean cycle, due to [10], to find the best such cycle.…”

Section: Techniquesmentioning

confidence: 99%

“…[7,6]). It is well-known that minimal edge-cuts in a plane graph correspond to simple cycles in the dual (plane) graph; this correspondence was exploited in [19,20,18] to devise algorithms that find approximate (or even optimal) quotient edge-cuts and edge-separators in planar graphs. In contrast, there is no exact correspondence between simple cycles in GT and vertex-cuts in G0 (see e.g.…”

Section: Figure 3: (A) a Plane Graph (B) Its Dual And (C) Its Face-mentioning

confidence: 99%

“…The algorithms of [19,20,18] for finding (or approximating) an edge-cut of minimum quotient all rely on the correspondence between minimum quotient edge cuts and simple cycles in the dual. In the vertex-cut problem such a correspondence does not hold.…”

Section: Where Is the Problem?mentioning

confidence: 99%

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Constant factor approximation of vertex-cuts in planar graphs

Amir

Krauthgamer

Rao

2003

Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing - STOC '03

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We devise the first constant factor approximation algorithm for minimum quotient vertex-cuts in planar graphs. Our algorithm achieves approximation ratio 1+ (1 + + o(1)) if there is an optimal quotient vertex-cut (A * , B * , C * ) where the weight of C * is of low order compared to those of A * and B * ; this holds, for example, when the input graph has uniform weights and costs. The ratio further im-W . We use our algorithm for quotient vertex-cuts to achieve the first constant-factor pseudo-approximation for vertex separators in planar graphs.Our technical contribution is two-fold. First, we prove a structural theorem for planar graphs, showing the existence of a nearoptimal quotient vertex-cut whose high-level structure is that of a bounded-depth tree. Second, we develop an algorithm that optimizes over such complex structures in running time that depends (exponentially) not on the size of the structure, but rather only on its depth. These techniques may be applicable in other problems.

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