## RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS (1999)

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Citations: | 75 - 11 self |

### BibTeX

@MISC{Karger99randomsampling,

author = {David R. Karger},

title = { RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS},

year = {1999}

}

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### Abstract

We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to 1 � O(�log n)/k).

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Citation Context ...how the skeleton approach can be applied to minimum cuts and maximumsows. In unweighted graphs, the s-t maximumsow problem is tosnd a maximum set, or packing, of edge-disjoint s-t paths. It is known (=-=Ford and Fulkerson 1962-=-) that the value of thissow is equal to the value of the minimum s-t cut. In fact, the only known algorithms forsnding an s-t minimum cut simply identify a cut that is saturated by an s-t maximumsow. ... |

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Citation Context ...werful general tool in algorithm design. It appears in a fast and elegant algorithm forsnding the median of an ordered set (Floyd and Rivest 1975). It has many applications in computational geometry (=-=Clarkson 1987-=-; Clarkson and Shor 1987) and in particular insxed-dimension linear and integer programming (Clarkson 1995). Random sampling drives thesrst linear-time minimum spanning tree algorithm (Karger, Klein, ... |

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Citation Context ...to O log f max log n f min . 1.4 Related Work Random sampling is a powerful general tool in algorithm design. It appears in a fast and elegant algorithm forsnding the median of an ordered set (=-=Floyd and Rivest 1975-=-). It has many applications in computational geometry (Clarkson 1987; Clarkson and Shor 1987) and in particular insxed-dimension linear and integer programming (Clarkson 1995). Random sampling drives ... |

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Citation Context ... paradigm also applies to the problem of packing bases in a matroid. Our approach also applies to certain covering problems. From random sampling, it is a small step to show that randomized rounding (=-=Raghavan and Thompson 1987-=-) can be eectively applied to graphs with fractional edge weights, yielding integrally weighted graphs with roughly the same cut values. This makes randomized rounding a useful tool in network design... |

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Citation Context ...tool in algorithm design. It appears in a fast and elegant algorithm forsnding the median of an ordered set (Floyd and Rivest 1975). It has many applications in computational geometry (Clarkson 1987; =-=Clarkson and Shor 1987-=-) and in particular insxed-dimension linear and integer programming (Clarkson 1995). Random sampling drives thesrst linear-time minimum spanning tree algorithm (Karger, Klein, and Tarjan 1995). This a... |

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Citation Context ...ost graph with minimum cut k. The minimum cost 1-connected subgraph is just the minimum spanning tree, but for larger values of k the problem is NP-complete even when all edge costs are 1 or innity (=-=Eswaran and Tarjan 1976-=-). Agrawal, Klein, and Ravi (1995) studied a special case of network design called the generalized Steiner problem,srst formulated by Krarup (see Winter (1987)). In this version, the demands are speci... |

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Citation Context ...f an ordered set (Floyd and Rivest 1975). It has many applications in computational geometry (Clarkson 1987; Clarkson and Shor 1987) and in particular insxed-dimension linear and integer programming (=-=Clarkson 1995-=-). Random sampling drives thesrst linear-time minimum spanning tree algorithm (Karger, Klein, and Tarjan 1995). This author (Karger 1998b) shows how it can speed up algorithms for matroid optimization... |

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Citation Context ... 2.2 (Karger and Stein (1996)). In an undirected graph, the number of -minimum cuts is less than n 2 . Proof. A proof appears in the appendix. It is a minor variant of one that appeared previously (=-=Karger and Stein 1996-=-). A quite dierent proof has also been developed (Karger 1996). Lemma 2.3 (Cherno (1952), cf. Motwani and Raghavan (1995)). Let X be a sum of independent Bernoulli (that is, 0/1) random variables wi... |

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Citation Context ...ion is whether these subgraphs can be constructed deterministically in polynomial time. In the case of complete graphs, this has been accomplished through the deterministic construction of expanders (=-=Gabber and Galil 1981-=-). Indeed, just as the expander of (Gabber and Galil 1981) has constant degree, it may be possible to deterministically construct a (1+)-accurate skeleton with a constant minimum cut, rather than the... |

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Citation Context ...n Pr[ERR] is O(1=n d ). Let S denote the number of edges actually chosen in such a sample. Note that S has the binomial distribution and that its so-called central term Pr[S = dpme] =s(1= p pm) (cf. (=-=Feller 1968-=-)). We can evaluate ERR conditioning on the value of S: 1=n d Pr[ERR] = X k Pr[S = k] Pr[ERR j S = k] Pr[S = dpme] Pr[ERR j S = dpme] =s( 1 p pm ) Pr[ERR j S = dpme]: In other words, Pr[ERR ... |

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(Show Context)
Citation Context ...ts with high probability in ~ O(n 2 ) time. This author used sampling in an algorithm tosnd an exact minimum cut in any (weighted or unweighted) undirected graph with high probability in ~ O(m) time (=-=Karger 1996-=-). More recently, this author gave a faster, sampling-based algorithm thatsnds a maximumsow of value v in ~ O( p mnv) time with high probability (Karger 1998a). Karger and Levine (1998) gave an even f... |

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(Show Context)
Citation Context ... This would most obviously imply an O(m log 1=p) time bound for generating a skeleton. However, even in this model, it is possible to perform the m biased coinsips in O(m) time with high probability (=-=Knuth and Yao 1976-=-), cf. Karger (1994b). 11 2.2 Determining the Right p Our approximation algorithms are based upon constructing p-skeletons. In these algorithms, given a desired approximation bound , we will want to ... |

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Citation Context ...ected subgraph problem (where edges cannot be reused, all connectivity demands are k, and edge costs are 1 or innity). For suciently large k this improves on a previous approximation ratio of 1:85 (=-=Khuller and Raghavachari 1995-=-). We also improve bounds for various other single-edge-use problems. All of our techniques apply only to undirected graphs, as cuts in directed graphs do not appear to have the same predictable behav... |

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Citation Context ... or maximumsows, although blockingsows can be used to achieve a certain large absolute error bound. This work relates to several previous algorithms forsnding minimum cuts. The Contraction Algorithm (=-=Karger and Stein 1993-=-) runs in O(n 2 log 3 n) time on undirected (weighted or unweighted) graphs. Gabow's Round Robin Algorithm (Gabow 1995) runs in O(mc log(n 2 =m)) time on unweighted (directed or undirected) graphs. 6 ... |

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Citation Context ...ph problem). Assuming edges can be used repeatedly, they gave an O(log f max )-approximation algorithm, where f max is the maximum demand across any cut (i.e. max d ij ). This extended previous work (=-=Goemans and Bertsimas 1993-=-) on the special case where d ij = min(d i ; d j ) for given \connectivity types" d i . Aggarwal and Garg (1994) gave an algorithm with performance ratio O(log k), where k is the number of sites with ... |

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Citation Context ...te. It runs in O(m) time on unweighted graphs. In weighted graphs, where the resulting certicate has total weight kn and preserves cuts of value up to k, the running time increases to O(m + n logn) (=-=Nagamochi and Ibaraki 1992-=-a). If we are looking for cuts orsows of value less than k, we cansnd them in the certicate, taking less time since the certicate has fewer edges. For example a sparse certicate can be constructed ... |

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(Show Context)
Citation Context ... only to undirected graphs, as cuts in directed graphs do not appear to have the same predictable behavior under random sampling. Preliminary versions of this work appeared in conference proceedings (=-=Karger 1994-=-a; Karger 1994c). A more extensive treatment is provided in the author's dissertation (Karger 1994b). The remainder of this introduction includes a more detailed description of our results as well as ... |

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21 |
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Citation Context ...irected graph that makes it k-connected as a directed graph. Cuts also play an important role in multicommoditysow problems, though the connection is not as tight as for the standard max- ow problem (=-=Leighton and Rao 1988-=-; Linial, London, and Rabinovich 1995; Aumann and Rabani 1998). Random sampling helps us solve cut-dependent undirected graph problems. We dene and use a graph skeleton. Given a graph, a skeleton is ... |

17 | Using randomized sparsification to approximate minimum cuts - Karger - 1994 |

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13 |
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Citation Context ...that each survives. But this probability is at most one. Thus, if there are k minimum cuts, we have k n 2 1 1. This corollary has been proven in the past (Dinitz, Karzanov, and Lomonosov 1976; =-=Lomonosov and Polesskii 1971-=-). This bound is tight. In a cycle on n vertices, there are n 2 minimum cuts, one for each pair of edges in the graph. Each of these minimum cuts is produced by the Contraction Algorithm with equa... |

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Citation Context ... of when an integral multicommoditysow can be found (the problem is discussed in (Ford and Fulkerson 1962); more recent discussions include (Grotschel, Lovasz, and Schrijver 1988, Section 8.6) and (=-=Frank 1990-=-)). Our sampling theorem gives new results on the existence of integralsows and fast algorithms forsnding them. Rather than assigning a fraction of each edge to each graph, assign each edge to a graph... |