## Experimental Study of Minimum Cut Algorithms (1997)

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Venue: | PROCEEDINGS OF THE EIGHTH ANNUAL ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA) |

Citations: | 40 - 2 self |

### BibTeX

@INPROCEEDINGS{Chekuri97experimentalstudy,

author = {Chandra S. Chekuri and Andrew V. Goldberg and David R. Karger and Matthew S. Levine and Cliff Stein},

title = {Experimental Study of Minimum Cut Algorithms},

booktitle = {PROCEEDINGS OF THE EIGHTH ANNUAL ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA)},

year = {1997},

pages = {324--333},

publisher = {}

}

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### Abstract

Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worst-case time bounds for the problem. We conduct experimental evaluation the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem.

### Citations

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Citation Context ... instances in our study. Note that for dense graphs, the number of increase key operations of the algorithm is relatively large. A brief study showed that for very large dense graphs, Fibonacci heaps =-=[17]-=- sometimes outperform 4-heaps. Fibonacci heaps or other priority queues with cheap increase key operations should be considered in applications involving such graphs. 4.3 Incorporating the Padberg-Rin... |

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Citation Context ...to the minimum cut problem. The classical Gomory-Hu algorithm [20] solves the minimum cut problem using n \Gamma 1 minimum s-t cut computations. The fastest current algorithms for the s-t cut problem =-=[1, 6, 7, 18, 28]-=- use flow techniques, in particular the push-relabel method [18], and run in !(nm) time. For the minimum cut problem, Hao and Orlin [22, 23] have given an algorithm (ho), based on the pushrelabel meth... |

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Citation Context ...the problem of finding a minimum cut in a graph with real-valued edge weights. Thus, cutting plane algorithms for the traveling salesman problem must solve a large number of minimum cut problems (see =-=[29] for a sur-=-vey of the area). We obtained some of the minimum cut instances that were solved by Applegate and Cook. Table 2 gives a summary of these instances, including their "names" which correspond t... |

297 |
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Citation Context ...eaningless. Our times are hundreds of times smaller. However, we think that the pr code would be much faster if its maximum flow subroutine (which is an implementation of the Sleator-Tarjan algorithm =-=[38]-=-) is replaced by a good implementation if the push-relabel method, such as that of [10]. Also, our hardware is faster than that used by Padberg and Rinaldi. Tables 44---51 give data for the PR1--PR8 f... |

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Citation Context ...: (4) The value of a flow is the net flow into the sink, i.e., jf j = X v2V f(v; t): The maximum flow problem is to determine a flow f for which jf j is maximum. The well-known maxflow-mincut theorem =-=[16, 15]-=- states that the value of the maximum s-t flow is equal to the value of the minimum s-t cut, i.e., jf j =ss;t (G). An s-t maximum flow algorithm can thus be used to find an s-t minimum cut, and minimi... |

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Citation Context ...emented the former. Packing spanning trees. Karger suggests two entirely different ways to pack spanning trees. We chose to use Gabow's algorithm [17]. It would be interesting to try the alternative (=-=[36]-=-) too. We tried checking to see if the tree packing contained multiple copies of the same tree, but this did not appear to help unless our sampling probability was too high, so we disabled it. 36 It w... |

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Citation Context ... result, our code is always competitive with hybrid, and sometimes outperforms it by a wide margin. Implementations of the push-relabel method for the maximum flow problem have been wellstudied,se.g. =-=[2, 10, 13, 14, 34]-=-. A maximum flow code of Cherkassky and Goldberg [10] was the starting point of our implementation, ho. The implementation uses the heuristics global update and gap relabeling heuristics that are used... |

79 |
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Citation Context ...y are theoretically more efficient -- time bounds for these algorithms are competitive with or better than the best time bounds for the minimum s-t cut problem. The algorithm of Nagamochi and Ibaraki =-=[32]-=- (ni) runs in O(n(m + n log n)) time. The algorithm of Karger and Stein [26] (ks) runs in O(n 2 log 3 n) expected time. Two closely related algorithms of Karger [25] (k) run in O(m log 3 n) and O(n 2 ... |

75 |
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Citation Context ...tope until the optimum solution to the relaxed problem is integral. The inequalities that have been very useful are subtour elimination constraints, first introduced by Dantzig, Fulkerson and Johnson =-=[12]-=-. The problem of identifying a subtour elimination constraint can be rephrased as the problem of finding a minimum cut in a graph with real-valued edge weights. Thus, cutting plane algorithms for the ... |

71 | Minimum cuts in near-linear time
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Citation Context ...algorithm of Nagamochi and Ibaraki [32] (ni) runs in O(n(m + n log n)) time. The algorithm of Karger and Stein [26] (ks) runs in O(n 2 log 3 n) expected time. Two closely related algorithms of Karger =-=[25]-=- (k) run in O(m log 3 n) and O(n 2 log n) expected time. These algorithms are based on new techniques which do not use flows. These algorithms do not extend to directed graphs, while the flow based al... |

71 | Random sampling in cut, flow, and network design problems. Mathematics of Operations Research (Preliminary version appeared
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Citation Context ...h, this algorithm finds a packing of c trees in O(mc log m=n) time. This is fine for small values of c but expensive for large values. To ensure a small value of c, we use random sampling: Lemma 7.3 (=-=[27]-=-) Given any graph G, in linear time we can construct a skeleton graph H on the same vertices with the following properties: 35 ffl H has m 0 = O(ffl \Gamma2 n log n) edges, ffl the minimum cut of H is... |

69 |
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Citation Context ...: (4) The value of a flow is the net flow into the sink, i.e., jf j = X v2V f(v; t): The maximum flow problem is to determine a flow f for which jf j is maximum. The well-known maxflow-mincut theorem =-=[16, 15]-=- states that the value of the maximum s-t flow is equal to the value of the minimum s-t cut, i.e., jf j =ss;t (G). An s-t maximum flow algorithm can thus be used to find an s-t minimum cut, and minimi... |

67 |
A matroid approach to finding edge connectivity and packing arborescences
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- 1995
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Citation Context ... Karger's Algorithm Karger's algorithm [25] is based on the following two observations: ffl Any undirected graph with minimum cut c has a packing of c spanning trees that uses each edge at most twice =-=[17]-=-. 34 ffl Given any packing of c spanning trees that uses each edge at most twice, at least some of the trees only cross the minimum cut twice. The second fact follows from a simple averaging argument:... |

67 |
Finding minimum-cost circulations by canceling negative cycles
- Goldberg, Tarjan
- 1989
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Citation Context ...is amortized by the cost of the preceding phase. As we shall see, the PR heuristic is crucial to ni's efficiency. 5 Hao-Orlin Algorithm 5.1 Push-Relabel Method First we review the push-relabel method =-=[19]-=- for finding minimum s-t cuts (as well as maximum flows) in directed graphs. We omit many details of the push-relabel methods, as they are not directly related to our implementation of the Hao-Orlin a... |

49 |
Tarjan, “A faster deterministic maximum flow algorithm
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Citation Context ...to the minimum cut problem. The classical Gomory-Hu algorithm [20] solves the minimum cut problem using n \Gamma 1 minimum s-t cut computations. The fastest current algorithms for the s-t cut problem =-=[1, 6, 7, 18, 28]-=- use flow techniques, in particular the push-relabel method [18], and run in !(nm) time. For the minimum cut problem, Hao and Orlin [22, 23] have given an algorithm (ho), based on the pushrelabel meth... |

48 |
A faster algorithms for finding the minimum cut in a graph, in
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(Show Context)
Citation Context ...fastest current algorithms for the s-t cut problem [1, 6, 7, 18, 28] use flow techniques, in particular the push-relabel method [18], and run in !(nm) time. For the minimum cut problem, Hao and Orlin =-=[22, 23]-=- have given an algorithm (ho), based on the pushrelabel method, that shows how to perform all n \Gamma 1 minimum s-t cuts in time asymptotically equal to that needed to perform one s-t minimum cut com... |

46 | An O(n ) algorithm for minimum cuts - Karger, Stein - 1993 |

38 | Improved Time Bounds for the Maximum Flow Problem
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Citation Context ...to the minimum cut problem. The classical Gomory-Hu algorithm [20] solves the minimum cut problem using n \Gamma 1 minimum s-t cut computations. The fastest current algorithms for the s-t cut problem =-=[1, 6, 7, 18, 28]-=- use flow techniques, in particular the push-relabel method [18], and run in !(nm) time. For the minimum cut problem, Hao and Orlin [22, 23] have given an algorithm (ho), based on the pushrelabel meth... |

37 |
Cluster analysis for hypertext systems
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Citation Context ...ts so that the total weight of the set of edges with endpoints in different sets is minimized. This problem has many applications, including network reliability theory [24, 37], information retrieval =-=[4]-=-, compilers for parallel languages [5], and as a subroutine in cutting-plane algorithms for the Traveling Salesman problem (TSP) [3]. The problem of finding a minimum capacity cut between two specifie... |

34 |
An efficient algorithm for the minimum capacity cut problem
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- 1990
(Show Context)
Citation Context ...sure that the heuristics do not significantly increase the running time when they fail. The most efficient implementation of the Gomory-Hu algorithm that we are aware of is due to Padberg and Rinaldi =-=[35]-=-. In order to reduce the number of maximum flow computations needed, which is n \Gamma 1 for the Gomory-Hu algorithm, they developed a set of heuristics which contract certain edges during the computa... |

31 | A faster algorithm for finding the minimum cut in a directed graph
- Hao, Orlin
- 1994
(Show Context)
Citation Context ...fastest current algorithms for the s-t cut problem [1, 6, 7, 18, 28] use flow techniques, in particular the push-relabel method [18], and run in !(nm) time. For the minimum cut problem, Hao and Orlin =-=[22, 23]-=- have given an algorithm (ho), based on the pushrelabel method, that shows how to perform all n \Gamma 1 minimum s-t cuts in time asymptotically equal to that needed to perform one s-t minimum cut com... |

28 |
Implementing Goldberg's Max-Flow Algorithm: A
- DERIGS, MEIER
- 1989
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Citation Context ... result, our code is always competitive with hybrid, and sometimes outperforms it by a wide margin. Implementations of the push-relabel method for the maximum flow problem have been wellstudied,se.g. =-=[2, 10, 13, 14, 34]-=-. A maximum flow code of Cherkassky and Goldberg [10] was the starting point of our implementation, ho. The implementation uses the heuristics global update and gap relabeling heuristics that are used... |

27 |
Very simple methods for all pairs network flow analysis
- Gusfield
- 1990
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Citation Context ...f the algorithms. We also develop several new heuristics that significantly improve performance of our implementations. Implementations using graph contraction are usually difficult to code (see e.g. =-=[21]-=-) and may be inefficient. Our fastest implementations of all the algorithms we study use contraction. Although indeed difficult to code, these implementations are efficient because of the graph data s... |

25 | Array distribution in data-parallel programs
- Chatterjee, Gilbert, et al.
(Show Context)
Citation Context ... of edges with endpoints in different sets is minimized. This problem has many applications, including network reliability theory [24, 37], information retrieval [4], compilers for parallel languages =-=[5]-=-, and as a subroutine in cutting-plane algorithms for the Traveling Salesman problem (TSP) [3]. The problem of finding a minimum capacity cut between two specified vertices, s and t, is called the min... |

25 |
Analysis of Preflow-Push Algorithms for Maximum Network Flow
- CHERIYAN, MAHESHWARI
- 1989
(Show Context)
Citation Context ...ghest distance label. This strategy, in combination with appropriate heuristics, seems to give the best results in practice [10]. This strategy also reduces the number of push operations. Theorem 5.4 =-=[8]-=- The push-relabel algorithm with the highest label selection performs O(n 2 p m) push operations and runs in O(n 2 p m) time. Sophisticated data structures, such as dynamic trees, improve running time... |

19 |
Goldberg's Algorithm for Maximum Flow in Perspective: A Computational Study, Algorithms for Network Flows and Matching, Edited by D
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(Show Context)
Citation Context ... result, our code is always competitive with hybrid, and sometimes outperforms it by a wide margin. Implementations of the push-relabel method for the maximum flow problem have been wellstudied,se.g. =-=[2, 10, 13, 14, 34]-=-. A maximum flow code of Cherkassky and Goldberg [10] was the starting point of our implementation, ho. The implementation uses the heuristics global update and gap relabeling heuristics that are used... |

18 |
Implementations of the GoldbergTarjan Maximum Flow Algorithm, Algorithms for Network Flows and Matching, Edited by D
- NGUYEN, VENKATESWARAN
- 1993
(Show Context)
Citation Context ...all vertices with distance labels d(v) or greater from the graph, as the sink is not reachable from these vertices. This heuristic often speeds up push-relabel algorithms for the maximum flow problem =-=[2, 10, 13, 34]-=- and is essential for the analysis of the Hao-Orlin algorithm. We use a standard implementation of gap relabeling that maintains an array of buckets, B[0 : : : 2n \Gamma 1], with bucket B[i] containin... |

16 |
A randomized maximum-flow algorithm
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- 1995
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Citation Context |

15 | Can a maximum flow be computed in O(nm) time
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Citation Context |

15 | Implementing an efficient minimum capacity cut algorithm
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(Show Context)
Citation Context ...cted graphs, while the flow based algorithms, including ho, do. Theoretical considerations suggest that some of these new algorithms should be practical, and an experimental study of Nagamochi et al. =-=[33]-=- confirms this for ni. The question of practical performance of the other new algorithms, however, has not been addressed. In this paper we study the practical performance of recent minimum cut algori... |

14 |
A randomized fully polynomial approximation scheme for the all terminal network reliability problem
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(Show Context)
Citation Context ...ted undirected graph into two sets so that the total weight of the set of edges with endpoints in different sets is minimized. This problem has many applications, including network reliability theory =-=[24, 37]-=-, information retrieval [4], compilers for parallel languages [5], and as a subroutine in cutting-plane algorithms for the Traveling Salesman problem (TSP) [3]. The problem of finding a minimum capaci... |

12 |
Counting almost minimum cutsets with reliability applications
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(Show Context)
Citation Context ...ted undirected graph into two sets so that the total weight of the set of edges with endpoints in different sets is minimized. This problem has many applications, including network reliability theory =-=[24, 37]-=-, information retrieval [4], compilers for parallel languages [5], and as a subroutine in cutting-plane algorithms for the Traveling Salesman problem (TSP) [3]. The problem of finding a minimum capaci... |

11 |
Lower bound of network reliability
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(Show Context)
Citation Context ...s with a single component. In other words: we consider deleting every edge of G with probability 2 \Gammad=c , and ask whether the remaining (contracted) edges connected G. The following is proven in =-=[30]-=- (see also [24]), using the fact that among all graph with minimum cut c, the graph most likely to become disconnected under random edge failures is a cycle: Lemma 6.2 Let G have n edges and minimum c... |

10 |
A fast algorithm for computing maximum flow in a network
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(Show Context)
Citation Context ...allest distance label of a vertex reachable from v via a residual arc. Lemma 5.1 [19] The relabel operation increases d(v). We assume that a relabel operation always uses the gap relabeling heuristic =-=[9, 13]-=-. Just before relabeling v, the heuristic checks if any other vertex has a label of d(v). If the answer is yes, then v is relabeled. Otherwise, the heuristic deletes all vertices with distance labels ... |

6 |
A linear time 2 + ffl approximation algorithm for edge connectivity
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Citation Context ...mum Cut In order to perform the sampling step correctly, the algorithm needs to know the value of the minimum cut. In [27] Karger gives two ways to resolve this problem. The first is to run Matula's (=-=[31]-=-) linear time 3-approximation algorithm; using that value divided by 3 gives a probability that doesn't affect correctness and at worst multiplies the number of edges in the sample by 3. The other opt... |

5 |
An Evaluation of Algorithmic Refinements and Proper Data-Structures for the Preflow-Push Approach for Maximum Flow
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Citation Context |