## The Partial Augment–Relabel Algorithm for the Maximum Flow Problem

Venue: | In Proc. 16th Annual European Symposium Algorithms |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Goldberg_thepartial,

author = {Andrew V. Goldberg},

title = {The Partial Augment–Relabel Algorithm for the Maximum Flow Problem},

booktitle = {In Proc. 16th Annual European Symposium Algorithms},

year = {},

pages = {466--477}

}

### OpenURL

### Abstract

Abstract. The maximum flow problem is a classical optimization problem with many applications. For a long time, HI-PR, an efficient implementation of the highest-label push-relabel algorithm, has been a benchmark due to its robust performance. We propose another variant of the push-relabel method, the partial augment-relabel (PAR) algorithm. Our experiments show that PAR is very robust. It outperforms HI-PR on all problem families tested, asymptotically in some cases. 1

### Citations

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Citation Context ...ssible path from v to x. To extend the path, the algorithm uses the current arc data structure to find an admissible arc (x,y). If such an arc exists, the algorithm extends the path and makes y the 2 =-=[1]-=- refers to this algorithm as the shortest augmenting path algorithm. However, this is only one of the methods that augment along the shortest paths. Methods of [14, 15] are also shortest augmenting pa... |

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(Show Context)
Citation Context ... maximum flow algorithms developed for specific applications outperform HI-PR in these applications. For example, minimum cuts are being used extensively in vision applications. Boykov and Kolmogorov =-=[5]-=- developed an algorithm that is superior to the highest level and FIFO push-relabel implementations on many vision problems. Their implementation is extensively used by the vision community. See [5, 6... |

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Citation Context ...le outperforms other popular selection rules. This implementation, PRF, and its later version, HI-PR, has been used for over a decade in many applications. Several subsequent implementations, such as =-=[26]-=-, use the combination of gap and global update and differ by low-level data structures or initialization strategies. A number of attempts have been made to develop an implementation that is more robus... |

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Citation Context ...ertices and arcs in the input network by n and m, respectively. Theoretical line of research led to the development of augmenting path [16], network simplex [12], blocking flow [14], and push-relabel =-=[20]-=- methods. The best currently known time bounds appear in [25, 19]. From the practical point of view, good implementations of Dinic’s blocking flow method [10, 21] proved superior to the network simple... |

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Citation Context ...m extends the path and makes y the 2 [1] refers to this algorithm as the shortest augmenting path algorithm. However, this is only one of the methods that augment along the shortest paths. Methods of =-=[14, 15]-=- are also shortest augmenting path algorithms.current vertex. Otherwise the algorithm shrinks the path and relabels x. The search terminates if x = t, or the length of the path reaches k, or v is the... |

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Citation Context ...ation [24] of the blocking flow method uses preflows and the push operation.flow excesses to the source by reducing arc flows in the reverse topological order with respect to this acyclic graph. See =-=[29]-=-. Both in theory and in practice, the first stage of the algorithm dominates the running time. The current arc data structure [20] is important for algorithm efficiency; it works as follows. Each vert... |

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Citation Context ...cities, the source, and the sink. Below we denote the number of vertices and arcs in the input network by n and m, respectively. Theoretical line of research led to the development of augmenting path =-=[16]-=-, network simplex [12], blocking flow [14], and push-relabel [20] methods. The best currently known time bounds appear in [25, 19]. From the practical point of view, good implementations of Dinic’s bl... |

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Citation Context ...p in practice, where its overhead exceeds the corresponding performance gains [4]. The Challenge also led to the development of the DIMACS problem families.Subsequent work of Cherkassky and Goldberg =-=[11]-=- showed that, with a proper choice of data structures, the combination of global update and gap heuristics is more robust than the individual heuristics, and the high-level selection rule outperforms ... |

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(Show Context)
Citation Context ...r a decade. In this context, the superior performance of PAR is significant. Furthermore, the push-relabel algorithm is the basis for efficient implementations of algorithms for the minimum-cost flow =-=[18]-=-, global minimum cut [8], and parametric flow [3] problems. The ideas presented in this paper apply to these problems, and may improve performance of the corresponding implementations. 2 Definitions a... |

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Citation Context ...d the sink. Below we denote the number of vertices and arcs in the input network by n and m, respectively. Theoretical line of research led to the development of augmenting path [16], network simplex =-=[12]-=-, blocking flow [14], and push-relabel [20] methods. The best currently known time bounds appear in [25, 19]. From the practical point of view, good implementations of Dinic’s blocking flow method [10... |

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Citation Context ...y. Theoretical line of research led to the development of augmenting path [16], network simplex [12], blocking flow [14], and push-relabel [20] methods. The best currently known time bounds appear in =-=[25, 19]-=-. From the practical point of view, good implementations of Dinic’s blocking flow method [10, 21] proved superior to the network simplex and the augmenting path algorithms. The blocking flow method re... |

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Citation Context ...econd stage is to first reduce flow around flow cycles to make the flow acyclic, and then to return 1 Sometimes it is referred to as preflow-push, which is misleading: e.g., Karzanov’s implementation =-=[24]-=- of the blocking flow method uses preflows and the push operation.flow excesses to the source by reducing arc flows in the reverse topological order with respect to this acyclic graph. See [29]. Both... |

41 | Experimental study of minimum cut algorithms
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(Show Context)
Citation Context ...xt, the superior performance of PAR is significant. Furthermore, the push-relabel algorithm is the basis for efficient implementations of algorithms for the minimum-cost flow [18], global minimum cut =-=[8]-=-, and parametric flow [3] problems. The ideas presented in this paper apply to these problems, and may improve performance of the corresponding implementations. 2 Definitions and Notation Input to the... |

32 |
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Citation Context ...12], blocking flow [14], and push-relabel [20] methods. The best currently known time bounds appear in [25, 19]. From the practical point of view, good implementations of Dinic’s blocking flow method =-=[10, 21]-=- proved superior to the network simplex and the augmenting path algorithms. The blocking flow method remained the method of choice until the development of the push-relabel method [20], which was quic... |

25 |
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(Show Context)
Citation Context ...y that a vertex v ̸= s,t is active if d(v) < n and ef > 0. The highest-label variant of the method, which at each step selects an active vertex with the highest distance label, runs in O(n 2√ m) time =-=[9, 30]-=-. HI-PR Implementation. Next we review the HI-PR implementation [11] of the push-relabel algorithm. It uses the highest-label selection rule, and global update and gap heuristics. To facilitate implem... |

24 | Graph cuts in vision and graphics: Theories and applications
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(Show Context)
Citation Context ...v [5] developed an algorithm that is superior to the highest level and FIFO push-relabel implementations on many vision problems. Their implementation is extensively used by the vision community. See =-=[5, 6]-=- for surveys of the vision applications. In this paper we describe a new partial augment-relabel (PAR) variant of the push-relabel method. In our comparison of PAR to HI-PR, the former code is consist... |

21 |
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(Show Context)
Citation Context ... speedup heuristics applicable in the sequential context as well, has been studied in [17]. Implementations of variants of the method has been studied during the First DIMACS Implementation Challenge =-=[23, 2, 27]-=-. This work showed that using the global update and gap heuristics (discussed in detail in Section 3), one gets an implementation superior to the efficient implementations of Dinic’s algorithm. Anothe... |

18 |
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Citation Context ... speedup heuristics applicable in the sequential context as well, has been studied in [17]. Implementations of variants of the method has been studied during the First DIMACS Implementation Challenge =-=[23, 2, 27]-=-. This work showed that using the global update and gap heuristics (discussed in detail in Section 3), one gets an implementation superior to the efficient implementations of Dinic’s algorithm. Anothe... |

7 | Experimental evaluation of a parametric flow algorithm
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(Show Context)
Citation Context ...nce of PAR is significant. Furthermore, the push-relabel algorithm is the basis for efficient implementations of algorithms for the minimum-cost flow [18], global minimum cut [8], and parametric flow =-=[3]-=- problems. The ideas presented in this paper apply to these problems, and may improve performance of the corresponding implementations. 2 Definitions and Notation Input to the maximum flow problem is ... |

5 |
An Evaluation of Algorithmic Refinements and Proper Data-Structures for the Preflow-Push Approach for Maximum Flow
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Citation Context ...the layer. A pointer is null if the corresponding bucket is empty. To implement the highest-level selection, we maintain the index of the highest layer with non-empty active bucket. The gap heuristic =-=[13]-=- is based on the following observation. Suppose for 0 < i < n, no vertex has a distance label of i but some vertices w have distance labels j : i < j < n. The validity of d implies that such w’s canno... |

5 |
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Citation Context ...he method of choice until the development of the push-relabel method [20], which was quickly recognized as practical. Within a year of its invention, the method had been used in a physics application =-=[28]-=-. A parallel implementation of the method, including some speedup heuristics applicable in the sequential context as well, has been studied in [17]. Implementations of variants of the method has been ... |

4 |
Experimental study of the pseudoflow push-relabel algorithm
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(Show Context)
Citation Context ...f our experiments is to compare PAR with HI-PR. We use the latest version, 3.6, of HI-PR and the current version, 0.23, of PAR. We also make a comparison to an implementation of Chandran and Hochbaum =-=[7]-=-. A paper describing this implementation is listed on authors’ web sites as “submitted for publication” and no preprint is publicly available. The authors do make their code available, and gave severa... |

4 |
Metod porazryadnogo sokrashcheniya nevyazok i transportnye zadachi. Issledovaniya po Diskretnoi
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(Show Context)
Citation Context ... denote the number of vertices and arcs in the input network by n and m, respectively. Theoretical line of research led to the development of augmenting path [16], network simplex [12], blocking flow =-=[14]-=-, and push-relabel [20] methods. The best currently known time bounds appear in [25, 19]. From the practical point of view, good implementations of Dinic’s blocking flow method [10, 21] proved superio... |

3 |
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(Show Context)
Citation Context ...pt in degenerate cases, the dynamic tree data structure, used in the most theoretically efficient algorithms, does not help in practice, where its overhead exceeds the corresponding performance gains =-=[4]-=-. The Challenge also led to the development of the DIMACS problem families.Subsequent work of Cherkassky and Goldberg [11] showed that, with a proper choice of data structures, the combination of glo... |

3 |
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(Show Context)
Citation Context ...y that a vertex v ̸= s,t is active if d(v) < n and ef > 0. The highest-label variant of the method, which at each step selects an active vertex with the highest distance label, runs in O(n 2√ m) time =-=[9, 30]-=-. HI-PR Implementation. Next we review the HI-PR implementation [11] of the push-relabel algorithm. It uses the highest-label selection rule, and global update and gap heuristics. To facilitate implem... |

2 | Träff, An implementation of the binary blocking flow algorithm
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(Show Context)
Citation Context ...A number of attempts have been made to develop an implementation that is more robust than HI-PR. The ideas behind the binary blocking flow algorithm in particular appear practical, and Hagerup et al. =-=[22]-=- show that this algorithm outperforms Dinic’ blocking flow algorithm. However, although the algorithm has a better worst-case time bound and performs better on specific bad instances, all attempts so ... |