Cut Tree Construction from Massive Graphs

Abstract

Abstract-The construction of cut trees (also known as GomoryHu trees) for a given graph enables the minimum-cut size of the original graph to be obtained for any pair of vertices. Cut trees are a powerful back-end for graph management and mining, as they support various procedures related to the minimum cut, maximum flow, and connectivity. However, the crucial drawback with cut trees is the computational cost of their construction. In theory, a cut tree is built by applying a maximum flow algorithm for n times, where n is the number of vertices. Therefore, naive implementations of this approach result in cubic time complexity, which is obviously too slow for today's large-scale graphs. To address this issue, in the present study, we propose a new cut-tree construction algorithm tailored to real-world networks. Using a series of experiments, we demonstrate that the proposed algorithm is several orders of magnitude faster than previous algorithms and it can construct cut trees for billion-scale graphs.

Practical Minimum Cut Algorithms

“…If for two of the vertices the incident triangle edges together are at least as heavy as all other incident edges, we can contract those, as shown in Figure 8.3. (5). Each of the continuous lines between v 1 and v 2 can be replaced with the dashed line without increasing the value of the cut.…”

“…MAIN CONTRIBUTIONS AND OUTLINE 3 take up ideas from FPT or kernelization theory, e.g. for independent sets (or equivalently vertex cover) [27,35,44,97,98,122], for cut tree construction [5], for treewidth computations [16,117,183], for the feedback vertex set problem [62,114], for the dominating set problem [2], for the maximum cut problem [57], for the cluster editing problem [23], and the matching problem [116]. In this dissertation, we make heavy use of data reduction techniques to improve the performance of algorithms for the minimum cut problem and the multiterminal cut problem.…”

“…A closely related problem is the cut tree problem (or Gomory-Hu tree problem), which aims to find a tree T = (V, E T , c T ), such that for each two vertices u, v ∈ V , the weight of the minimum u-v-cut is equal to the lightest edge weight on the unique path from u to v on T . This problem was first solved by Gomory and Hu [78] using n − 1 minimum s-t-cuts and has been studied experimentally by Goldberg and Tsioutsiouliklis [76] and Akiba et al [6], who solve the cut tree problem for graphs with millions of vertices and up to one billion edges in a few hours. Hartmann and Wagner [83] give a fully-dynamic algorithm to construct and maintain a cut tree under edge insertions, deletions, and weight changes.…”

“…6: Improvement of existing partitions from the Walshaw benchmark with = 1% using our ILP approach. In each k-column the results computed by our approach are on the left and the current Walshaw cuts are on the right.…”

Finding All Global Minimum Cuts In Practice

“…There are several experimental studies in the area that take up ideas from FPT or kernelization theory, e.g. for independent sets (or equivalently vertex cover) [2,11,14,41,29,30], for cut tree construction [3], for treewidth computations [7,54,39], for the feedback vertex set problem [37,18], for the dominating set problem [1], for the minimum cut [27,28], for the multiterminal cut problem [26], for the maximum cut problem [17] and for the cluster editing problem [8]. Recently, this type of data reduction techniques is also applied for problem in P such as matching [38],…”