Faster and Scalable Algorithms for Densest Subgraph and Decomposition
Faster and Scalable Algorithms for Densest Subgraph and Decomposition
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发表时间：
2022
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通讯作者：
Elfarouk Harb;Kent Quanrud;C. Chekuri
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作者：
Elfarouk Harb;Kent Quanrud;C. Chekuri
We study the densest subgraph problem (DSG) and the densest subgraph local decomposition problem (DSG-LD) in undirected graphs. We also consider su-permodular generalizations of these problems. For large scale graphs simple iterative algorithms perform much better in practice than theoretical基于网络流量或LP求解器的快速算法在[2]中，它会收敛到a（1✏2 -（g）⇤）的最佳密度的相对近似值，其中 - （g）是最大程度，而最佳密度是最佳密度。 Danisch等人。 m是本文中的边缘数量。在O（M）时间内实施，我们描述了具有强大的经验性能和理论保证的分数剥离技术。我们在实际和合成数据集上测试了我们的算法，并表明它对以前的算法提供了显着的好处。
We study the densest subgraph problem (DSG) and the densest subgraph local decomposition problem (DSG-LD) in undirected graphs. We also consider su-permodular generalizations of these problems. For large scale graphs simple iterative algorithms perform much better in practice than theoretically fast algorithms based on network-flow or LP solvers. Boob et al. [1] recently gave a fast iterative algorithm called G REEDY ++ for DSG. It was shown in [2] that it converges to a (1 � ✏ ) relative approximation to the optimum density in O ( 1 ✏ 2 � ( G ) � ⇤ ) iterations where � ( G ) is the maximum degree and � ⇤ is the optimum density. Danisch et al. [3] gave an iterative algorithm based on the Frank-Wolfe algorithm for DSG-LD that takes O ( m � ( G ) ✏ 2 ) iterations to converge to an ✏ -additive approximate local decomposition vector ˆ b , where m is number of edges in the graph. In this paper we give a new iterative algorithm for both problems that takes at most O ( p m � ( G ) ✏ ) iterations to converge to an ✏ -additive approximate local decomposition vector; each iteration can be implemented in O ( m ) time. We describe a fractional peeling technique which has strong empirical performance as well as theoretical guarantees. The algorithm is scalable and simple, and can be applied to graphs with hundreds of millions of edges. We test our algorithm on real and synthetic data sets and show that it provides a significant benefit over previous algorithms. The algorithm and analysis extends to hypergraphs.