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Densest Subgraph Problem (DSP) is an important primitive problem with a widerange of applications, including fraud detection, community detection and DNAmotif discovery. Edge-based density is one of the most common metrics in DSP.Although a maximum flow algorithm can exactly solve it in polynomial time, theincreasing amount of data and the high complexity of algorithms motivatescientists to find approximation algorithms. Among these, its duality of linearprogramming derives several iterative algorithms including Greedy++,Frank-Wolfe and FISTA which redistribute edge weights to find the densestsubgraph, however, these iterative algorithms vibrate around the optimalsolution, which are not satisfactory for fast convergence. We propose our mainalgorithm Locally Optimal Weight Distribution (LOWD) to distribute theremaining edge weights in a locally optimal operation to converge to theoptimal solution monotonically. Theoretically, we show that it will reach theoptimal state of a specific linear programming which is called locally-densedecomposition. Besides, we show that it is not necessary to consider most ofthe edges in the original graph. Therefore, we develop a pruning algorithmusing a modified Counting Sort to prune graphs by removing unnecessary edgesand nodes, and then we can search the densest subgraph in a much smaller graph.

Fast Searching The Densest Subgraph And Decomposition With Local Optimality