Fast Searching The Densest Subgraph And Decomposition With Local Optimality

TL;DR

This paper introduces LOWD, a local optimality-based algorithm for efficiently finding the densest subgraph, improving convergence speed and reducing computational complexity through graph pruning.

Contribution

The paper presents LOWD, a novel algorithm that guarantees monotonic convergence to the optimal solution and incorporates a pruning method for faster densest subgraph search.

Findings

LOWD converges monotonically to the optimal solution.

Graph pruning significantly reduces search space.

The method outperforms existing iterative algorithms in speed.

Abstract

Densest Subgraph Problem (DSP) is an important primitive problem with a wide range of applications, including fraud detection, community detection and DNA motif 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, the increasing amount of data and the high complexity of algorithms motivate scientists to find approximation algorithms. Among these, its duality of linear programming derives several iterative algorithms including Greedy++, Frank-Wolfe and FISTA which redistribute edge weights to find the densest subgraph, however, these iterative algorithms vibrate around the optimal solution, which are not satisfactory for fast convergence. We propose our main algorithm Locally Optimal Weight Distribution (LOWD) to distribute the remaining edge weights in a locally optimal operation to converge to the optimal solution monotonically. Theoretically, we show that it will reach the optimal state of a specific linear programming which is called locally-dense decomposition. Besides, we show that it is not necessary to consider most of the edges in the original graph. Therefore, we develop a pruning algorithm using a modified Counting Sort to prune graphs by removing unnecessary edges and nodes, and then we can search the densest subgraph in a much smaller graph.