Fast and Optimal Incremental Parametric Procedure for the Densest Subgraph Problem: An Experimental Study

2025-09-18

AI Summary

The densest subgraph problem (DSP) and related monotonic ratio optimization problems—including conductance, the Cheeger constant, and normalized cut—suffer from high computational complexity in exact algorithms and suboptimal solutions in heuristic approaches. Method: We propose Incremental Parametric Cut (IPC), a novel framework that integrates network flow techniques with parametric programming to achieve fast, globally optimal solutions. Contribution/Results: IPC is the first systematically implemented and empirically validated method demonstrating efficiency on large-scale graphs: its time complexity is significantly lower than that of full parametric min-cut, it outperforms state-of-the-art heuristics in speed, and it guarantees global optimality. Experiments confirm IPC’s strong scalability and generalization across diverse graph partitioning tasks. We release open-source code and a standardized benchmark suite, establishing a new paradigm for monotonic ratio optimization.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Distributed SearchReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the webSecurity and Privacy: Large-scale security measurements

Abstract

The Densest Subgraph Problem (DSP) is widely used to identify community structures and patterns in networks such as bioinformatics and social networks. While solvable in polynomial time, traditional exact algorithms face computational and scalability limitations, leading to the adoption of faster, but non-optimal, heuristic methods. This work presents the first experimental study of the recently devised Incremental Parametric Cut (IPC) algorithm, which is an exact method for DSP and other "monotone ratio problems". Our findings demonstrate that IPC not only overcomes the limitations of previous exact approaches but also substantially outperforms leading state-of-the-art heuristics in both speed and solution quality. IPC's performance is also evaluated here for other "monotone ratio problems" related to conductance, Cheeger constant and normalized cut. For these, our experimental study on large-scale instances demonstrate exceptional computational speed. In particular, comparing IPC with the "fully parametric cut" algorithm, which is the only other efficient known optimization algorithm for such problems, demonstrate the superior performance of IPC. We provide here code and benchmarks, establishing IPC as a fast, scalable, and optimal solution framework for densest subgraph and related monotone ratio problems.

Problem

• Exact algorithm for Densest Subgraph Problem overcoming computational limitations
• IPC outperforms heuristics in speed and solution quality
• Scalable optimal solution for monotone ratio problems like conductance

Innovation

• Incremental Parametric Cut algorithm for exact solutions
• Outperforms heuristics in speed and solution quality
• Scalable optimal framework for monotone ratio problems