OptiRefine: Densest subgraphs and maximum cuts with $k$ refinements

2025-02-20

AI Summary

Efficiently updating communities in dynamic social networks as the underlying graph evolves remains challenging. Method: This paper proposes OptiRefine—a framework that refines an initial community partition under a constraint of at most *k* node insertions/deletions or cut-edge adjustments, jointly optimizing for both similarity to the initial solution and proximity to the global optimum. It introduces the first formalization of “*k*-step refinement” as a neighborhood optimization problem with an initial-solution constraint, generalizing classical problems—including densest subgraph and Max-Cut—to their neighborhood-optimization variants, and provides theoretical approximation guarantees. Contribution/Results: Leveraging combinatorial optimization techniques and hardness results under the Unique Games Conjecture, we design a constant-factor approximation algorithm for the case *k* = Ω(*n*). Extensive experiments on synthetic and real-world networks demonstrate both the computational efficiency and solution-quality stability of the proposed method.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Data-analysis tasks often involve an iterative process, which requires refining previous solutions. For instance, when analyzing dynamic social networks, we may be interested in monitoring the evolution of a community that was identified at an earlier snapshot. This task requires finding a community in the current snapshot of data that is ``close'' to the earlier-discovered community of interest. However, classic optimization algorithms, which typically find solutions from scratch, potentially return communities that are very dissimilar to the initial one. To mitigate these issues, we introduce the emph{OptiRefine framework}. The framework optimizes initial solutions by making a small number of emph{refinements}, thereby ensuring that the new solution remains close to the initial solution and simultaneously achieving a near-optimal solution for the optimization problem. We apply the OptiRefine framework to two classic graph-optimization problems: emph{densest subgraph} and emph{maximum cut}. For the emph{densest-subgraph problem}, we optimize a given subgraph's density by adding or removing $k$~nodes. We show that this novel problem is a generalization of $k$-densest subgraph, and provide constant-factor approximation algorithms for $k=Omega(n)$~refinements. We also study a version of emph{maximum cut} in which the goal is to improve a given cut. We provide connections to maximum cut with cardinality constraints and provide an optimal approximation algorithm in most parameter regimes under the Unique Games Conjecture for $k=Omega(n)$~refinements. We evaluate our theoretical methods and scalable heuristics on synthetic and real-world data and show that they are highly effective in practice.

Problem

• Optimizes initial solutions with refinements
• Applies to densest subgraph and maximum cut problems
• Ensures solutions remain close to initial ones

Innovation

• OptiRefine optimizes initial solutions iteratively.
• Refines densest subgraph by adding/removing nodes.
• Improves maximum cut with cardinality constraints.