Optimal Neighborhood Exploration for Dynamic Independent Sets

TL;DR

The paper tackles the fully dynamic maximum weight and cardinality independent set problems on large-scale graphs. It introduces Optimum Neighborhood Exploration (ONE), which builds small induced subgraphs around updates and solves them to optimality with the KaMIS branch-and-reduce framework, enabling high-quality updates with tunable subproblem size parameters ($d$ and $\nu_{\max}$). By combining fast greedy updates (DegGreedy) with expensive but targeted local optimizations (DynamicONE) and several pruning strategies (pinching) and rare-update schemes, the method achieves strong empirical performance across diverse dynamic benchmarks, often outperforming state-of-the-art dynamic solvers. The approach demonstrates practical scalability and offers a flexible trade-off between running time and solution quality, with clear pathways for extending to other dynamic graph problems and parallelization.

Abstract

A dynamic graph algorithm is a data structure that supports edge insertions, deletions, and specific problem queries. While extensive research exists on dynamic algorithms for graph problems solvable in polynomial time, most of these algorithms have not been implemented or empirically evaluated. This work addresses the NP-complete maximum weight and cardinality independent set problems in a dynamic setting, applicable to areas like dynamic map-labeling and vehicle routing. Real-world instances can be vast, with millions of vertices and edges, making it challenging to find near-optimal solutions quickly. Exact solvers can find optimal solutions but have exponential worst-case runtimes. Conversely, heuristic algorithms use local search techniques to improve solutions by optimizing vertices. In this work, we introduce a novel local search technique called optimal neighborhood exploration. This technique creates independent subproblems that are solved to optimality, leading to improved overall solutions. Through numerous experiments, we assess the effectiveness of our approach and compare it with other state-of-the-art dynamic solvers. Our algorithm features a parameter, the subproblem size, that balances running time and solution quality. With this parameter, our configuration matches state-of-the-art performance for the cardinality independent set problem. By increasing the parameter, we significantly enhance solution quality.

Figures (5)

• Figure 1: Starting from a node $u$, the algorithms find a locally independent subgraph $G[H]$ by exploring the neighborhood up to a certain depth and afterwards adding adjacent nodes from the current solution to the subgraph. The subgraph $G[H]$ is then solved by a solver of our choice.
• Figure 2: Induced subgraph with BFS of depth 1 starting at $u$. Current solution vertices are orange. The set $H = N[u]\cup \{v\}$. The new optimal solution on the subgraph is green. When changing to this solution, the independent set is not maximal.
• Figure 3: Performance profiles (top) and running times in total for all updates (bottom) for various depths $d$ with $\nu_\text{max}=2500$.
• Figure 4: Performance profile for solution quality (top) and total update time comparison (bottom) against state-of-the-art algorithms on unweighted instances.
• Figure 5: Performance profile for solution quality (top) and running time box-plot comparison (bottom) against greedy algorithms on weighted instances.