Core Concepts
The FLIP algorithm for the local Max-Cut problem has a smoothed polynomial running time on graphs with bounded arboricity, improving over the best known results for general graphs.
Abstract
The paper analyzes the smoothed complexity of the FLIP algorithm for the local Max-Cut problem on graphs with bounded arboricity.
Key highlights:
For graphs with arboricity α = O(log^(1-ε) n), FLIP terminates in φpoly(n) iterations with high probability, where φ is the smoothing parameter. This improves over the previous results which only showed polynomial smoothed running time for complete graphs and graphs with logarithmic maximum degree.
For arbitrary values of arboricity α, the running time of FLIP is bounded by φn^O(α/log n + log α), which is significantly faster than the previous best bound of φn^O(√log n) for α = o(log^1.5 n). Specifically, when α = O(log n), the running time is φn^O(log log n).
The analysis uses a hierarchical partition of the node set based on the graph's arboricity. It shows that certain "good" sequences of node movements lead to a significant increase in the cut weight. The paper then proves that such good sequences must appear frequently in any sufficiently long execution of FLIP, leading to the final smoothed complexity bounds.
Stats
The paper does not provide any specific numerical data or metrics. It focuses on analyzing the smoothed running time of the FLIP algorithm as a function of the graph's arboricity.