Local Max-Cut on Sparse Graphs

TL;DR

This work analyzes the smoothed running time of the FLIP local-search algorithm for the local Max-Cut problem on graphs with bounded arboricity $α$, under edge weights perturbed with density at most $φ$. The authors develop a hierarchical vertex-partitioning framework exploiting sparsity and prove that two consecutive vertex flips within a layer yield a guaranteed improvement, enabling a bound on progress under smoothed perturbations. They obtain two main results: (i) for $α=O(\log^{1-δ} n)$, FLIP runs in polynomial time $φ\,n^{O(1/δ)}$ both in expectation and w.h.p.; (ii) for arbitrary $α$, FLIP runs in $φ\,n^{O(\frac{α}{\log n}+\log α)}$ iterations, improving prior $φ\,n^{O(\sqrt{\log n})}$ bounds and achieving $φ\,n^{O(\log\log n)}$ when $α=O(\log n)$. The results extend polynomial smoothed-time guarantees from complete graphs and low-degree graphs to the broader class of bounded-arboricity graphs, advancing understanding of local-search dynamics in sparse networks under perturbations. The approach combines hierarchical decompositions, probabilistic bounds on edge-weight linear combinations, and a potential-function argument to quantify progress. ∎

Abstract

We bound the smoothed running time of the FLIP algorithm for local Max-Cut as a function of $α$, the arboricity of the input graph. We show that, with high probability and in expectation, the following holds (where $n$ is the number of nodes and $φ$ is the smoothing parameter): 1) When $α= O(\log^{1-δ} n)$ FLIP terminates in $φpoly(n)$ iterations, where $δ\in (0,1]$ is an arbitrarily small constant. Previous to our results the only graph families for which FLIP was known to achieve a smoothed polynomial running time were complete graphs and graphs with logarithmic maximum degree. 2) For arbitrary values of $α$ we get a running time of $φn^{O(\fracα{\log n} + \log α)}$. This improves over the best known running time for general graphs of $φn^{O(\sqrt{ \log n })}$ for $α= o(\log^{1.5} n)$. Specifically, when $α= O(\log n)$ we get a significantly faster running time of $φn^{O(\log \log n)}$.

Figures (1)

• Figure 1: We consider the gain in the cut weight when $v$ is flipped for the first time plus the gain when it is flipped for the second time. We note that weights of edges to nodes that were flipped an even number of times between the two movement of $v$ get cancelled out and do not appear in the sum, while nodes that were flipped an odd number of times have a coefficient in $\left\{ -2,2 \right\}$.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 3
• proof