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Superpolynomial smoothed complexity of 3-FLIP in Local Max-Cut

Lukas Michel, Alex Scott

TL;DR

The paper addresses the smoothed complexity of local-search Max-Cut, focusing on the 3-FLIP algorithm. It demonstrates two main results: (i) there exist $\mathcal{O}(n)$-vertex graphs with max degree $4$ for which FLIP requires exponential time to terminate under any pivot rule, and (ii) the smoothed runtime of 3-FLIP can be as large as $2^{\Omega(\sqrt{n})}$ when edge weights are randomly perturbed, marking the first superpolynomial smoothed-runtime example for a Max-Cut local-search method. A key technical tool is a linear-combination approximation lemma showing that sums of randomly weighted edges with coefficients in {−1,1} can approximate large intervals, enabling long improving sequences even under perturbations. These results highlight that smoothed performance can be highly sensitive to the specific local-search rule (e.g., 3-FLIP) and raise open questions about pivot rules that avoid multi-vertex moves or restricted graph classes that may retain efficient smoothed runtimes.

Abstract

Local search algorithms for NP-hard problems such as Max-Cut frequently perform much better in practice than worst-case analysis suggests. Smoothed analysis has proved an effective approach to understanding this: a substantial literature shows that when a small amount of random noise is added to input data, local search algorithms typically run in polynomial or quasi-polynomial time. In this paper, we provide the first example where a local search algorithm for the Max-Cut problem fails to be efficient in the framework of smoothed analysis. Specifically, we construct a graph with $n$ vertices where the smoothed runtime of the 3-FLIP algorithm can be as large as $2^{Ω(\sqrt{n})}$. Additionally, for the setting without random noise, we give a new construction of graphs where the runtime of the FLIP algorithm is $2^{Ω(n)}$ for any pivot rule. These graphs are much smaller and have a simpler structure than previous constructions.

Superpolynomial smoothed complexity of 3-FLIP in Local Max-Cut

TL;DR

The paper addresses the smoothed complexity of local-search Max-Cut, focusing on the 3-FLIP algorithm. It demonstrates two main results: (i) there exist -vertex graphs with max degree for which FLIP requires exponential time to terminate under any pivot rule, and (ii) the smoothed runtime of 3-FLIP can be as large as when edge weights are randomly perturbed, marking the first superpolynomial smoothed-runtime example for a Max-Cut local-search method. A key technical tool is a linear-combination approximation lemma showing that sums of randomly weighted edges with coefficients in {−1,1} can approximate large intervals, enabling long improving sequences even under perturbations. These results highlight that smoothed performance can be highly sensitive to the specific local-search rule (e.g., 3-FLIP) and raise open questions about pivot rules that avoid multi-vertex moves or restricted graph classes that may retain efficient smoothed runtimes.

Abstract

Local search algorithms for NP-hard problems such as Max-Cut frequently perform much better in practice than worst-case analysis suggests. Smoothed analysis has proved an effective approach to understanding this: a substantial literature shows that when a small amount of random noise is added to input data, local search algorithms typically run in polynomial or quasi-polynomial time. In this paper, we provide the first example where a local search algorithm for the Max-Cut problem fails to be efficient in the framework of smoothed analysis. Specifically, we construct a graph with vertices where the smoothed runtime of the 3-FLIP algorithm can be as large as . Additionally, for the setting without random noise, we give a new construction of graphs where the runtime of the FLIP algorithm is for any pivot rule. These graphs are much smaller and have a simpler structure than previous constructions.
Paper Structure (6 sections, 3 theorems, 6 equations, 2 figures)

This paper contains 6 sections, 3 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Local Search
  3. Smoothed Analysis
  4. Preliminaries
  5. Runtime of the FLIP algorithm
  6. Smoothed runtime of the 3-FLIP algorithm

Key Result

Theorem 1.1

For all $n \in \mathnormal{\mathbb{N}}$ there exist graphs $G$ with $\cO(n)$ vertices such that the following holds. Let $a < b$ be real numbers. Suppose that the edge weights of $G$ are chosen uniformly at random in $[a, b]$ and consider a uniformly random initial cut of $G$. Then, with high probab

Figures (2)

  • Figure 1: A graph $G_n$ with $\cO(n)$ vertices where the FLIP algorithm runs in time $\Omega(3^n)$.
  • Figure :

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • proof