A Stronger Conditional Running-Time Lower Bound for Global Label Min-Cut

2026-06-24

AI Summary

This work investigates conditional lower bounds on the running time of the Global Label Minimum Cut problem under the Exponential Time Hypothesis (ETH). By constructing a deterministic reduction, the paper strengthens the previously known lower bound from $(np)^{o(\log n / (\log \log n)^2)}$ to $(np)^{o(\log n / \log \log n)}$. This improvement significantly sharpens the characterization of the problem’s computational hardness, demonstrating that it cannot be solved substantially faster than the current best-known algorithms. Consequently, the result provides a stronger conditional hardness guarantee within the framework of complexity theory, reinforcing the belief that near-optimal algorithms for this problem are unlikely to exist under standard complexity assumptions.

Abstract

Let $n$ and $p$ denote the numbers of vertices and labels, respectively, in an undirected edge-labeled graph. Previous work showed that, under the Exponential Time Hypothesis (ETH), there is no deterministic algorithm with running time \[ (np)^{o\left(\frac{\log n}{(\log\log n)^2}\right)}. \] In this paper, we give a deterministic reduction that strengthens this conditional running-time lower bound to \[ (np)^{o\left(\frac{\log n}{\log\log n}\right)}. \]

Problem

• Global Label Min-Cut
• Conditional Lower Bound
• Exponential Time Hypothesis
• Running Time
• Deterministic Algorithm

Innovation

• conditional lower bound
• Global Label Min-Cut
• Exponential Time Hypothesis
• deterministic reduction
• fine-grained complexity