Improved Time Complexity of Bandwidth Approximation in Dense Graphs

TL;DR

This paper presents a faster randomized algorithm for approximating the bandwidth of dense graphs, reducing the time complexity significantly while maintaining approximation quality, and extends applicability to broader graph classes.

Contribution

The paper introduces a novel, faster randomized algorithm for bandwidth approximation in dense graphs, improving previous time bounds and extending the class of graphs where the approach is effective.

Findings

Reduced time complexity from $O(n^6 ( ext{log} n)^{O(1)})$ to $O(n^{4+o(1)})$

Reformulated the perfect matching phase with a smaller maximum flow problem, achieving $O(n^2 ext{log} ext{log} n)$ complexity

Extended the algorithm's polynomial time applicability to graphs with $ ext{min degree} o O(( ext{log} ext{log} n)^2 / ext{log} n)$

Abstract

Given a graph $G=(V, E)$ and and a proper labeling $f$ from $V$ to $\{1, ..., n\}$, we define $B(f)$ as the maximum absolute difference between $f(u)$ and $f(v)$ where $(u,v)\in E$. The bandwidth of $G$ is the minimum $B(f)$ for all $f$. Say $G$ is $\delta$-dense if its minimum degree is $\delta n$. In this paper, we investigate the trade-off between the approximation ratio and the time complexity of the classical approach of Karpinski {et al}.\cite{Karpin97}, and present a faster randomized algorithm for approximating the bandwidth of $\delta$-dense graphs. In particular, by removing the polylog factor of the time complexity required to enumerate all possible placements for balls to bins, we reduce the time complexity from $O(n^6\cdot (\log n)^{O(1)})$ to $O(n^{4+o(1)})$. In advance, we reformulate the perfect matching phase of the algorithm with a maximum flow problem of smaller size and reduce the time complexity to $O(n^2\log\log n)$. We also extend the graph classes could be applied by the original approach: we show that the algorithm remains polynomial time as long as $\delta$ is $O({(\log\log n)}^2 / {\log n})$.