Improved Time Complexity of Bandwidth Approximation in Dense Graphs

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 $O({(\log\log n)}² / {\log n})$.