A polynomial-time algorithm for recognizing high-bandwidth graphs

TL;DR

This paper presents a polynomial-time algorithm for recognizing high-bandwidth graphs by reformulating the problem as bipartite matching, enabling efficient recognition when either the bandwidth or its complement is small.

Contribution

It introduces a novel reformulation of the bandwidth recognition problem for large k, using bipartite matching and Hall's theorem to achieve polynomial-time complexity.

Findings

Algorithm runs in O(n^{n - k + 1}) time for large k

Polynomial-time recognition when k or n - k is small

Reformulation leverages Hall's marriage theorem

Abstract

An unweighted, undirected graph $G$ on $n$ nodes is said to have \emph{bandwidth} at most $k$ if its nodes can be labelled from $0$ to $n - 1$ such that no two adjacent nodes have labels that differ by more than $k$. It is known that one can decide whether the bandwidth of $G$ is at most $k$ in $O(n^k)$ time and $O(n^k)$ space using dynamic programming techniques. For small $k$ close to $0$, this approach is effectively polynomial, but as $k$ scales with $n$, it becomes superexponential, requiring up to $O(n^{n - 1})$ time (where $n - 1$ is the maximum possible bandwidth). In this paper, we reformulate the problem in terms of bipartite matching for sufficiently large $k \ge \lfloor (n - 1)/2 \rfloor$, allowing us to use Hall's marriage theorem to develop an algorithm that runs in $O(n^{n - k + 1})$ time and $O(n)$ auxiliary space (beyond storage of the input graph). This yields polynomial complexity for large $k$ close to $n - 1$, demonstrating that the bandwidth recognition problem is solvable in polynomial time whenever either $k$ or $n - k$ remains small.