A polynomial-time algorithm for recognizing high-bandwidth graphs
Luis M. B. Varona
TL;DR
The paper addresses the problem of recognizing whether a graph has bandwidth at most $k$, focusing on the high-bandwidth regime where $k \ge \left\lfloor (n-1)/2 \right\rfloor$. It reframes the task as a bipartite-matching problem and leverages Hall's marriage theorem to achieve an algorithm with time $O(n^{n-k+1})$ and space $O(n)$, making recognition polynomial when $n-k$ is small. Key ideas include left/right partial layouts, the feasible-pair condition, and an efficient construction of a compatible right layout via a nested family of sets $A_j$. The results demonstrate that this approach outperforms existing methods on affirmative high-bandwidth cases and provides a principled polynomial-time solution for the high-bandwidth regime, with potential extensions to pruning, weighted graphs, and low-bandwidth recognition.
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.