低带宽图的禁用子图
Forbidden Subgraphs of Graphs with Low Bandwidth
- Princeton University（普林斯顿大学）
- Department of Computer Science, University of California Santa Barbara（加州大学圣塔芭芭拉分校计算机科学系）
- Hebrew University of Jerusalem（耶路撒冷希伯来大学）
- Universidad de Valladolid（巴利亚多利德大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对低带宽图的禁用子图问题，证明一般图的带宽问题是FPT可近似的，给出对应算法并正面解决Chung和Seymour的开放问题，得到有界带宽图的禁用子图刻画。
AI中文摘要：
图G的布局是一个单射函数$f: V(G) \rightarrow Z$，布局$f$的带宽为$bw(G,f) = max_{uv \in E(G)} |f(u) - f(v)|$，图G的带宽$bw(G)$是其所有布局中带宽的最小值。计算图的带宽是一个公认的难题：假设P≠NP，即使在非常受限的树类上也不存在多项式时间算法[Monien, SIAM Journal on Algebraic Discrete Methods, 1986]，且即使在树类上也不存在常数因子近似算法[Dubey et al., JCSS 2011]。假设指数时间假设成立，不存在运行时间为$f(k)n^{o(k)}$的算法来判定输入图的带宽是否至多为k，即使在非常受限的树类上亦是如此[Dregi和Lokshtanov, ICALP 2014]。本文中，我们证明一般图上的带宽问题（Bandwidth）是FPT可近似的。具体而言，我们给出一个算法，输入为图G和整数k，运行时间为$2^{O(9^k)}n^{O(1)}$，输出G的一个子树T，满足$bw(T) \geq k$，或输出G的一个带宽至多为$(10^{85} k^{28})^{4^k}$的布局。这正面解决了Chung和Seymour[Discrete Mathematics, 1989]提出的开放问题，该问题询问是否每个图G的带宽都可由其某个子树的最大带宽上界来界定。我们的定理给出了有界带宽图的禁用子图刻画，可视为带宽对应于树宽的经典网格子式定理、路径宽的禁用子树定理、树深度的禁用子路径定理的类似结果。
英文摘要：
A layout of a graph G is an injective function $f : V(G) \rightarrow Z$, and the bandwidth of a layout f is $bw(G,f) = max_{uv \in E(G)} |f(u) - f(v)|$. The bandwidth bw(G) of G is the minimum bandwidth of a layout of G. Computing the bandwidth of a graph is a notoriously hard problem: assuming P != NP, there is no polynomial time algorithm, even on very restricted classes of trees [Monien, SIAM Journal on Algebraic Discrete Methods, 1986], and no constant factor approximation, even on trees [Dubey et al., JCSS 2011]. Assuming the Exponential Time Hypothesis, there is no algorithm with running time $f(k)n^{o(k)}$ to determine whether an input graph has bandwidth at most k, even on very restricted classes of trees [Dregi and Lokshtanov, ICALP 2014]. In this paper we show that {\sc Bandwidth} on general graphs is FPT-approximable. In particular we give an algorithm that takes as input a graph G and an integer k, runs in time $2^{O(9^k)}n^{O(1)}$, and outputs a subtree T of G such that $bw(T) \geq k$ or a layout of G of bandwidth at most $(10^{85} k^{28})^{4^k}$. This resolves in the affirmative an open problem of Chung and Seymour [Discrete Mathematics, 1989], who asked whether the bandwidth of every graph G is upper bounded in terms of the maximum bandwidth of a subtree of G. Our theorem leads to a forbidden subgraph characterization for graphs of bounded bandwidth, and can be seen as an analog for bandwidth of the classic grid minor theorem for treewidth, the forbidden subtree theorem for pathwidth, and the forbidden subpath theorem for treedepth.
