Bandwidth Parameterized by Cluster Vertex Deletion Number

TL;DR

This paper advances the parameterized complexity of the Bandwidth problem by showing it is fixed-parameter tractable (FPT) when parameterized by the cluster vertex deletion number plus the clique number, $cvd(G) + \omega(G)$, and by establishing W[1]-hardness when parameterized by $cvd(G)$ alone. The core technique encodes Bandwidth as a feasibility instance of integer linear programming (ILP) over a compact set of variables derived from a small augmented deletion set and a representative set of clusters, together with a refined notion of bucket-distribution and cluster-types. A key contribution is the development of distribution-types and nice orderings (Pi1–Pi3) that allow all relevant placement constraints to be captured by linear inequalities, reducing the problem to ILP feasibility in a parameter-bounded space. These results generalize prior tractability by vertex cover and narrow gaps between tractable and intractable parameter regimes, shedding light on how local neighborhood structure governs bandwidth in graphs near cluster graphs. The methods have potential to inform design of ILP-based fixed-parameter algorithms for related graph-layout problems.

Abstract

Given a graph $G$ and an integer $b$, Bandwidth asks whether there exists a bijection $π$ from $V(G)$ to $\{1, \ldots, |V(G)|\}$ such that $\max_{\{u, v \} \in E(G)} | π(u) - π(v) | \leq b$. This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the tree-depth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number. In this paper we make progress in understanding the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number $ω$, thus significantly strengthening the previously mentioned result for vertex cover number. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results develop and generalize some of the methods of argumentation of the previous results and narrow some of the complexity gaps.

Figures (1)

• Figure 1: Our results and hierarchy of some related structural graph parameters, where $\omega$ and $\Delta$ denote the clique number and the maximum degree of the input graph, respectively. Arrows between parameters indicate generalization relations, that is, for any graph, if the parameter at the tail of an arrow is a constant then the parameter at the head of the arrow is also a constant. The reverse does not hold in this figure. The framed green, frameless orange, and double framed red rectangles indicate fixed-parameter tractable, W[$\ast$]-hard, and NP-complete cases, respectively.

Theorems & Definitions (8)

• Remark 1
• Theorem 2
• Definition 5: Property $(\Pi_1)$
• Lemma 6
• Definition 7: Property $(\Pi_2)$
• Lemma 8
• Remark 9
• Definition 10: Property $(\Pi_3)$