Parameterized complexity of Bandwidth of Caterpillars and Weighted Path Emulation

TL;DR

This paper proves that the Bandwidth problem remains computationally hard for all levels of the W[t] hierarchy, even for simple caterpillar graphs, by introducing and analyzing the Weighted Path Emulation problem.

Contribution

It introduces the Weighted Path Emulation problem and demonstrates its W[t]-hardness and NP-completeness, extending hardness results to directed bandwidth on caterpillars.

Findings

Bandwidth is W[t]-hard for all t, even for caterpillars with hair length at most three.

Weighted Path Emulation is W[t]-hard and strongly NP-complete.

Directed Bandwidth remains W[t]-hard for certain directed acyclic graphs.

Abstract

In this paper, we show that Bandwidth is hard for the complexity class $W[t]$ for all $t\in {\bf N}$, even for caterpillars with hair length at most three. As intermediate problem, we introduce the Weighted Path Emulation problem: given a vertex-weighted path $P_N$ and integer $M$, decide if there exists a mapping of the vertices of $P_N$ to a path $P_M$, such that adjacent vertices are mapped to adjacent or equal vertices, and such that the total weight of the image of a vertex from $P_M$ equals an integer $c$. We show that {\sc Weighted Path Emulation}, with $c$ as parameter, is hard for $W[t]$ for all $t\in {\bf N}$, and is strongly NP-complete. We also show that Directed Bandwidth is hard for $W[t]$ for all $t\in {\bf N}$, for directed acyclic graphs whose underlying undirected graph is a caterpillar.