Parameterized complexity of Bandwidth of Caterpillars and Weighted Path Emulation

12/02/2020

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by   Hans L. Bodlaender, et al.

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In this paper, we show that Bandwidth is hard for the complexity class W[t] for all t∈ 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 Weighted Path Emulation, with c as parameter, is hard for W[t] for all t∈ N, and is strongly NP-complete. We also show that Directed Bandwidth is hard for W[t] for all t∈ N, for directed acyclic graphs whose underlying undirected graph is a caterpillar.