Hamiltonicity Parameterized by Mim-Width is (Indeed) Para-NP-Hard
🤖 AI Summary
This work resolves the computational complexity of the Hamiltonian Path and Hamiltonian Cycle problems parameterized by mim-width. Prior to this, the para-NP-hardness of these problems for constant mim-width lacked a rigorous proof, and existing attempts contained flaws. We construct a carefully designed polynomial-time reduction that, for the first time, rigorously establishes NP-completeness of both problems even when the input graph is accompanied by a linear mim-width decomposition of width 26. This result pinpoints the precise computational hardness threshold for Hamiltonicity with respect to mim-width, thereby settling a long-standing open question in parameterized complexity theory and correcting previously informal or incomplete arguments.
Application Category
📝 Abstract
We prove that Hamiltonian Path and Hamiltonian Cycle are NP-hard on graphs of linear mim-width 26, even when a linear order of the input graph with mim-width 26 is provided together with input. This fills a gap left by a broken proof of the para-NP-hardness of Hamiltonicity problems parameterized by mim-width.
Problem
Research questions and friction points this paper is trying to address.
Proves Hamiltonian Path NP-hard on linear mim-width 26 graphs
Shows Hamiltonian Cycle NP-hard with given mim-width order
Fixes broken para-NP-hardness proof for mim-width parameter
Innovation
Methods, ideas, or system contributions that make the work stand out.
Proves NP-hardness for Hamiltonian Path
Uses linear mim-width 26 graphs
Fixes gap in para-NP-hardness proof
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