Hamiltonicity Parameterized by Mim-Width is (Indeed) Para-NP-Hard

TL;DR

The paper resolves the para-NP-hardness of Hamiltonian Path and Hamiltonian Cycle parameterized by mim-width, showing hardness for graphs of linear mim-width $26$ even when a witnessing linear order is provided. The authors present a gadget-based reduction from a $3$-CNF formula, introducing variable gadgets, clause gadgets, and gamma_k structures, together with dummy edges to tightly bound mim-width. They first achieve a graph with mim-width at most $25$ and then augment it with a degree-two vertex to reach mim-width $26$, preserving the equivalence between satisfiability and the existence of a Hamiltonian path or cycle. This establishes the para-NP-hardness result and clarifies the boundary between tractability and intractability for Hamiltonicity with respect to mim-width, while leaving open precise thresholds near mim-width $1$–$2$ for related problems.

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.

Figures (6)

• Figure 1: Counterexample to the construction from PP08. The three dots symbolize adjacency to all vertices to the left on the middle layer.
• Figure 2: Illustration of the variable gadget. Bottom left (resp., right): a traversal of the sequence corresponding to setting the variable to false (resp., true).
• Figure 3: The clause gadgets due to Cygan, Kratsch, and Nederlof CKN18. On the left, a gadget such that each Hamiltonian path that enters it via an edge labeled $a$ (resp., $b$) must immediately collect all vertices in the gadget and leave via the other edge labeled $a$ (resp, $b$). This gadget can be used for clauses of size two. On the right, a gadget for a clause of size three, with three entry-exit pairs and the analogous functionality.
• Figure 4: Overview of the construction. This is the example for the formula $(x_1 \lor x_2 \lor x_3) \land (x_1 \lor \overline{x_3}) \land (\overline{x_1} \lor x_3 \lor x_4)$. The dummy edges are depicted in blue.
• Figure 5: Example of the Hamiltonian path---of the graph presented in \ref{['fig:construction']}---constructed in the proof of \ref{['lem:cor-ass-to-path']} from the satisfying assignment $\alpha$ with $\alpha(x_1)=\alpha(x_3)= 0$ and $\alpha(x_2)=\alpha(x_4)=1$. We omit the dummy edges to improve the legibility.

Theorems & Definitions (13)

• Theorem 1
• Lemma 1: Cygan, Kratsch, and Nederlof CKN18
• Lemma 2
• proof
• Definition 1
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof