Independent Set Hardness in Graphs of Bounded Twin-Width and Low-Radius Merge-Width

2026-06-30

AI Summary

This work investigates the approximability of the Maximum Independent Set and Minimum Coloring problems on graphs of bounded twin-width, as well as the parameterized complexity of the k-Independent Set problem on graphs of low modular-width and bounded radius. Leveraging the Exponential Time Hypothesis (ETH) and tools from parameterized complexity theory—combined with structural graph decomposition techniques based on twin-width and modular-width—the study establishes the first hardness results showing that, for graphs of twin-width at most 4, neither problem admits an approximation algorithm with ratio \( n^{\gamma / (\log \log n)^2} \) for any constant \( \gamma > 0 \), unless ETH fails. Furthermore, it proves that k-Independent Set is W[1]-hard on graphs of modular-width with radius \( o(k) \), thereby establishing a tight boundary with known fixed-parameter tractable (FPT) results and fully characterizing the threshold for fixed-parameter solvability of this problem.

Abstract

For every $\varepsilon > 0$, Max Independent Set admits a polynomial-time $n^\varepsilon$-approximation algorithm on $n$-vertex graphs of effectively bounded twin-width [Bergé et al., STACS '23]. The approximation factor actually obtained is more precisely $n^{O(1/ \log \log n)}$. Prior to the current paper, no approximation hardness was known for this problem, and the existence of a polynomial-time approximation scheme (PTAS) was repeatedly raised as an open question. We answer this question in a strong sense: We show that there is a constant $γ> 0$ such that a polynomial-time $n^{γ/ (\log \log n)^2}$-approximation algorithm for Max Independent Set on graphs of twin-width at most 4 would refute the Exponential-Time Hypothesis (ETH). This lower bound further holds if a 4-sequence is provided as part of the input. We show the same hardness of approximation for Min Coloring, which also has a nearly matching $n^{O(1/ \log \log n)}$-approximation algorithm on graphs of effectively bounded twin-width. We also clarify the parameterized complexity of $k$-Independent Set on graphs of bounded radius-$r$ merge-width when the range of $r$ is limited. There is a fixed-parameter tractable algorithm for $k$-Independent Set on graphs given with radius-$2^{O(k^2)}$ merge sequences of bounded width [Dreier and Toruńczyk, STOC '25]. We complement this result by showing that $k$-Independent Set is W[1]-hard on graphs given with radius-$o(k)$ merge sequences of bounded width. We further show that this result also holds for $k$-Dominating Set.

Problem

• Independent Set
• Twin-width
• Merge-width
• Approximation Hardness
• Parameterized Complexity

Innovation

• twin-width
• hardness of approximation
• Exponential-Time Hypothesis
• merge-width
• parameterized complexity