Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915218188140544 |
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| author | Hirahara, Shuichi Ohsaka, Naoto |
| author_facet | Hirahara, Shuichi Ohsaka, Naoto |
| contents | $k$-Coloring Reconfiguration is one of the most well-studied reconfiguration problems, which asks to transform a given proper $k$-coloring of a graph to another by repeatedly recoloring a single vertex. Its approximate version, Maxmin $k$-Cut Reconfiguration, is defined as an optimization problem of maximizing the minimum fraction of bichromatic edges during the transformation between (not necessarily proper) $k$-colorings. In this paper, we prove that the optimal approximation factor of this problem is $1 - Θ\left(\frac{1}{k}\right)$ for every $k \ge 2$. Specifically, we show the $\mathsf{PSPACE}$-hardness of approximating the objective value within a factor of $1 - \frac{\varepsilon}{k}$ for some universal constant $\varepsilon > 0$, whereas we present a deterministic polynomial-time algorithm that achieves the approximation factor of $1 - \frac{2}{k}$.
To prove the hardness result, we develop a new probabilistic verifier that tests a ``striped'' pattern. Our polynomial-time algorithm is based on ``a random reconfiguration via a random solution,'' i.e., the transformation that goes through one random $k$-coloring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration Hirahara, Shuichi Ohsaka, Naoto Computational Complexity Discrete Mathematics Data Structures and Algorithms $k$-Coloring Reconfiguration is one of the most well-studied reconfiguration problems, which asks to transform a given proper $k$-coloring of a graph to another by repeatedly recoloring a single vertex. Its approximate version, Maxmin $k$-Cut Reconfiguration, is defined as an optimization problem of maximizing the minimum fraction of bichromatic edges during the transformation between (not necessarily proper) $k$-colorings. In this paper, we prove that the optimal approximation factor of this problem is $1 - Θ\left(\frac{1}{k}\right)$ for every $k \ge 2$. Specifically, we show the $\mathsf{PSPACE}$-hardness of approximating the objective value within a factor of $1 - \frac{\varepsilon}{k}$ for some universal constant $\varepsilon > 0$, whereas we present a deterministic polynomial-time algorithm that achieves the approximation factor of $1 - \frac{2}{k}$. To prove the hardness result, we develop a new probabilistic verifier that tests a ``striped'' pattern. Our polynomial-time algorithm is based on ``a random reconfiguration via a random solution,'' i.e., the transformation that goes through one random $k$-coloring. |
| title | Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration |
| topic | Computational Complexity Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2410.03416 |