Tight Inapproximability of Target Set Reconfiguration
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929254371950592 |
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| author | Ohsaka, Naoto |
| author_facet | Ohsaka, Naoto |
| contents | Given a graph $G$ with a vertex threshold function $τ$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $τ(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and its two target sets $X$ and $Y$, we wish to transform $X$ into $Y$ by repeatedly adding or removing a single vertex, using only target sets of $G$, so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of $2-o\left(\frac{1}{\operatorname{polylog} n}\right)$, where $n$ is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a $2$-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15076 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tight Inapproximability of Target Set Reconfiguration Ohsaka, Naoto Data Structures and Algorithms Computational Complexity Discrete Mathematics Given a graph $G$ with a vertex threshold function $τ$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $τ(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and its two target sets $X$ and $Y$, we wish to transform $X$ into $Y$ by repeatedly adding or removing a single vertex, using only target sets of $G$, so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of $2-o\left(\frac{1}{\operatorname{polylog} n}\right)$, where $n$ is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a $2$-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016). |
| title | Tight Inapproximability of Target Set Reconfiguration |
| topic | Data Structures and Algorithms Computational Complexity Discrete Mathematics |
| url | https://arxiv.org/abs/2402.15076 |