Beyond 2-approximation for k-Center in Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917953930264576 |
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| author | Jin, Ce Kirkpatrick, Yael Williams, Virginia Vassilevska Wein, Nicole |
| author_facet | Jin, Ce Kirkpatrick, Yael Williams, Virginia Vassilevska Wein, Nicole |
| contents | We consider the classical $k$-Center problem in undirected graphs. The problem is known to have a polynomial-time 2-approximation. There are even $(2+\varepsilon)$-approximations running in near-linear time. The conventional wisdom is that the problem is closed, as $(2-\varepsilon)$-approximation is NP-hard when $k$ is part of the input, and for constant $k\geq 2$ it requires $n^{k-o(1)}$ time under SETH.
Our first set of results show that one can beat the multiplicative factor of $2$ in undirected unweighted graphs if one is willing to allow additional small additive error, obtaining $(2-\varepsilon,O(1))$ approximations. We provide several algorithms that achieve such approximations for all integers $k$ with running time $O(n^{k-δ})$ for $δ>0$. For instance, for every $k\geq 2$, we obtain an $O(mn + n^{k/2+1})$ time $(2 - \frac{1}{2k-1}, 1 - \frac{1}{2k-1})$-approximation to $k$-Center. For $2$-Center we also obtain an $\tilde{O}(mn^{ω/3})$ time $(5/3,2/3)$-approximation algorithm. Notably, the running time of this $2$-Center algorithm is faster than the time needed to compute APSP.
Our second set of results are strong fine-grained lower bounds for $k$-Center. We show that our $(3/2,O(1))$-approximation algorithm is optimal, under SETH, as any $(3/2-\varepsilon,O(1))$-approximation algorithm requires $n^{k-o(1)}$ time. We also give a time/approximation trade-off: under SETH, for any integer $t\geq 1$, $n^{k/t^2-1-o(1)}$ time is needed for any $(2-1/(2t-1),O(1))$-approximation algorithm for $k$-Center. This explains why our $(2-\varepsilon,O(1))$ approximation algorithms have $k$ appearing in the exponent of the running time. Our reductions also imply that, assuming ETH, the approximation ratio 2 of the known near-linear time algorithms cannot be improved by any algorithm whose running time is a polynomial independent of $k$, even if one allows additive error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09468 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond 2-approximation for k-Center in Graphs Jin, Ce Kirkpatrick, Yael Williams, Virginia Vassilevska Wein, Nicole Data Structures and Algorithms We consider the classical $k$-Center problem in undirected graphs. The problem is known to have a polynomial-time 2-approximation. There are even $(2+\varepsilon)$-approximations running in near-linear time. The conventional wisdom is that the problem is closed, as $(2-\varepsilon)$-approximation is NP-hard when $k$ is part of the input, and for constant $k\geq 2$ it requires $n^{k-o(1)}$ time under SETH. Our first set of results show that one can beat the multiplicative factor of $2$ in undirected unweighted graphs if one is willing to allow additional small additive error, obtaining $(2-\varepsilon,O(1))$ approximations. We provide several algorithms that achieve such approximations for all integers $k$ with running time $O(n^{k-δ})$ for $δ>0$. For instance, for every $k\geq 2$, we obtain an $O(mn + n^{k/2+1})$ time $(2 - \frac{1}{2k-1}, 1 - \frac{1}{2k-1})$-approximation to $k$-Center. For $2$-Center we also obtain an $\tilde{O}(mn^{ω/3})$ time $(5/3,2/3)$-approximation algorithm. Notably, the running time of this $2$-Center algorithm is faster than the time needed to compute APSP. Our second set of results are strong fine-grained lower bounds for $k$-Center. We show that our $(3/2,O(1))$-approximation algorithm is optimal, under SETH, as any $(3/2-\varepsilon,O(1))$-approximation algorithm requires $n^{k-o(1)}$ time. We also give a time/approximation trade-off: under SETH, for any integer $t\geq 1$, $n^{k/t^2-1-o(1)}$ time is needed for any $(2-1/(2t-1),O(1))$-approximation algorithm for $k$-Center. This explains why our $(2-\varepsilon,O(1))$ approximation algorithms have $k$ appearing in the exponent of the running time. Our reductions also imply that, assuming ETH, the approximation ratio 2 of the known near-linear time algorithms cannot be improved by any algorithm whose running time is a polynomial independent of $k$, even if one allows additive error. |
| title | Beyond 2-approximation for k-Center in Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2503.09468 |