Dynamic $((1+ε)\ln n)$-Approximation Algorithms for Minimum Set Cover and Dominating Set
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866911743450546176 |
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| author | Solomon, Shay Uzrad, Amitai |
| author_facet | Solomon, Shay Uzrad, Amitai |
| contents | The minimum set cover (MSC) problem admits two classic algorithms: a greedy $\ln n$-approximation and a primal-dual $f$-approximation, where $n$ is the universe size and $f$ is the maximum frequency of an element. Both algorithms are simple and efficient, and remarkably -- one cannot improve these approximations under hardness results by more than a factor of $(1+ε)$, for any constant $ε> 0$.
In their pioneering work, Gupta et al. [STOC'17] showed that the greedy algorithm can be dynamized to achieve $O(\log n)$-approximation with update time $O(f \log n)$. Building on this result, Hjuler et al. [STACS'18] dynamized the greedy minimum dominating set (MDS) algorithm, achieving a similar approximation with update time $O(Δ\log n)$ (the analog of $O(f \log n)$), albeit for unweighted instances. The approximations of both algorithms, which are the state-of-the-art, exceed the static $\ln n$-approximation by a rather large constant factor. In sharp contrast, the current best dynamic primal-dual MSC algorithms achieve fast update times together with an approximation that exceeds the static $f$-approximation by a factor of (at most) $1+ε$, for any $ε> 0$.
This paper aims to bridge the gap between the best approximation factor of the dynamic greedy MSC and MDS algorithms and the static $\ln n$ bound. We present dynamic algorithms for weighted greedy MSC and MDS with approximation $(1+ε)\ln n$ for any $ε> 0$, while achieving the same update time (ignoring dependencies on $ε$) of the best previous algorithms (with approximation significantly larger than $\ln n$). Moreover, [...] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17625 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamic $((1+ε)\ln n)$-Approximation Algorithms for Minimum Set Cover and Dominating Set Solomon, Shay Uzrad, Amitai Data Structures and Algorithms The minimum set cover (MSC) problem admits two classic algorithms: a greedy $\ln n$-approximation and a primal-dual $f$-approximation, where $n$ is the universe size and $f$ is the maximum frequency of an element. Both algorithms are simple and efficient, and remarkably -- one cannot improve these approximations under hardness results by more than a factor of $(1+ε)$, for any constant $ε> 0$. In their pioneering work, Gupta et al. [STOC'17] showed that the greedy algorithm can be dynamized to achieve $O(\log n)$-approximation with update time $O(f \log n)$. Building on this result, Hjuler et al. [STACS'18] dynamized the greedy minimum dominating set (MDS) algorithm, achieving a similar approximation with update time $O(Δ\log n)$ (the analog of $O(f \log n)$), albeit for unweighted instances. The approximations of both algorithms, which are the state-of-the-art, exceed the static $\ln n$-approximation by a rather large constant factor. In sharp contrast, the current best dynamic primal-dual MSC algorithms achieve fast update times together with an approximation that exceeds the static $f$-approximation by a factor of (at most) $1+ε$, for any $ε> 0$. This paper aims to bridge the gap between the best approximation factor of the dynamic greedy MSC and MDS algorithms and the static $\ln n$ bound. We present dynamic algorithms for weighted greedy MSC and MDS with approximation $(1+ε)\ln n$ for any $ε> 0$, while achieving the same update time (ignoring dependencies on $ε$) of the best previous algorithms (with approximation significantly larger than $\ln n$). Moreover, [...] |
| title | Dynamic $((1+ε)\ln n)$-Approximation Algorithms for Minimum Set Cover and Dominating Set |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2312.17625 |