Separating $k$-Median from the Supplier Version
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914652474048512 |
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| author | Anand, Aditya Lee, Euiwoong |
| author_facet | Anand, Aditya Lee, Euiwoong |
| contents | Given a metric space $(V, d)$ along with an integer $k$, the $k$-Median problem asks to open $k$ centers $C \subseteq V$ to minimize $\sum_{v \in V} d(v, C)$, where $d(v, C) := \min_{c \in C} d(v, c)$. While the best-known approximation ratio of $2.613$ holds for the more general supplier version where an additional set $F \subseteq V$ is given with the restriction $C \subseteq F$, the best known hardness for these two versions are $1+1/e \approx 1.36$ and $1+2/e \approx 1.73$ respectively, using the same reduction from Max $k$-Coverage. We prove the following two results separating them.
First, we show a $1.546$-parameterized approximation algorithm that runs in time $f(k) n^{O(1)}$. Since $1+2/e$ is proved to be the optimal approximation ratio for the supplier version in the parameterized setting, this result separates the original $k$-Median from the supplier version.
Next, we prove a $1.416$-hardness for polynomial-time algorithms assuming the Unique Games Conjecture. This is achieved via a new fine-grained hardness of Max-$k$-Coverage for small set sizes.
Our upper bound and lower bound are derived from almost the same expression, with the only difference coming from the well-known separation between the powers of LP and SDP on (hypergraph) vertex cover. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Separating $k$-Median from the Supplier Version Anand, Aditya Lee, Euiwoong Data Structures and Algorithms Given a metric space $(V, d)$ along with an integer $k$, the $k$-Median problem asks to open $k$ centers $C \subseteq V$ to minimize $\sum_{v \in V} d(v, C)$, where $d(v, C) := \min_{c \in C} d(v, c)$. While the best-known approximation ratio of $2.613$ holds for the more general supplier version where an additional set $F \subseteq V$ is given with the restriction $C \subseteq F$, the best known hardness for these two versions are $1+1/e \approx 1.36$ and $1+2/e \approx 1.73$ respectively, using the same reduction from Max $k$-Coverage. We prove the following two results separating them. First, we show a $1.546$-parameterized approximation algorithm that runs in time $f(k) n^{O(1)}$. Since $1+2/e$ is proved to be the optimal approximation ratio for the supplier version in the parameterized setting, this result separates the original $k$-Median from the supplier version. Next, we prove a $1.416$-hardness for polynomial-time algorithms assuming the Unique Games Conjecture. This is achieved via a new fine-grained hardness of Max-$k$-Coverage for small set sizes. Our upper bound and lower bound are derived from almost the same expression, with the only difference coming from the well-known separation between the powers of LP and SDP on (hypergraph) vertex cover. |
| title | Separating $k$-Median from the Supplier Version |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2401.13819 |