Breaching the 2 LMP Approximation Barrier for Facility Location with Applications to k-Median
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908793573474304 |
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| author | Cohen-Addad, Vincent Grandoni, Fabrizio Lee, Euiwoong Schwiegelshohn, Chris |
| author_facet | Cohen-Addad, Vincent Grandoni, Fabrizio Lee, Euiwoong Schwiegelshohn, Chris |
| contents | The Uncapacitated Facility Location (UFL) problem is one of the most fundamental clustering problems: Given a set of clients $C$ and a set of facilities $F$ in a metric space $(C \cup F, dist)$ with facility costs $open : F \to \mathbb{R}^+$, the goal is to find a set of facilities $S \subseteq F$ to minimize the sum of the opening cost $open(S)$ and the connection cost $d(S) := \sum_{p \in C} \min_{c \in S} dist(p, c)$. An algorithm for UFL is called a Lagrangian Multiplier Preserving (LMP) $α$ approximation if it outputs a solution $S\subseteq F$ satisfying $open(S) + d(S) \leq open(S^*) + αd(S^*)$ for any $S^* \subseteq F$. The best-known LMP approximation ratio for UFL is at most $2$ by the JMS algorithm of Jain, Mahdian, and Saberi based on the Dual-Fitting technique.
We present a (slightly) improved LMP approximation algorithm for UFL. This is achieved by combining the Dual-Fitting technique with Local Search, another popular technique to address clustering problems. From a conceptual viewpoint, our result gives a theoretical evidence that local search can be enhanced so as to avoid bad local optima by choosing the initial feasible solution with LP-based techniques. Using the framework of bipoint solutions, our result directly implies a (slightly) improved approximation for the $k$-Median problem from 2.6742 to 2.67059. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_05150 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Breaching the 2 LMP Approximation Barrier for Facility Location with Applications to k-Median Cohen-Addad, Vincent Grandoni, Fabrizio Lee, Euiwoong Schwiegelshohn, Chris Data Structures and Algorithms The Uncapacitated Facility Location (UFL) problem is one of the most fundamental clustering problems: Given a set of clients $C$ and a set of facilities $F$ in a metric space $(C \cup F, dist)$ with facility costs $open : F \to \mathbb{R}^+$, the goal is to find a set of facilities $S \subseteq F$ to minimize the sum of the opening cost $open(S)$ and the connection cost $d(S) := \sum_{p \in C} \min_{c \in S} dist(p, c)$. An algorithm for UFL is called a Lagrangian Multiplier Preserving (LMP) $α$ approximation if it outputs a solution $S\subseteq F$ satisfying $open(S) + d(S) \leq open(S^*) + αd(S^*)$ for any $S^* \subseteq F$. The best-known LMP approximation ratio for UFL is at most $2$ by the JMS algorithm of Jain, Mahdian, and Saberi based on the Dual-Fitting technique. We present a (slightly) improved LMP approximation algorithm for UFL. This is achieved by combining the Dual-Fitting technique with Local Search, another popular technique to address clustering problems. From a conceptual viewpoint, our result gives a theoretical evidence that local search can be enhanced so as to avoid bad local optima by choosing the initial feasible solution with LP-based techniques. Using the framework of bipoint solutions, our result directly implies a (slightly) improved approximation for the $k$-Median problem from 2.6742 to 2.67059. |
| title | Breaching the 2 LMP Approximation Barrier for Facility Location with Applications to k-Median |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2207.05150 |