## Online Facility Location with Predictions

### Shaofeng Jiang · Erzhi Liu · You Lyu · Zhihao Tang · Yubo Zhang

Keywords: [ prediction ]

[ Abstract ]
Wed 27 Apr 2:30 a.m. PDT — 4:30 a.m. PDT

Abstract: We provide nearly optimal algorithms for online facility location (OFL) with predictions. In OFL, $n$ demand points arrive in order and the algorithm must irrevocably assign each demand point to an open facility upon its arrival. The objective is to minimize the total connection costs from demand points to assigned facilities plus the facility opening cost. We further assume the algorithm is additionally given for each demand point $x_i$ a natural prediction $f_{x_i}^{\mathrm{pred}}$ which is supposed to be the facility $f_{x_i}^{\mathrm{opt}}$ that serves $x_i$ in the offline optimal solution.Our main result is an $O(\min\{\log {\frac{n\eta_\infty}{\mathrm{OPT}}}, \log{n} \})$-competitive algorithm where $\eta_\infty$ is the maximum prediction error (i.e., the distance between $f_{x_i}^{\mathrm{pred}}$ and $f_{x_i}^{\mathrm{opt}}$). Our algorithm overcomes the fundamental $\Omega(\frac{\log n}{\log \log n})$ lower bound of OFL (without predictions) when $\eta_\infty$ is small, and it still maintains $O(\log n)$ ratio even when $\eta_\infty$ is unbounded. Furthermore, our theoretical analysis is supported by empirical evaluations for the tradeoffs between $\eta_\infty$ and the competitive ratio on various real datasets of different types.

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