Abstract
We consider the online k-taxi problem, a generalization of the k-server problem, in which k servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger.
We show that the classic Double Coverage algorithm has competitive ratio \(2^k-1\) on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive ratio turns out to be much better and we obtain tight bounds. When the depth is \(d\ll k\), these bounds are approximately \(k^d/d!\). By standard embedding results, we obtain a randomized algorithm for arbitrary n-point metrics with (polynomial) competitive ratio \(O(k^c\Delta ^{1/c}\log _{\Delta } n)\), where \(\Delta \) is the aspect ratio and \(c\ge 1\) is an arbitrary positive integer constant. The only previous known bound was \(O(2^k\log n)\). For general (weighted) tree metrics, we prove the competitive ratio of Double Coverage to be \(\Theta (k^d)\) for any fixed depth d, but unlike on HSTs it is not bounded by \(2^k-1\).
We obtain our results by a dual fitting analysis where the dual solution is constructed step-by-step backwards in time. Unlike the forward-time approach typical of online primal-dual analyses, this allows us to combine information from the past and the future when assigning dual variables. We believe this method can be useful also for other problems. Using this technique, we also provide a dual fitting proof of the k-competitiveness of Double Coverage for the k-server problem on trees.
This research was supported in part by US-Israel BSF grant 2018352, by ISF grant 2233/19 (2027511) and by NWO VICI grant 639.023.812.
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Notes
- 1.
See Sect. 2 for an exact definition of HSTs.
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- 3.
We note that our LP for the k-server problem is different from LPs used in the context of polylogarithmically-competitive randomized algorithms for the k-server problem. In our context of deterministic algorithms for k-taxi (and k-server), we show that we can work with this simpler formulation.
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Buchbinder, N., Coester, C., Naor, J.(. (2021). Online k-Taxi via Double Coverage and Time-Reverse Primal-Dual. In: Singh, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2021. Lecture Notes in Computer Science(), vol 12707. Springer, Cham. https://doi.org/10.1007/978-3-030-73879-2_2
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