# Serving Online Requests with Mobile Servers

## Abstract

We study an online problem in which mobile servers have to be moved in order to efficiently serve at set of online requests. More formally, there is a set of *n* nodes and a set of *k* mobile servers that are placed at some of the nodes. Each node can potentially host several servers and the servers can be moved between the nodes. There are requests \(1,2,\ldots \) that are adversarially issued at nodes one at a time, where a request issued at time *t* needs to be served at all times \(t' \ge t\). The cost for serving the requests is a function of the number of servers and requests at the different nodes. The requirements on how to serve the requests are governed by two parameters \(\alpha \ge 1\) and \(\beta \ge 0\). An algorithm needs to guarantee that at all times, the total service cost remains within a multiplicative factor \(\alpha \) and an additive term \(\beta \) of the current optimal service cost.

We consider online algorithms for two different minimization objectives. We first consider the natural problem of minimizing the total number of server movements. We show that in this case for every *k*, the competitive ratio of every deterministic online algorithm needs to be at least \(\varOmega (n)\). Given this negative result, we then extend the minimization objective to also include the current service cost. We give almost tight bounds on the competitive ratio of the online problem where one needs to minimize the sum of the total number of movements and the current service cost. In particular, we show that at the cost of an additional additive term which is roughly linear in *k*, it is possible to achieve a multiplicative competitive ratio of \(1+\varepsilon \) for every constant \(\varepsilon >0\).

## Keywords

Movement minimization Competitive analysis General cost function## References

- 1.Ahmadian, S., Friggstad, Z., Swamy, C.: Local-search based approximation algorithms for mobile facility location problems. In: Proceedings of the 24th Symposium on Discrete Algorithms (SODA), pp. 1607–1621 (2013)Google Scholar
- 2.Arora, S., Hazan, E., Kale, S.: The multiplicative weights update method: a meta-algorithm and applications. Theory Comput.
**8**(1), 121–164 (2012)MathSciNetCrossRefMATHGoogle Scholar - 3.Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k-median and facility location problems. J. Comput.
**33**(3), 544–562 (2004)MathSciNetMATHGoogle Scholar - 4.Bansal, N., Buchbinder, N., Madry, A., Naor, J.S.: A polylogarithmic-competitive algorithm for the k-server problem. In: Proceedings of the 52nd Symposium on Foundations of Computer Science (FOCS), pp. 267–276 (2011)Google Scholar
- 5.Byrka, J., Aardal, K.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. J. Comput.
**39**(6), 2212–2231 (2010)MathSciNetMATHGoogle Scholar - 6.Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. In: Proceedings of the 29th Symposium on Theory of Computing (STOC), pp. 626–635 (1997)Google Scholar
- 7.Demaine, E.D., Hajiaghayi, M., Mahini, H., Sayedi-Roshkhar, A.S., Oveisgharan, S., Zadimoghaddam, M.: Minimizing movement. Tran. Algorithms (TALG)
**5**(3), 30 (2009)MathSciNetMATHGoogle Scholar - 8.Drezner, Z., Hamacher, H.W.: Facility Location: Applications and Theory. Springer Science & Business Media, Heidelberg (2004)MATHGoogle Scholar
- 9.Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proceedings of the 35th Symposium on Theory of Computing (STOC), pp. 448–455 (2003)Google Scholar
- 10.Fotakis, D.: Online and incremental algorithms for facility location. SIGACT News
**42**(1), 97–131 (2011)CrossRefGoogle Scholar - 11.Friggstad, Z., Salavatipour, M.R.: Minimizing movement in mobile facility location problems. Trans. Algorithms (TALG)
**7**(3), 28 (2011)MathSciNetMATHGoogle Scholar - 12.Ghodselahi, A., Kuhn, F.: Serving online demands with movable centers. arXiv preprint arXiv:1404.5510 (2014)
- 13.Guha, S., Khuller, S.: Greedy strikes back: improved facility location algorithms. In: Proceedings of the 9th Symposium on Discrete Algorithms (SODA), pp. 649–657 (1998)Google Scholar
- 14.Hajiaghayi, M.T., Mahdian, M., Mirrokni, V.S.: The facility location problem with general cost functions. Networks
**42**(1), 42–47 (2003)MathSciNetCrossRefMATHGoogle Scholar - 15.Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM
**50**(6), 795–824 (2003)MathSciNetCrossRefMATHGoogle Scholar - 16.Koutsoupias, E., Papadimitriou, C.H.: On the k-server conjecture. J. ACM
**42**(5), 971–983 (1995)MathSciNetCrossRefMATHGoogle Scholar - 17.Littlestone, N., Warmuth, M.K.: The weighted majority algorithm. Inf. Comput.
**108**(2), 212–261 (1994)MathSciNetCrossRefMATHGoogle Scholar - 18.Manasse, M.S., McGeoch, L.A., Sleator, D.D.: Competitive algorithms for server problems. J. Algorithms
**11**(2), 208–230 (1990)MathSciNetCrossRefMATHGoogle Scholar - 19.Meyerson, A.: Online facility location. In: Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS), p. 426 (2001)Google Scholar
- 20.Shalev-Shwartz, S.: Online learning and online convex optimization. Found. Trends Mach. Learn.
**4**(2), 107–194 (2011)CrossRefMATHGoogle Scholar - 21.Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM
**28**(2), 202–208 (1985)MathSciNetCrossRefGoogle Scholar