We consider an online facility location problem where clients arrive over time and their demands have to be served by opening facilities and assigning the clients to opened facilities. When opening a facility we must choose one of K different lease types to use. A lease type k has a certain lease length lk. Opening a facility i using lease type k causes a cost of $f_i^k$ and ensures that i is open for the next lk time steps. In addition to costs for opening facilities, we have to take connection costs cij into account when assigning a client j to facility i. We develop and analyze the first online algorithm for this problem that has a time-independent competitive factor.
This variant of the online facility location problem was introduced by [7] and is strongly related to both the online facility problem by [5] and the parking permit problem by [6]. Nagarajan and Williamson gave a 3-approximation algorithm for the offline problem and an O (Klogn)-competitive algorithm for the online variant. Here, n denotes the total number of clients arriving over time. We extend their result by removing the dependency on n (and thereby on the time). In general, our algorithm is $O (\ensuremath{l_{\text{max}}} \log(\ensuremath{l_{\text{max}}}))$-competitive. Here $\ensuremath{l_{\text{max}}} $ denotes the maximum lease length. Moreover, we prove that it is $O (\log^2(\ensuremath{l_{\text{max}}}))$-competitive for many "natural" cases. Such cases include, for example, situations where the number of clients arriving in each time step does not vary too much, or is non-increasing, or is polynomially bounded in $l_{\text{max}}$.
[1]
Dimitris Fotakis.
A primal-dual algorithm for online non-uniform facility location
,
2007,
J. Discrete Algorithms.
[2]
Adam Meyerson,et al.
Online facility location
,
2001,
Proceedings 2001 IEEE International Conference on Cluster Computing.
[3]
Sriram V. Pemmaraju,et al.
Rapid randomized pruning for fast greedy distributed algorithms
,
2010,
PODC.
[4]
Vijay V. Vazirani,et al.
Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation
,
2001,
JACM.
[5]
Guy E. Blelloch,et al.
Parallel approximation algorithms for facility-location problems
,
2010,
SPAA '10.
[6]
Barbara M. Anthony,et al.
Infrastructure Leasing Problems
,
2007,
IPCO.
[7]
Adam Meyerson.
The parking permit problem
,
2005,
46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).
[8]
David P. Williamson,et al.
Offline and Online Facility Leasing
,
2008,
IPCO.