A Randomized $$\mathrm {O}(\log n)$$ O ( log n ) -Competitive Algorithm for the Online Connected Facility Location Problem

Abstract

The Connected Facility Location (CFL) is a network design problem that arises from a combination of the Uncapacitated Facility Location (FL) and the Steiner Tree (ST) problems. The Online Connected Facility Location problem (OCFL) is an online version of the CFL. San Felice et al. (2014) presented a randomized algorithm for the OCFL and proved that it is $$\mathrm {O}(\log ^2 n)$$O(log2n)-competitive, where n is the number of clients. That algorithm combines the sample-and-augment framework of Gupta, Kumar, Pal, and Roughgarden with previous algorithms for the Online Facility Location (OFL) and the Online Steiner Tree (OST) problems. In this paper we use a more precise analysis to show that the same algorithm is $$\mathrm {O}(\log n)$$O(logn)-competitive. Since there is a lower bound of $$\mathrm {\Omega }(\log n)$$Ω(logn) for this problem, our result achieves the best possible competitive ratio, asymptotically.