# Stochastic k-Server: How Should Uber Work?

## TL;DR

This paper develops approximation algorithms for a stochastic variant of the k-server problem, including the Uber problem, providing near-optimal solutions in various metric spaces with applications to correlated request arrivals.

## Contribution

It introduces a structural characterization and approximation algorithms for the stochastic k-server problem, extending to the Uber problem and correlated request settings.

## Key findings

- Achieves a 3-approximation for the stochastic k-server problem in line and circle metrics.
- Provides an O(log n) approximation in general metrics.
- Extends results to correlated request arrivals.

## Abstract

In this paper, we study a stochastic variant of the celebrated k-server problem. In the k-server problem, we are required to minimize the total movement of k servers that are serving an online sequence of t requests in a metric. In the stochastic setting we are given t independent distributions <P_1, P_2,..., P_t> in advance, and at every time step i a request is drawn from Pi. Designing the optimal online algorithm in such setting is NP-hard, therefore the emphasis of our work is on designing an approximately optimal online algorithm. We first show a structural characterization for a certain class of non-adaptive online algorithms. We prove that in general metrics, the best of such algorithms has a cost of no worse than three times that of the optimal online algorithm. Next, we present an integer program that finds the optimal algorithm of this class for any arbitrary metric. Finally, by rounding the solution of the linear relaxation of this program, we present an online algorithm for the stochastic k-server problem with the approximation factor of 3 in the line and circle metrics and O(log n) in a general metric of size n. Moreover, we define the Uber problem, in which each demand consists of two endpoints, a source and a destination. We show that given an a-approximation algorithm for the k-server problem, we can obtain an (a+2)-approximation algorithm for the Uber problem. Motivated by the fact that demands are usually highly correlated with the time we study the stochastic Uber problem. Furthermore, we extend our results to the correlated setting where the probability of a request arriving at a certain point depends not only on the time step but also on the previously arrived requests.