On Randomized Memoryless Algorithms for the Weighted kk-server Problem

The weighted $k$-server problem is a generalization of the $k$-server problem in which the cost of moving a server of weight $\beta_i$ through a distance $d$ is $\beta_i\cdot d$. The weighted server problem on uniform spaces models caching where caches have different write costs. We prove tight bounds on the performance of randomized memoryless algorithms for this problem on uniform metric spaces. We prove that there is an $\alpha_k$-competitive memoryless algorithm for this problem, where $\alpha_k=\alpha_{k-1}^2+3\alpha_{k-1}+1$; $\alpha_1=1$. On the other hand we also prove that no randomized memoryless algorithm can have competitive ratio better than $\alpha_k$. To prove the upper bound of $\alpha_k$ we develop a framework to bound from above the competitive ratio of any randomized memoryless algorithm for this problem. The key technical contribution is a method for working with potential functions defined implicitly as the solution of a linear system. The result is robust in the sense that a small change in the probabilities used by the algorithm results in a small change in the upper bound on the competitive ratio. The above result has two important implications. Firstly this yields an $\alpha_k$-competitive memoryless algorithm for the weighted $k$-server problem on uniform spaces. This is the first competitive algorithm for $k>2$ which is memoryless. Secondly, this helps us prove that the Harmonic algorithm, which chooses probabilities in inverse proportion to weights, has a competitive ratio of $k\alpha_k$.Comment: Published at the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013

Metrical Service Systems with Multiple Servers

We study the problem of metrical service systems with multiple servers (MSSMS), which generalizes two well-known problems -- the $k$-server problem, and metrical service systems. The MSSMS problem is to service requests, each of which is an $l$-point subset of a metric space, using $k$ servers, with the objective of minimizing the total distance traveled by the servers. Feuerstein initiated a study of this problem by proving upper and lower bounds on the deterministic competitive ratio for uniform metric spaces. We improve Feuerstein's analysis of the upper bound and prove that his algorithm achieves a competitive ratio of $k({{k+l}\choose{l}}-1)$. In the randomized online setting, for uniform metric spaces, we give an algorithm which achieves a competitive ratio $\mathcal{O}(k^3\log l)$, beating the deterministic lower bound of ${{k+l}\choose{l}}-1$. We prove that any randomized algorithm for MSSMS on uniform metric spaces must be $\Omega(\log kl)$-competitive. We then prove an improved lower bound of ${{k+2l-1}\choose{k}}-{{k+l-1}\choose{k}}$ on the competitive ratio of any deterministic algorithm for $(k,l)$-MSSMS, on general metric spaces. In the offline setting, we give a pseudo-approximation algorithm for $(k,l)$-MSSMS on general metric spaces, which achieves an approximation ratio of $l$ using $kl$ servers. We also prove a matching hardness result, that a pseudo-approximation with less than $kl$ servers is unlikely, even for uniform metric spaces. For general metric spaces, we highlight the limitations of a few popular techniques, that have been used in algorithm design for the $k$-server problem and metrical service systems.Comment: 18 pages; accepted for publication at COCOON 201

Simple Pricing Schemes for the Cloud

The problem of pricing the cloud has attracted much recent attention due to the widespread use of cloud computing and cloud services. From a theoretical perspective, several mechanisms that provide strong efficiency or fairness guarantees and desirable incentive properties have been designed. However, these mechanisms often rely on a rigid model, with several parameters needing to be precisely known in order for the guarantees to hold. In this paper, we consider a stochastic model and show that it is possible to obtain good welfare and revenue guarantees with simple mechanisms that do not make use of the information on some of these parameters. In particular, we prove that a mechanism that sets the same price per time step for jobs of any length achieves at least 50% of the welfare and revenue obtained by a mechanism that can set different prices for jobs of different lengths, and the ratio can be improved if we have more specific knowledge of some parameters. Similarly, a mechanism that sets the same price for all servers even though the servers may receive different kinds of jobs can provide a reasonable welfare and revenue approximation compared to a mechanism that is allowed to set different prices for different servers.Comment: To appear in the 13th Conference on Web and Internet Economics (WINE), 2017. A preliminary version was presented at the 12th Workshop on the Economics of Networks, Systems and Computation (NetEcon), 201