The Generalized Work Function Algorithm Is Competitive for the Generalized 2-Server Problem

The generalized 2-server problem is an online optimization problem where a sequence of requests has to be served at minimal cost. Requests arrive one by one and need to be served instantly by at least one of two servers. We consider the general model where the cost function of the two servers may be different. Formally, each server moves in its own metric space and a request consists of one point in each metric space. It is served by moving one of the two servers to its request point. Requests have to be served without knowledge of future requests. The objective is to minimize the total traveled distance. The special case where both servers move on the real line is known as the CNN problem. We show that the generalized work function algorithm, $\mathrm{WFA}_{\lambda}$, is constant competitive for the generalized 2-server problem. Further, we give an outline for a possible extension to $k\geqslant2$ servers and discuss the applicability of our techniques and of the work function algorithm in general. We conclude with a discussion on several open problems in online optimization

The generalized work function algorithm is competitive for the generalized 2-server problem

The generalized 2-server problem is an online optimization problem where a sequence of requests has to be served at minimal cost. Requests arrive one by one and need to be served instantly by at least one of two servers. We consider the general model where the cost function of the two servers may be different. Formally, each server moves in its own metric space and a request consists of one point in each metric space. It is served by moving one of the two servers to its request point. Requests have to be served without knowledge of the future requests. The objective is to minimize the total traveled distance. The special case where both servers move on the real line is known as the CNN-problem. We show that the generalized work function algorithm is constant competitive for the generalized 2-server problem

Weighted k-Server Bounds via Combinatorial Dichotomies

The weighted $k$-server problem is a natural generalization of the $k$-server problem where each server has a different weight. We consider the problem on uniform metrics, which corresponds to a natural generalization of paging. Our main result is a doubly exponential lower bound on the competitive ratio of any deterministic online algorithm, that essentially matches the known upper bounds for the problem and closes a large and long-standing gap. The lower bound is based on relating the weighted $k$-server problem to a certain combinatorial problem and proving a Ramsey-theoretic lower bound for it. This combinatorial connection also reveals several structural properties of low cost feasible solutions to serve a sequence of requests. We use this to show that the generalized Work Function Algorithm achieves an almost optimum competitive ratio, and to obtain new refined upper bounds on the competitive ratio for the case of $d$ different weight classes.Comment: accepted to FOCS'1

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

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

On the Continuous CNN Problem

In the (discrete) CNN problem, online requests appear as points in $\mathbb{R}^2$. Each request must be served before the next one is revealed. We have a server that can serve a request simply by aligning either its $x$ or $y$ coordinate with the request. The goal of the online algorithm is to minimize the total $L_1$ distance traveled by the server to serve all the requests. The best known competitive ratio for the discrete version is 879 (due to Sitters and Stougie). We study the continuous version, in which, the request can move continuously in $\mathbb{R}^2$ and the server must continuously serve the request. A simple adversarial argument shows that the lower bound on the competitive ratio of any online algorithm for the continuous CNN problem is 3. Our main contribution is an online algorithm with competitive ratio $3+2 \sqrt{3} \approx 6.464$. Our analysis is tight. The continuous version generalizes the discrete orthogonal CNN problem, in which every request must be $x$ or $y$ aligned with the previous request. Therefore, Our result improves upon the previous best competitive ratio of 9 (due to Iwama and Yonezawa)