The (h,k)(h,k)-Server Problem on Bounded Depth Trees

We study the $k$-server problem in the resource augmentation setting i.e., when the performance of the online algorithm with $k$ servers is compared to the offline optimal solution with $h \leq k$ servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic $k$-server algorithms are roughly $(1+1/\epsilon)$-competitive when $k=(1+\epsilon) h$, for any $\epsilon >0$. Surprisingly however, no $o(h)$-competitive algorithm is known even for HSTs of depth 2 and even when $k/h$ is arbitrarily large. We obtain several new results for the problem. First we show that the known $k$-server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio $\Omega(h)$ irrespective of the value of $k$, even for depth-2 HSTs. Similarly the Work Function Algorithm, that is believed to be optimal for all metric spaces when $k=h$, has competitive ratio $\Omega(h)$ on depth-3 HSTs even if $k=2h$. Our main result is a new algorithm that is $O(1)$-competitive for constant depth trees, whenever $k =(1+\epsilon )h$ for any $\epsilon > 0$. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when $k/h$ is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the $(h,k)$-server problem, and gives the strongest known lower bound for the problem on general metrics.Comment: Appeared in SODA 201

The Adwords Problem with Strict Capacity Constraints

We study an online assignment problem where the offline servers have capacities, and the objective is to obtain a maximum-weight assignment of requests that arrive online. The weight of edges incident to any server can be at most the server capacity. Our problem is related to the adwords problem, where the assignment to a server is allowed to exceed its capacity. In many applications, however, server capacities are strict and partially-served requests are of no use, motivating the problem we study. While no deterministic algorithm can be competitive in general for this problem, we give an algorithm with competitive ratio that depends on the ratio of maximum weight of any edge to the capacity of the server it is incident to. If this ratio is 1/2, our algorithm is tight. Further, we give a randomized algorithm that is 6-competitive in expectation for the general problem. Most previous work on the problem and its variants assumes that the edge weights are much smaller than server capacities. Our guarantee, in contrast, does not require any assumptions about job weights. We also give improved lower bounds for both deterministic and randomized algorithms. For the special case of parallel servers, we show that a load-balancing algorithm is tight and near-optimal

The Infinite Server Problem

We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which implies the same lower bound for the (h,k)-server problem even when k>>h and holds also for the line metric; the previous known bounds were 2.4 for general metric spaces and 2 for the line. For weighted trees and layered graphs we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval away from the original position of the servers. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case

Competitive algorithms for generalized k-server in uniform metrics

The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server si lies in its own metric space Mi. A request is a k-tuple r = (r1, r2, … , rk), which is served by moving some server si to the point ri ∈ Mi, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines. Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio k · 2k and O(k3log k) respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of 2k − 1. We also give a (2^{2^{O(k)}} ) -competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem

Algorithms for the on-line travelling salesman

In this paper the problem of efficiently serving a sequence of requests presented in an on-line fashion located at points of a metric space is considered. We call this problem the On-Line Travelling Salesman Problem (OLTSP). It has a variety of relevant applications in logistics and robotics. We consider two versions of the problem. In the first one the server is not required to return to the departure point after all presented requests have been served. For this problem we derive a lower bound on the competitive ratio of 2 on the real line. Besides, a 2.5-competitive algorithm for a wide class of metric spaces, and a 7/3-competitive algorithm for the real line are provided. For the other version of the problem, in which returning to the departure point is required, we present an optimal 2-competitive algorithm for the above mentioned general class of metric spaces. If in this case the metric space is the real line we present a 1.75-competitive algorithm that compares with a \approx 1.64 lower bound

The infinite server problem

We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the resource augmentation version of the k-server problem, also known as the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which holds even for the line and some simple weighted stars. It implies the same lower bound for the (h,k)-server problem on the line, even when k/h → ∞, improving on the previous known bounds of 2 for the line and 2.4 for general metrics. For weighted trees and layered graphs, we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case

Competitive Algorithms for Generalized k-Server in Uniform Metrics

The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server $s_i$ lies in its own metric space $M_i$. A request is a k-tuple $r = (r_1,r_2,\dotsc,r_k)$ and to serve it, we need to move some server $s_i$ to the point $r_i \in M_i$, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines. Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio $O(k 2^k)$ and $O(k^3 \log k)$ respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of $2^k-1$. We also give a $2^{2^{O(k)}}$-competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem

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

Robust Online Speed Scaling With Deadline Uncertainty

A speed scaling problem is considered, where time is divided into slots, and jobs with payoff v arrive at the beginning of the slot with associated deadlines d. Each job takes one slot to be processed, and multiple jobs can be processed by the server in each slot with energy cost g(k) for processing k jobs in one slot. The payoff is accrued by the algorithm only if the job is processed by its deadline. We consider a robust version of this speed scaling problem, where a job on its arrival reveals its payoff v, however, the deadline is hidden to the online algorithm, which could potentially be chosen adversarially and known to the optimal offline algorithm. The objective is to derive a robust (to deadlines) and optimal online algorithm that achieves the best competitive ratio. We propose an algorithm (called min-LCR) and show that it is an optimal online algorithm for any convex energy cost function g(.). We do so without actually evaluating the optimal competitive ratio, and give a general proof that works for any convex g, which is rather novel. For the popular choice of energy cost function g(k) = k^alpha, alpha >= 2, we give concrete bounds on the competitive ratio of the algorithm, which ranges between 2.618 and 3 depending on the value of alpha. The best known online algorithm for the same problem, but where deadlines are revealed to the online algorithm has competitive ratio of 2 and a lower bound of sqrt{2}. Thus, importantly, lack of deadline knowledge does not make the problem degenerate, and the effect of deadline information on the optimal competitive ratio is limited