In the Convex Body Chasing problem, we are given an initial point @math in @math and an online sequence of @math convex bodies @math . When we receive @math , we are required to move inside @math . Our goal is to minimize the total distance travelled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an @math lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: @math . The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give the first @math -competitive algorithm for chasing nested convex bodies in @math .
We consider three related online problems: Online Convex Optimization, Convex Body Chasing, and Lazy Convex Body Chasing. In Online Convex Optimization the input is an online sequence of convex functions over some Euclidean space. In response to a function, the online algorithm can move to any destination point in the Euclidean space. The cost is the total distance moved plus the sum of the function costs at the destination points. Lazy Convex Body Chasing is a special case of Online Convex Optimization where the function is zero in some convex region, and grows linearly with the distance from this region. And Convex Body Chasing is a special case of Lazy Convex Body Chasing where the destination point has to be in the convex region. We show that these problems are equivalent in the sense that if any of these problems have an O(1)-competitive algorithm then all of the problems have an O(1)-competitive algorithm. By leveraging these results we then obtain the first O(1)-competitive algorithm for Online Convex Optimization in two dimensions, and give the first O(1)-competitive algorithm for chasing linear subspaces. We also give a simple algorithm and O(1)-competitiveness analysis for chasing lines. A player moving in the plane is given a sequence of instructions of the following type: at stepi a planar convex setFi is specified, and the player has to move to a point inFi. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequenceFi the cost to the player is within a constant (multiplicative) factor of the "off-line" cost (i.e., the least possible cost when allFi are known in advance). We conjecture that similar strategies can be developed for this game in any Euclidean space and perhaps even in all metric spaces. The analogous statement where convex sets are replaced by more general families of sets in a metric space includes many on-line off-line problems such as thek-server problem; we make some remarks on these more general problems. 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, @math , is constant competitive for the generalized 2-server problem. Further, we give an outline for a possible extension to @math servers and discuss the applicability of our techniques and of the work function algorithm in general. We co...