Online k-taxi via double coverage and time-reverse primal-dual

We consider the online k-taxi problem, a generalization of the k-server problem, in which k servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger. We show that the classic Double Coverage algorithm has competitive ratio 2 k- 1 on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive ratio turns out to be much better and we obtain tight bounds. When the depth is d≪ k, these bounds are approximately kd/ d!. By standard embedding results, we obtain a randomized algorithm for arbitrary n-point metrics with (polynomial) competitive ratio O(kcΔ 1/clog Δn), where Δ is the aspect ratio and c≥ 1 is an arbitrary positive integer constant. The only previous known bound was O(2 klog n). For general (weighted) tree metrics, we prove the competitive ratio of Double Coverage to be Θ (kd) for any fixed depth d, but unlike on HSTs it is not bounded by 2 k- 1. We obtain our results by a dual fitting analysis where the dual solution is constructed step-by-step backwards in time. Unlike the forward-time approach typical of online primal-dual analyses, this allows us to combine information from the past and the future when assigning dual variables. We believe this method can be useful also for other problems. Using this technique, we also provide a dual fitting proof of the k-competitiveness of Double Coverage for the k-server problem on trees

Online k-taxi via Double Coverage and time-reverse primal-dual

We consider the online k-taxi problem, a generalization of the k-server problem, in which k servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger. We show that the classic Double Coverage algorithm has competitive ratio 2k−1 on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive ratio turns out to be much better and we obtain tight bounds. When the depth is d≪k, these bounds are approximately kd/d! . By standard embedding results, we obtain a randomized algorithm for arbitrary n-point metrics with (polynomial) competitive ratio O(kcΔ1/clogΔn), where Δ is the aspect ratio and c≥1 is an arbitrary positive integer constant. The previous known bound was O(2klogn). For general (weighted) tree metrics, we prove the competitive ratio of Double Coverage to be Θ(kd) for any fixed depth d, and in contrast to HSTs it is not bounded by 2k−1. We obtain our results by a dual fitting analysis where the dual solution is constructed step-by-step backwards in time. Unlike the forward-time approach typical of online primal-dual analyses, this allows us to combine information from both the past and the future when assigning dual variables. We believe this method can also be useful for other problems. Using this technique, we also provide a dual fitting proof of the k-competitiveness of Double Coverage for the k-server problem on trees

Online Service with Delay

In this paper, we introduce the online service with delay problem. In this problem, there are $n$ points in a metric space that issue service requests over time, and a server that serves these requests. The goal is to minimize the sum of distance traveled by the server and the total delay in serving the requests. This problem models the fundamental tradeoff between batching requests to improve locality and reducing delay to improve response time, that has many applications in operations management, operating systems, logistics, supply chain management, and scheduling. Our main result is to show a poly-logarithmic competitive ratio for the online service with delay problem. This result is obtained by an algorithm that we call the preemptive service algorithm. The salient feature of this algorithm is a process called preemptive service, which uses a novel combination of (recursive) time forwarding and spatial exploration on a metric space. We hope this technique will be useful for related problems such as reordering buffer management, online TSP, vehicle routing, etc. We also generalize our results to $k > 1$ servers.Comment: 30 pages, 11 figures, Appeared in 49th ACM Symposium on Theory of Computing (STOC), 201

Online Metric Allocation and Time-Varying Regularization

We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let be an -point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of . At each time , a convex monotone cost function : [0, 1] → ℝ+ appears at some point ∈ . In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost ( ), where is the fraction of the resource at at the end of time . For example, when the cost functions are () = , this is equivalent to randomized MTS, and when the cost functions are () = ∞·<1/, this is equivalent to fractional -server. Because of an inherent scale-freeness property of the problem, existing techniques for MTS and -server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an (log)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm. We also consider the case of non-convex cost functions. Using a simple ₂²-regularizer, we give tight bounds of Θ() on tree metrics, which imply deterministic and randomized competitive ratios of (2) and ( log ) respectively on arbitrary metrics