1,744 research outputs found
Tight bounds for online TSP on the line
We consider the online traveling salesperson problem (TSP), where requests appear online over time on the real line and need to be visited by a server initially located at the origin. We distinguish between closed and open online TSP, depending on whether the server eventually needs to return to the origin or not. While online TSP on the line is a very natural online problem that was introduced more than two decades ago, no tight competitive analysis was known to date. We settle this problem by providing tight bounds on the competitive ratios for both the closed and the open variant of the problem. In particular, for closed online TSP, we provide a 1.64-competitive algorithm,thus matching a known lower bound. For open online TSP, we give a new upper bound as well as a matching lower bound that establish the remarkable competitive ratio of 2.04. Additionally, we consider the online Dial-A-Ride problem on the line, where each request needs to be transported to a specified destination. We provide an improved non-preemptive lower bound of 1.75 for this setting, as well as an improved preemptive algorithm with competitive ratio 2.41.Finally, we generalize known and give new complexity results for the underlying offline problems. In particular, we give an algorithm with running time O(n2) for closed offline TSP on the line with release dates and show that both variants of offline Dial-A-Ride on the line are NP-hard for any capacity c≥2 of the server
Improved Bounds for Open Online Dial-a-Ride on the Line
We consider the open, non-preemptive online Dial-a-Ride problem on the real line, where transportation requests appear over time and need to be served by a single server. We give a lower bound of 2.0585 on the competitive ratio, which is the first bound that strictly separates online Dial-a-Ride on the line from online TSP on the line in terms of competitive analysis, and is the best currently known lower bound even for general metric spaces. On the other hand, we present an algorithm that improves the best known upper bound from 2.9377 to 2.6662. The analysis of our algorithm is tight
Learning-Augmented Online TSP on Rings, Trees, Flowers and (Almost) Everywhere Else
We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line [Gouleakis et al., 2023], in which the predictions correspond to the locations of the requests and extend it to more general metric spaces.
We first propose an oracle-based algorithmic framework, inspired by previous work [Bampis et al., 2023]. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness).
Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results
Power Strip Packing of Malleable Demands in Smart Grid
We consider a problem of supplying electricity to a set of
customers in a smart-grid framework. Each customer requires a certain amount of
electrical energy which has to be supplied during the time interval . We
assume that each demand has to be supplied without interruption, with possible
duration between and , which are given system parameters (). At each moment of time, the power of the grid is the sum of all the
consumption rates for the demands being supplied at that moment. Our goal is to
find an assignment that minimizes the {\it power peak} - maximal power over
- while satisfying all the demands. To do this first we find the lower
bound of optimal power peak. We show that the problem depends on whether or not
the pair belongs to a "good" region . If it does - then
an optimal assignment almost perfectly "fills" the rectangle with being the sum of all the energy demands - thus
achieving an optimal power peak . Conversely, if do not belong to
, we identify the lower bound on the optimal value of
power peak and introduce a simple linear time algorithm that almost perfectly
arranges all the demands in a rectangle
and show that it is asymptotically optimal
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