Abstract
We consider the on-line dial-a-ride problem, where a server fulfills requests that arrive over time. Each request has a source, destination, and release time. We study a variation of this problem where each request also has a revenue that the server earns for fulfilling the request. The goal is to serve requests within a time limit while maximizing the total revenue. We first prove that no deterministic online algorithm can be competitive unless the input graph is complete and edge weights are unit. We therefore focus on these graphs and present a 2-competitive algorithm for this problem. We also consider two variations of this problem: (1) the input graph is complete bipartite and (2) there is a single node that is the source for every request, and present a 1-competitive algorithm for the former and an optimal algorithm for the latter. We also provide experimental results for the complete and complete bipartite graphs. Our simulations support our theoretical findings and demonstrate that our algorithms perform well under settings that reflect realistic dial-a-ride systems.
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Notes
Note that since \(v_{last}\) can be at most the maximum allowed revenue, this proof shows non-competitiveness for b equal to all possible revenue values.
We note that there are at least two enhancements that can improve the performance of grf without improving the competitive ratio: (1) In steps 2 and 7, instead of simply moving to the request that earns the greatest revenue, the algorithm can serve a request if there is one available while performing this move. (2) In steps 3 and 8, the algorithm can check if a request with higher revenue has been released since the previous step and if so, serve this request instead of r.
If at any time t, there is no request issued, we can generate a “dummy” request of the form (s, d, t, 0), where s and d are nodes in the input graph, since neither grf nor any optimal algorithm would accept this request.
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A preliminary version of this work appeared in the proceedings of Conference of Combinatorial Optimization and Application, 2014.
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Christman, A., Forcier, W. & Poudel, A. From theory to practice: maximizing revenues for on-line dial-a-ride. J Comb Optim 35, 512–529 (2018). https://doi.org/10.1007/s10878-017-0188-z
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DOI: https://doi.org/10.1007/s10878-017-0188-z