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Register a new userOnline Matching in a Ride-Sharing Platform
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Added by
Chinmoy Dutta
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We propose a formal graph-theoretic model for studying the problem of matching rides online in a ride-sharing platform. Unlike most of the literature on online matching, our model, that we call {em Online Windowed Non-Bipartite Matching} ($mbox{OWNBM}$), pertains to online matching in {em non-bipartite} graphs. We show that the edge-weighted and vertex-weight
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The unmanned aerial vehicle (UAV) has emerged as a promising solution to provide delivery and other mobile services to customers rapidly, yet it drains its stored energy quickly when travelling on the way and (even if solar-powered) it takes time for charging power on the way before reaching the destination. To address this issue, existing works focus more on UAVs path planning with designated system vehicles providing charging service. However, in some emergency cases and rural areas where system vehicles are not available, public trucks can provide more feasible and cost-saving services and hence a silver lining. In this paper, we explore how a single UAV can save flying distance by exploiting public trucks, to minimize the travel time of the UAV. We give the first theoretical work studying online algorithms for the problem, which guarantees a worst-case performance. We first consider the offline problem knowing future truck trip information far ahead of time. By delicately transforming the problem into a graph satisfying both time and power constraints, we present a shortest-path algorithm that outputs the optimal solution of the problem. Then, we proceed to the online setting where trucks appear in real-time and only inform the UAV of their trip information some certain time $Delta t$ beforehand. As a benchmark, we propose a well-constructed lower bound that an online algorithm could achieve. We propose an online algorithm MyopicHitching that greedily takes truck trips and an improved algorithm $Delta t$-Adaptive that further tolerates a waiting time in taking a ride. Our theoretical analysis shows that $Delta t$-Adaptive is asymptotically optimal in the sense that its ratio approaches the proposed lower bounds as $Delta t$ increases.
We study the greedy-based online algorithm for edge-weighted matching with (one-sided) vertex arrivals in bipartite graphs, and edge arrivals in general graphs. This algorithm was first studied more than a decade ago by Korula and Pal for the bipartite case in the random-order model. While the weighted bipartite matching problem is solved in the random-order model, this is not the case in recent and exciting online models in which the online player is provided with a sample, and the arrival order is adversarial. The greedy-based algorithm is arguably the most natural and practical algorithm to be applied in these models. Despite its simplicity and appeal, and despite being studied in multiple works, the greedy-based algorithm was not fully understood in any of the studied online models, and its actual performance remained an open question for more than a decade. We provide a thorough analysis of the greedy-based algorithm in several online models. For vertex arrivals in bipartite graphs, we characterize the exact competitive-ratio of this algorithm in the random-order model, for any arrival order of the vertices subsequent to the sampling phase (adversarial and random orders in particular). We use it to derive tight analysis in the recent adversarial-order model with a sample (AOS model) for any sample size, providing the first result in this model beyond the simple secretary problem. Then, we generalize and strengthen the black box method of converting results in the random-order model to single-sample prophet inequalities, and use it to derive the state-of-the-art single-sample prophet inequality for the problem. Finally, we use our new techniques to analyze the greedy-based algorithm for edge arrivals in general graphs and derive results in all the mentioned online models. In this case as well, we improve upon the state-of-the-art single-sample prophet inequality.
We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching. Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight $O(log n)$-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of $O(log n)$ has long been conjectured and remains a tantalizing open question. In this paper, we show improved results in the i.i.d arrival model. We show how the i.i.d model can be used to give substantially better algorithms: our main result is an $O((log log log n)^2)$-competitive algorithm in this model. Along the way we give a $9$-competitive algorithm for the line and tree metrics. Both results imply a strict separation between the i.i.d model and the adversarial and random order models, both for general metrics and these much-studied metrics.
Over three decades ago, Karp, Vazirani and Vazirani (STOC90) introduced the online bipartite matching problem. They observed that deterministic algorithms competitive ratio for this problem is no greater than $1/2$, and proved that randomized algorithms can do better. A natural question thus arises: emph{how random is random}? i.e., how much randomness is needed to outperform deterministic algorithms? The textsc{ranking} algorithm of Karp et al.~requires $tilde{O}(n)$ random bits, which, ignoring polylog terms, remained unimproved. On the other hand, Pena and Borodin (TCS19) established a lower bound of $(1-o(1))loglog n$ random bits for any $1/2+Omega(1)$ competitive ratio. We close this doubly-exponential gap, proving that, surprisingly, the lower bound is tight. In fact, we prove a emph{sharp threshold} of $(1pm o(1))loglog n$ random bits for the randomness necessary and sufficient to outperform deterministic algorithms for this problem, as well as its vertex-weighted generalization. This implies the same threshold for the advice complexity (nondeterminism) of these problems. Similar to recent breakthroughs in the online matching literature, for edge-weighted matching (Fahrbach et al.~FOCS20) and adwords (Huang et al.~FOCS20), our algorithms break the barrier of $1/2$ by randomizing matching choices over two neighbors. Unlike these works, our approach does not rely on the recently-introduced OCS machinery, nor the more established randomized primal-dual method. Instead, our work revisits a highly-successful online design technique, which was nonetheless under-utilized in the area of online matching, namely (lossless) online rounding of fractional algorithms. While this technique is known to be hopeless for online matching in general, we show that it is nonetheless applicable to carefully designed fractional algorithms with additional (non-convex) constraints.
Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal competitive ratio of $1-1/e$. Later, Aggarwal et al. (SODA 2011) generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial $1/2$-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing $1/2$ barrier and achieves a competitive ratio of at least $0.5086$. In light of the hardness result of Kapralov, Post, and Vondrak (SODA 2013) that restricts beating a $1/2$ competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.
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