Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

Buchbinder, Niv; Segev, Danny; Tkach, Yevgeny

doi:10.4230/lipics.esa.2017.22

Cited by 3 publications

(1 citation statement)

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“…Any graphs Worst-case (unweighted) 1 2 (tight) bipartite & any graphs [24] 1 ´1 e (tight) [31] ě 1 2 `Ωp1q(any graph) [24] ď 0.591(bipartite graphs) [7] Bayesian (weighted) ě 1 3 (bipartite graphs) ď 4 9 (bipartite graphs) [26] ě 0.337 (any graph) (Theorem 5.3) k matroids. They found an adaptive 1 1 4k´2 -competitive algorithm for this setting.…”

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confidence: 99%

Online Stochastic Max-Weight Matching: prophet inequality for vertex and edge arrival models

Ezra¹,

Feldman²,

Gravin³

et al. 2020

Preprint

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We provide prophet inequality algorithms for online weighted matching in general (nonbipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. To study these settings, we introduce a novel unified framework of batched prophet inequalities that captures online settings where elements arrive in batches; in particular it captures matching under the two aforementioned arrival models. Our algorithms rely on the construction of suitable online contention resolution scheme (OCRS). We first extend the framework of OCRS to batched-OCRS, we then establish a reduction from batched prophet inequality to batched OCRS, and finally we construct batched OCRSs with selectable ratios of 0.337 and 0.5 for edge and vertex arrival models, respectively. Both results improve the state of the art for the corresponding settings. For the vertex arrival, our result is tight. Interestingly, a pricing-based prophet inequality with comparable competitive ratios is unknown.

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confidence: 99%

Online Stochastic Max-Weight Matching: prophet inequality for vertex and edge arrival models

Ezra¹,

Feldman²,

Gravin³

et al. 2020

Preprint

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Learning Augmented Methods for Matching: Improving Invasive Species Management and Urban Mobility

Björck

Shi

Brown‐Lima

et al. 2021

AAAI

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With the success of machine learning, integrating learned models into real-world systems has become a critical challenge. Naively applying predictions to combinatorial optimization problems can incur high costs, which has motivated researchers to consider learning augmented algorithms that can make use of faulty or incomplete predictions. Inspired by two matching problems in computational sustainability where data is abundant, we consider the learning augmented min weight matching problem where some nodes are revealed online while others are known a priori, e.g., by being predicted by machine learning. We develop an algorithm that is able to make use of this extra information and provably improves upon pessimistic online algorithms. We evaluate our algorithm on two settings from computational sustainability -- the coordination of unreliable citizen scientists for invasive species management, and the matching between taxis and riders under uncertain trip duration predictions. In both cases, we perform extensive experiments on real-world datasets and find that our method outperforms baselines, showing how learning augmented algorithms can reliably improve solutions for problems in computational sustainability.

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Fully Online Matching II: Beating Ranking and Water-filling

Huang

Tang

et al. 2020

Preprint

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Karp, Vazirani, and Vazirani (STOC 1990) initiated the study of online bipartite matching, which has held a central role in online algorithms ever since. Of particular importance are the Ranking algorithm for integral matching and the Water-filling algorithm for fractional matching. Most algorithms in the literature can be viewed as adaptations of these two in the corresponding models. Recently, Huang et al. (STOC 2018, SODA 2019 introduced a more general model called fully online matching, which considers general graphs and allows all vertices to arrive online. They also generalized Ranking and Water-filling to fully online matching and gave some tight analysis: Ranking is Ω ≈ 0.567-competitive on bipartite graphs where the Ω-constant satisfies Ωe Ω = 1, and Water-filling is 2 − √ 2 ≈ 0.585-competitive on general graphs. We propose fully online matching algorithms strictly better than Ranking and Water-filling. For integral matching on bipartite graphs, we build on the online primal dual analysis of Ranking and Water-filling to design a 0.569-competitive hybrid algorithm called Balanced Ranking. To our knowledge, it is the first integral algorithm in the online matching literature that successfully integrates ideas from Water-filling. For fractional matching on general graphs, we give a 0.592competitive algorithm called Eager Water-filling, which may match a vertex on its arrival. By contrast, the original Water-filling algorithm always matches vertices at their deadlines. Our result for fractional matching further shows a separation between fully online matching and the general vertex arrival model by Wang and Wong (ICALP 2015), due to an upper bound of 0.5914 in the latter model by Buchbinder, Segev, and Tkach (ESA 2017).

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Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

Cited by 3 publications

References 0 publications

Online Stochastic Max-Weight Matching: prophet inequality for vertex and edge arrival models

Online Stochastic Max-Weight Matching: prophet inequality for vertex and edge arrival models

Learning Augmented Methods for Matching: Improving Invasive Species Management and Urban Mobility

Fully Online Matching II: Beating Ranking and Water-filling

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