Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival

Machine learning-guided synthesis of advanced inorganic materials

Synthesis of advanced inorganic materials with minimum number of trials is of paramount importance towards the acceleration of inorganic materials development. The enormous complexity involved in existing multi-variable synthesis methods leads to high uncertainty, numerous trials and exorbitant cost. Recently, machine learning (ML) has demonstrated tremendous potential for material research. Here, we report the application of ML to optimize and accelerate material synthesis process in two representative multi-variable systems. A classification ML model on chemical vapor deposition-grown MoS2 is established, capable of optimizing the synthesis conditions to achieve higher success rate. While a regression model is constructed on the hydrothermal-synthesized carbon quantum dots, to enhance the process-related properties such as the photoluminescence quantum yield. Progressive adaptive model is further developed, aiming to involve ML at the beginning stage of new material synthesis. Optimization of the experimental outcome with minimized number of trials can be achieved with the effective feedback loops. This work serves as proof of concept revealing the feasibility and remarkable capability of ML to facilitate the synthesis of inorganic materials, and opens up a new window for accelerating material development

On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching

Ranking and Balance are arguably the two most important algorithms in the online matching literature. They achieve the same optimal competitive ratio of 1-1/e for the integral version and fractional version of online bipartite matching by Karp, Vazirani, and Vazirani (STOC 1990) respectively. The two algorithms have been generalized to weighted online bipartite matching problems, including vertex-weighted online bipartite matching and AdWords, by utilizing a perturbation function. The canonical choice of the perturbation function is f(x) = 1-e^{x-1} as it leads to the optimal competitive ratio of 1-1/e in both settings. We advance the understanding of the weighted generalizations of Ranking and Balance in this paper, with a focus on studying the effect of different perturbation functions. First, we prove that the canonical perturbation function is the unique optimal perturbation function for vertex-weighted online bipartite matching. In stark contrast, all perturbation functions achieve the optimal competitive ratio of 1-1/e in the unweighted setting. Second, we prove that the generalization of Ranking to AdWords with unknown budgets using the canonical perturbation function is at most 0.624 competitive, refuting a conjecture of Vazirani (2021). More generally, as an application of the first result, we prove that no perturbation function leads to the prominent competitive ratio of 1-1/e by establishing an upper bound of 1-1/e-0.0003. Finally, we propose the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets, and we design an optimal 1-1/e competitive algorithm by generalizing Balance