Srinivasan Raghuraman

Jay Bhan, Nicole Nobili, Srinivasan Raghuraman et al.

The Zarankiewicz number $\textbf{Z}(m, n, s, t)$ is the maximum number of edges in a bipartite graph $G_{m, n}$ such that there is no complete $K_{s, t}$ bipartite subgraph. We determine for the first time the exact values of three Zarankiewicz numbers: $\textbf{Z}(11, 21, 3, 3)=116$, $\textbf{Z}(11, 22, 3, 3)=121$, and $\textbf{Z}(12, 22, 3, 3)=132$. We further establish lower bounds for 41 more Zarankiewicz numbers, including several that are within one edge of the best known upper bound, and we match the established value in four more closed cases. Our results are obtained using OpenEvolve, an open-source evolutionary algorithm based on Large Language Models (LLMs) that iteratively improves algorithms for generating mathematical constructions by optimizing a reward signal which we tailored for this specific problem. These findings provide new extremal graph constructions and demonstrate the potential of LLM-guided evolutionary search to contribute to mathematical research. In addition to presenting the resulting constructions, we report the generation algorithms produced, describe the relevant implementation details, and provide our computational costs. Our costs are remarkably low, at less than \$30 for each Zarankiewicz parameter combination, showing that LLM-guided evolutionary search can be an inexpensive, reproducible, and accessible tool for discovering new combinatorial constructions.

Mihai Christodorescu, Erin English, Wanyun Catherine Gu et al.

With the innovation of distributed ledger technology (DLT), often known as blockchain technology, there has been significant growth of digital tokens in the form of cryptocurrencies, stablecoins, and central bank digital currencies. As the number of DLT networks increases, each with varying design characteristics, the likelihood that transacting parties are on the same network decreases. Thus, it is crucial to facilitate payments that are universal across networks, scalable to massive loads, and highly available. We envision a future payment network that may be built on top of DLT networks without being subject to their limitations on interoperability, scalability, and availability faced by DLT payment solutions today. Specifically, we propose a hub-and-spoke payment route, referred to here as Universal Payment Channels (UPC), that can be used to support digital token transfers of funds across different networks through payment channels. We further discuss the potential use cases of the UPC technology to support, and not complicate, an already robust digital payment ecosystem. Finally, through the paper, we share some future directions of the UPC technology.

Mihai Christodorescu, Wanyun Catherine Gu, Ranjit Kumaresan et al.

Digital payments traditionally rely on online communications with several intermediaries such as banks, payment networks, and payment processors in order to authorize and process payment transactions. While these communication networks are designed to be highly available with continuous uptime, there may be times when an end-user experiences little or no access to network connectivity. The growing interest in digital forms of payments has led central banks around the world to explore the possibility of issuing a new type of central-bank money, known as central bank digital currency (CBDC). To facilitate the secure issuance and transfer of CBDC, we envision a CBDC design under a two-tier hierarchical trust infrastructure, which is implemented using public-key cryptography with the central bank as the root certificate authority for generating digital signatures, and other financial institutions as intermediate certificate authorities. One important design feature for CBDC that can be developed under this hierarchical trust infrastructure is an offline capability to create secure point-to-point offline payments through the use of authorized hardware. An offline capability for CBDC as digital cash can create a resilient payment system for consumers and businesses to transact in any situation. We propose an offline payment system (OPS) protocol for CBDC that allows a user to make digital payments to another user while both users are temporarily offline and unable to connect to payment intermediaries (or even the Internet). OPS can be used to instantly complete a transaction involving any form of digital currency over a point-to-point channel without communicating with any payment intermediary, achieving virtually unbounded throughput and real-time transaction latency.