This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erd\H{o}s, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

翻译：本研究探讨了一个核心的极值图论问题，该问题源于Erdős在1975年提出的一个猜想，旨在寻找给定阶数（节点数量）下、在不包含3-环或4-环的条件下最大化边数的图。我们将该问题形式化为一个序列决策问题，并比较了两种方法：基于神经网络引导树搜索的AlphaZero与启发式局部搜索方法禁忌搜索。通过引入课程学习策略——利用在较小阶数中发现的高质量图来加速对更大图的搜索——两种方法均改进了多个阶数下的当前最优下界。此外，我们提出了一个灵活的图生成环境以及一种置换不变网络架构，用于学习在图空间中进行搜索。