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：本研究探讨了一个中心极值图论问题，该问题源于Erdős 1975年提出的一个猜想，旨在寻找给定规模（节点数）下边数最大且不含3环或4环的图。我们将该问题建模为序贯决策问题，并比较了AlphaZero（一种神经网络引导的树搜索）与禁忌搜索（一种启发式局部搜索方法）的性能。通过引入课程学习策略——利用较小规模下找到的优质图来引导更大规模图的搜索——我们使用这两种方法均改进了多个规模下的当前最优下界。此外，我们提出了一种灵活的图生成环境，以及一种用于学习在图空间中搜索的置换不变网络架构。