AI 中文总结
研究一般图上边集的无偏 Maker - Breaker 三角形博弈，通过边 - 三角形关联图提出新算法。针对不同类图给出不同时间复杂度算法，改进了现有结果，还给出从三角形检测到该博弈判定的线性归约。
AI 中文摘要
本文提出了用于确定在一般图的边集上进行的无偏三角形博弈获胜者的新多项式时间算法。为此，提议通过边 - 三角形关联图而非标准超图模型来审视该博弈。确定了 Maker 在边 - 三角形关联图方面的充要获胜条件，并表明实现此条件的获胜策略尽可能快地单调进行，即仅考虑边 - 三角形关联图的单调递减连通子图。针对不同类别的图给出了三种不同算法。对于一般图\(G\)，无偏三角形博弈的结果可在\(\mathcal{O}(n + m^{3.5})=\mathcal{O}(n^7)\)时间内判定，显著改进了 Galliot 等人（arXiv 2022）隐含的\(\mathcal{O}(n^{16})\)算法。对于包含\(K_4\)作为子图且边 - 三角形关联图连通的图\(G\)，获胜者可在\(\mathcal{O}(n+\min\{n^{\omega + 1},m^2\})=\mathcal{O}(n^{\omega + 1})\)时间内判定，其中\(\omega<2.372\)是矩阵乘法的指数（Alman 等人，SODA 2025）。对于边 - 三角形关联图为仙人掌图（即所有循环边不相交）的图\(G\)，获胜者可在\(\mathcal{O}(n + m^{1.5})=\mathcal{O}(n^3)\)时间内判定，此类\(G\)是不含\(K_4\)的。特殊情况的算法基于每个图类中 Maker 获胜的新颖结构特征。还给出了从三角形检测到判定无偏三角形博弈的线性时间归约。
英文摘要
In this paper, we present new polynomial-time algorithms for determining the winner of the unbiased triangle game played on the edge set of general graphs. To that end, we propose to view the game through the edge-triangle incidence graph instead of the standard hypergraph model. We identify a necessary and sufficient winning condition for Maker in terms of the edge-triangle incidence graph and show that winning strategies achieving this condition as fast as possible play monotonically in the sense that they only consider monotonically decreasing connected subgraphs of the edge-triangle incidence graph. We give three different algorithms for different classes of graphs. For general graphs $G$, the outcome of the unbiased triangle game can be decided in time $\mathcal{O}(n+m^{3.5})=\mathcal{O}(n^7)$. This significantly improves on the $\mathcal{O}(n^{16})$ algorithm implied by the work of Galliot, Gravier and Sivignon (arXiv 2022). For graphs $G$ which contain $K_4$, the complete graph on four vertices, as a subgraph and whose edge-triangle incidence graph is connected, the winner can be decided in time $\mathcal{O}(n+\min\{n^{ω+1},m^2\})=\mathcal{O}(n^{ω+1})$, where $ω<2.372$ is the exponent of matrix multiplication (Alman et. al., SODA 2025). For graphs $G$ whose edge-triangle incidence graph is a cactus graph, i.e. all its cycles are edge-disjoint, the winner can be decided in time $\mathcal{O}(n+m^{1.5})=\mathcal{O}(n^3)$. Such $G$ are $K_4$-free. The algorithms for the special cases are based on novel structural characterizations of Maker's win for each graph class. We also give a linear time reduction from triangle detection to deciding the unbiased triangle game.
Comments31 pages, 12 figures
