AI 中文总结
针对排除团子式和排除团浸入的图，提出两种最大团枚举算法，分别将时间复杂度优化为n·4^(2t/5+o(t))和n·3^(t/3+o(t))，并证明其指数基为渐近最优。
AI 中文摘要
我们研究由排除团子式和排除团浸入定义的图类中最大团的精确枚举。对于n个顶点的K_t-子式自由图，我们提出一种算法，可在n·4^(2t/5+o(t))时间内枚举所有最大团，显著改进了Eppstein、Löffler和Strash之前的n·2^(O(t log log t))时间复杂度。对于n个顶点的K_t-浸入自由图，我们建立了首个以浸入数为参数的精确枚举算法，运行时间为n·3^(t/3+o(t))。尽管两种算法采用共同的基于简并度的根分配方案，但它们的分析需要不同的结构机制。关键在于，我们的算法未应用通用稀疏性界，而是将特定结构障碍——子式的局部密度阈值、浸入的最小度分支——深度集成到枚举逻辑中。我们还通过专门构造证明了匹配的输出规模下界，直至t的次指数因子，因此指数基4^(2/5)和3^(1/3)是渐近最优的。
英文摘要
We study the exact enumeration of maximal cliques in graph classes defined by excluded clique minors and excluded clique immersions. For n-vertex K_t-minor-free graphs, we give an algorithm that lists all maximal cliques in n * 4^(2t/5+o(t)) time, significantly improving the previous n * 2^O(t log log t) bound of Eppstein, Löffler, and Strash. For n-vertex K_t-immersion-free graphs, we establish the first exact enumeration algorithm parameterized by immersion number, achieving a running time of n * 3^(t/3+o(t)). While both algorithms employ a common degeneracy-based root-assignment scheme, their analyses require distinct structural mechanisms. Crucially, rather than applying generic sparsity bounds, our algorithms deeply integrate the specific structural obstructions -- local density thresholds for minors and minimum-degree branchings for immersions -- directly into the enumeration logic. We also prove matching output-size lower bounds, up to sub-exponential factors in t, using specialized constructions. Consequently, the exponential bases 4^(2/5) and 3^(1/3) are asymptotically optimal.
Comments14 pages. Comments are welcome
