有界VC维锦标赛的同构性
Isomorphism of tournaments with bounded VC dimension
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中文总结 AI 辅助
本文解决Neuen与Grohe的开放问题，证明有界VC维锦标赛同构问题可在n^{O(d log d)}时间内判定，还证明有界色数锦标赛同构问题可多项式时间判定，开发了同构不变分解等方法。
中文摘要 AI 辅助
锦标赛同构问题是设计更优图同构算法的两大核心瓶颈之一。尽管该问题已被研究逾五十年，但与图同构问题相比，锦标赛同构问题的研究成果极少；对于大多数类别的锦标赛，既未明确其难解性也未明确其多项式时间可解性。有界VC维锦标赛是此类未被研究的类别之一，尽管VC维可说是组合驯顺性中最稳健、最核心的概念之一。本文解决了Neuen与Grohe提出的一个开放问题，证明VC维为d的锦标赛的同构问题可在n^{O(d log d)}时间内判定；进而，有界VC维锦标赛的自同构群可在多项式时间内计算。为此，本文开发了一种同构不变的锦标赛分解新方法；为便于递归，本文引入修补锦标赛（patched tournament）的概念，并分析修补锦标赛的有界VC维性质；本文设计了一种递归算法，平衡分解块的大小与数量，并利用近双胞胎（near twins）的结构。此外，已知遗传类锦标赛若具有无界VC维，则必包含所有2-可着色锦标赛；作为第二个结果，本文证明该类锦标赛也不构成多项式时间同构测试的障碍，且有界色数锦标赛的同构问题可在多项式时间内判定。
英文摘要
The tournament isomorphism problem is one of the two fundamental bottlenecks to designing better algorithms for the graph isomorphism problem. Though the problem has been investigated for more than five decades, compared to graphs, there are only very few results on the isomorphism problem of tournaments. For most classes of tournaments neither hardness nor polynomial-time solvability is known. Tournaments of bounded VC dimension are such a class for which no results are available, even though the VC dimension is arguably one of the most robust and central notions of combinatorial tameness. Resolving an open problem of Neuen and Grohe, we show that the isomorphism problem for tournaments of VC dimension $d$ can be decided in time $n^{O(d\log d)}$. Consequently, automorphism groups of tournaments of bounded VC dimension can be computed in polynomial time. To this end, we develop a new method to isomorphism-invariantly decompose tournaments. To facilitate recursion, we introduce the notion of a patched tournament and analyze bounded VC dimension in patched tournaments. We design a recursive algorithm that balances the size of the decomposed pieces against their number and makes use of the structure of near twins. In an orthogonal direction, it is known that a hereditary class of tournaments has unbounded VC dimension if and only if it contains all 2-colorable tournaments. As a second result, we show that also this class does not form an obstruction towards polynomial-time isomorphism testing and indeed show that isomorphism of tournaments of bounded chromatic number is polynomial-time decidable.
