AI 中文总结
研究仅依据锦标赛得分序列能解决的问题，给出统一框架，利用入度解决多种问题，刻画了由入度信息决定的问题类，还得到相关算法，部分算法达最优界，部分改进了技术水平。
AI 中文摘要
如果仅知道锦标赛的得分序列，能解决哪些问题？锦标赛是定向完全图，从组合和算法角度都是广泛研究的有向图类。多年来，研究人员已确定多个可仅从得分序列解决的经典有向图问题。本文给出一个简单统一框架，仅用入度就能解决所有这些问题，还完全刻画了由入度信息决定的问题类：答案在循环反转下不变的问题。这一刻画是更一般结果的特殊情况。作为结果的副产品，我们得到了在流、两人通信和割查询计算模型中关于锦标赛及“几乎锦标赛”的各种基于连通性、割和顶点排序问题的算法。部分算法达到现有最优界，其他算法改进了现有技术水平。
英文摘要
What problems can one solve on a tournament if only its score sequence is known? Tournaments are oriented complete graphs that form an extensively-studied class of directed graphs (digraphs), both from combinatorial and algorithmic perspectives. Over the years, researchers have identified multiple classical digraph problems that can be solved on a tournament from only its score sequence (indegree sequence). These problems include acyclicity testing and topological sorting [Chakrabarti, Ghosh, McGregor, and Vorotnikova; SODA'20], $s,t$-reachability, strong connectivity, and decomposition into strongly connected components (SCC) [Ghosh and Kuchlous; ESA'24], and vertex-ordering problems such as cutwidth and optimal linear arrangement [Barbero, Paul, and Pilipczuk; ICALP'17]. These prior works showed the sufficiency of the score sequence by designing distinct algorithms for the individual problems. In this work, we give a simple unified framework that solves all these problems using only indegrees and, in fact, completely characterises the class of problems that is determined by the indegree information: problems whose answers are invariant under cycle reversals. This characterisation is a special case of a much more general result that we establish: for any arbitrary digraph, the knowledge of its skeleton (underlying undirected graph) and the vertex indegrees completely determines its properties that are invariant under cycle reversal. As a byproduct of our results, we obtain algorithms for a variety of connectivity-based, cut-based, and vertex-ordering problems on tournaments and ``almost tournaments'' in the streaming, the two-player communication, and the cut-query models of computation. Some of these algorithms match existing optimal bounds and others provide bounds improving the state of the art.
CommentsTo appear in ESA 2026
