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. 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 new bounds improving the state of the art. Specifically, our polynomial-time algorithms for almost-tournaments improve upon the exponential-time algorithms of Ghosh and Kuchlous and have much simpler analysis. The said 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. In particular, this gives us an O(n²)-cut-query algorithm to solve directed minimum cut on n-node graphs in polynomial time, a significant result that has been observed in the literature but not concretely stated in this form. Our results also unveil interesting general connections between two well-studied sublinear models for graph problems: semi-streaming and cut-query.

We study fundamental directed graph (digraph) problems in the streaming model. An initial investigation by Chakrabarti, Ghosh, McGregor, and Vorotnikova [SODA'20] on streaming digraphs showed that while most of these problems are provably hard in general, some of them become tractable when restricted to the well-studied class of tournament graphs where every pair of nodes shares exactly one directed edge. Thus, we focus on tournaments and improve the state of the art for multiple problems in terms of both upper and lower bounds. Our primary upper bound is a deterministic single-pass semi-streaming algorithm (using $\tilde{O}(n)$ space for $n$-node graphs, where $\tilde{O}(.)$ hides polylog$(n)$ factors) for decomposing a tournament into strongly connected components (SCC). it improves upon the previously best-known algorithm by Baweja, Jia, and Woodruff [ITCS'22] in terms of both space and passes: for $p\geq 1$, they used $(p+1)$-passes and $\tilde{O}(n^{1+1/p})$-space. We further extend our algorithm to digraphs that are close to tournaments and establish tight bounds demonstrating that the problem's complexity grows smoothly with the "distance" from tournaments. Applying our framework, we obtain improved tournament algorithms for $s,t$-reachability, strong connectivity, Hamiltonian paths and cycles, and feedback arc set. On the other hand, we prove the first $Ω(n^2)$-space lower bounds for this class, exhibiting that some well-studied problems -- such as (exact) feedback arc set on tournaments (FAST) and $s,t$-distance -- remain hard here. We obtain a generalized lower bound on space-approximation tradeoffs for FAST: any single-pass $(1\pm \varepsilon)$-approximation algorithm requires $Ω(n/\sqrt{\varepsilon})$ space. As a whole, our collection of results contributes significantly to the growing literature on streaming digraphs.