We study the generalized sorting problem where only a subset of the pairs can be compared. Equivalently, we are given an $n$-vertex $m$-edge graph $G=(V, E)$ where the vertex set $V$ is to be sorted, and the edge set $E$ indicates possible comparisons. The goal is to sort $V$ with as few (adaptive) queries to $E$ as possible. When $G$ is the complete graph, the problem is equivalent to standard sorting for which $Θ(n \log n)$ queries are sufficient and necessary. All classical sorting algorithms crucially rely on the ability to compare any pair and do not imply an $o(n^2)$-query algorithm in the general case. This was improved 15 years ago by Huang, Kannan, and Khanna [FOCS'11] who achieved a bound of $\widetilde{O}(n\sqrt{n})$. Kuszmaul and Narayanan [FOCS'21] later gave a bound of $\widetilde{O}(\sqrt{m n})$, which is more efficient for sparse graphs, but matches the $\widetilde{O}(n\sqrt{n})$ barrier for dense graphs. In this paper, we present an algorithm for generalized sorting that makes at most $\widetilde{O}(n^{4/3})$ queries, therefore breaking the longstanding $\widetilde{O}(n\sqrt{n})$ barrier for all graphs. The key to our result is to query edges that belong to many (specific types of) four-cycles. We show that querying such edges allows the algorithm to verify the direction of many other edges (specifically, their opposing edges in the four-cycles) and thus achieve a much better amortized query cost.
The seminal result of Czumaj and Sohler [STOC'04] showed that a 2-approximation of the value of metric TSP can be obtained with sublinear queries. Whether sublinear algorithms can break the 2-approximation barrier has remained a major open problem. See problem 71 on sublinear.info. This paper resolves this open problem. We show there are fixed $\varepsilon, δ> 0$ such that a $(2-δ)$-approximation of metric TSP value can be obtained with $O(n^{2-\varepsilon})$ queries. Here $n$ is the number of vertices, and the input size is $Θ(n^2)$.
We study the stochastic matching problem, where a graph $G$ is known in advance, but each edge survives independently with probability $p$. The goal is to select a subgraph $H$ of small maximum degree before seeing the surviving edges, while preserving a $(1-ε)$ fraction of the expected maximum matching size. This problem has received significant attention due to its numerous applications in kidney exchange, labor markets, and online dating. In this paper, we show there is always a subgraph $H \subseteq G$ achieving a $(1-ε)$-approximation while having a maximum degree of at most $O\left(\frac{\log(1/ε)}{ε^4} \cdot \frac{1}{p}\right).$ This resolves a major conjecture of the literature, first stated by [BDH, STOC'20], that $poly(\frac{1}{εp})$ degrees suffice. The best previous bound was $(1/p)^{\exp((1/ε)\log(1/ε))}$ by [ABGR, STOC'25]. The key to this result is a lemma of independent interest that any distribution over the matchings of a stochastic graph can be transformed to another distribution which matches each vertex with exactly the same probability, but exhibits extremely low total pairwise correlation.