The optimal sample complexity of testing if an $n$-vertex graph has an independent set of size $\rho n$, or is $\varepsilon$-far from having an independent set of size $\rho n$, was established to be $\widetilde{O}(\rho^3/\varepsilon^2)$, in a notable result by Blais and Seth (SICOMP 2025). In contrast, for $q$-uniform hypergraphs, there is a significant gap between the best known upper and lower bounds, and there has been no progress on the problem for the last two decades. In this work, we prove a new upper bound of $\widetilde{O}\!\left(\frac{q\rho^{2q-3}}{\varepsilon^2 (q-2)!^2}\right)$ on the sample complexity of testing the $\rho$-independent set property. The previous best known upper bound was $\widetilde{O}\!\left(\frac{2^q q! \rho^{2q}}{\varepsilon^3}\right)$, due to Langberg (RANDOM 2004). This establishes the optimal dependence on $\varepsilon$ and gives an exponential improvement in the dependence on $q$. We prove our result via a new application of the hypergraph container method.
Elena Grigorescu, Shreya Nasa, Cameron Seth· 1 citation
We present a simple linear-time algorithm that outputs an Eulerian tour of an undirected multigraph with $n$ vertices and $m$ edges, if one exists, in $O(m)$ time and using $O(n)$ words of working memory. The input is given as read-only adjacency lists, and the output is written to an append-only stream in traversal order. Our algorithm first finds a sparse spanning circuit (a skeleton), then traverses the circuit step-by-step, repeatedly outputting further circuits rooted at the current vertex. This solves a problem left open by Ismaili Alaoui, Plump, and Wild (SOSA 2026): their space-efficient variant of Hierholzer's algorithm handles general directed multigraphs, but it is unclear how to generalize it to general undirected multigraphs. Our result completes the picture in the read-only model for space-efficient output of Eulerian tours.
Elena Grigorescu, Ziad Ismaili Alaoui, Tamio-Vesa Nakajima et al.· 0 citations
This paper provides the first truly linear-time approximation scheme for the Densest Subgraph Problem, and uses assignments arising from a flow-based formulation together with a structural carving lemma to progressively carve "sparse" parts of the graph while nearly preserving the densest subgraph.
