This work proves 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.
Abstract
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.
The soundness uses a result of Haeupler, Saha, and Srinivasan building on the proof of Moser and Tardos, to upper-bound the probability that a fixed relatively large subset is an independent set after the Moser-Tardos algorithm terminates.
We prove that for all fixed $k\geq 4$, any $N$ vertex graph with no independent set of size $n$ and $N\geq \Omega(n^{k-1}/\log^{k-2}n)$ contains at least $$ \Omega\bigg(\binom Nk \Big(\frac{\log n}{n}\Big)^{\binom k2}/\log n\bigg) $$ cliques of order $k$, and for $k\geq 5$ this is best possible conditional on the known...
Given two tournaments of order $n$, Bollob\'as and Scott defined their discrepancy as the largest deviation of their overlap from its random average under relabelling, and they asked whether the resulting discrepancy is always $\Omega (n^{3/2})$. We answer this question by proving that there is an absolute constant $c>...
We study graph‐theoretic properties of random 0/1$0/1$ polytopes. Specifically, let Qpn⊆{0,1}n$Q_p^n \subseteq \lbrace 0,1\rbrace ^n$ be a random subset where each point is included independently with probability p$p$ , and consider the graph Gp$G_p$ of the polytope conv(Qpn)$\operatorname{conv}(Q_p^n)$ . We provide a...
Catherine Babecki, Tycho Elling, A. Ferber· Bulletin of the London Mathe...· 0 citations
The main result is an algorithm that $(1+\epsilon)$-approximates distance to monotonicity in $\sqrt{n}^{1+o(1)}$ passes, and it is proved that this is the best pass-complexity one can hope for, for any $O(1)$-approximation, short of improving the state-of-the-art streaming algorithm for st-reachability.
Amir Azarmehr, Soheil Behnezhad, Lily Chung et al.· 0 citations
Erd\H{o}s et al. (JCT-B, 1989) conjectured that, for integers $r\ge 2$ and $\delta\ge 2$ with $3r-1\mid\delta$, every connected $K_{2r+1}$-free graph of order $n$ and minimum degree $\delta$ has diameter at most $ \frac{3r-1}{r}\cdot \frac{n}{\delta}+O(1)$. Czabarka et al. (JCT-B, 2021) later proposed the following gen...
Hang Chen, Yao-Jun Chen· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
