Nearly optimal independence oracle algorithms for edge estimation in hypergraphs

TL;DR

This work provides near-optimal independence-oracle algorithms for estimating the number of edges in $k$-uniform hypergraphs under two query models: uncoloured IND and colourful cIND. By introducing cost-aware oracle models and carefully analyzing both upper and unconditional lower bounds, the authors obtain tight, parameter-dependent overheads—$n^{g(k,eta)}$ for uncoloured and $n^{0}$–$n^{eta} ext{ polylog}(n)$-type refinements for colourful settings—while proving these bounds hold without rely­ing on conjectures such as SETH. The paper also connects approximate counting to decision via reductions, and shows how sampling-based counting can transfer into approximate sampling, yielding practical fine-grained reductions for a broad class of uniform-witness problems. Overall, the results substantially sharpen the landscape of oracle-based edge counting in hypergraphs, with unconditional lower bounds and nearly matching upper bounds that illuminate when polylogarithmic overheads are possible. The methods have potential impact on designing efficient sublinear-time estimators and on understanding the limits of decision-to-count reductions in fine-grained complexity.

Abstract

We study a query model of computation in which an n-vertex k-hypergraph can be accessed only via its independence oracle or via its colourful independence oracle, and each oracle query may incur a cost depending on the size of the query. In each of these models, we obtain oracle algorithms to approximately count the hypergraph's edges, and we unconditionally prove that no oracle algorithm for this problem can have significantly smaller worst-case oracle cost than our algorithms.

Figures (2)

• Figure 1: Left: If each $\mathtt{IND}$-query of size $x$ has cost $x^{\alpha_k}$, then up to $k^{\mathcal{O}(k)}\log^{\mathcal{O}(1)}n$ factors, we show in \ref{['thm:uncol-main-simple']} that $n^{g(k,\alpha_k)+\alpha_k}$ is the smallest possible $\mathtt{IND}$-oracle cost to (1/2)-approximate the number of edges. Plotted here in thick lines is the overhead $\alpha\mapsto g(k,\alpha)$ in the exponent of $n$ for $k\in\{2,3,4,5\}$, and in dashed lines is the larger overhead exponent $\alpha\mapsto(k-\alpha)/2$ obtained from a naive generalisation of the techniques of BHRRS-oracle-intro. Right: If each $\mathtt{cIND}$-query of size $x$ has cost $x^{\alpha_k}$, then up to $k^{\mathcal{O}(k)}(\log n)^{\mathcal{O}(1)}(\log\log n)^{\mathcal{O}(k-\alpha_k)}$ factors, we show in Theorem \ref{['thm:col-main-simple']} that $n^{\alpha_k}\log^{4(k-\lceil\alpha_k\rceil)+18}n$ is the smallest possible $\mathtt{cIND}$-oracle cost to $(1/2)$-approximate the number of edges. Plotted in thick lines is the overhead $\alpha\mapsto 4(k-\lceil \alpha\rceil ) + 18$ in the exponent of $\log n$ for $k=12$, and the dashed line depicts the overhead $\alpha\mapsto 3k+5$ obtained by using the bound on the number of queries by BBGM-runtime; our bound is better if $\lceil{}\alpha_k\rceil{}\ge k/4+\Theta(1)$.
• Figure 2: Depicted are the functions $i\mapsto \tfrac{1}{k} f_1(\hat{L}_i,\hat{\gamma}_i)$ () and $i\mapsto \tfrac{1}{k} f_2(\hat{L}_i,\hat{\gamma}_i)$ () from \ref{['lem:uncol-bound-ti']} for $n=2^{60}$ and $k=12$. For each fixed integer $L\in\{0,\dots,k-1\}$, the functions are linear in $\gamma$, which gives rise to the step artefacts; when $\gamma$ is fixed, the functions are degree-2 polynomials in $L$, which causes the overall parabolic shape. When $k$ and $\log n$ are even, the overall maximum is equal to $\frac{k}{4}$ and achieved at $i=\frac{\log n}{2}$.

Theorems & Definitions (37)

• Theorem 1: Uncoloured independence oracle, polynomial cost function
• Theorem 2: Colourful independence oracle, polynomial cost function
• Definition 1
• Theorem 3
• Definition 2
• Theorem 4
• Definition 3
• Lemma 5
• Definition 4
• Lemma 6