Algorithmic Phase Transition for Large Independent Sets in Dense Hypergraphs

TL;DR

This paper investigates the algorithmic limits of finding large independent sets in dense hypergraphs, establishing thresholds and designing online algorithms that are provably optimal within these bounds.

Contribution

It identifies the largest independent sets in dense hypergraphs and develops online algorithms with optimal approximation ratios, proving these bounds are tight.

Findings

Designed online algorithms achieving optimal approximation factors.

Established matching lower bounds proving the optimality of these algorithms.

Pinpointed the size thresholds for large independent sets in dense hypergraphs.

Abstract

We study the algorithmic tractability of finding large independent sets in dense random hypergraphs. In the sparse regime, much of the natural algorithms can be formulated within either the local or the low-degree polynomial (LDP) framework, and a rich literature has subsequently identified nearly sharp algorithmic thresholds within these classes by exploiting their stability. In the dense setting, however, the algorithmic paradigms are fundamentally different: they are online and thus need not be stable. Perhaps more crucially, even for the classical Erd\H{o}s-R\'enyi random graph $G(n,p)$, LDPs are conjectured to fail in the 'easy' regime accessible to online algorithms, thereby challenging their viability for dense models. Our focus is on two models: (i) finding large independent sets in dense $r$-uniform Erd\H{o}s-R\'enyi hypergraphs, and (ii) the more challenging problem of finding large $\gamma$-balanced independent sets in dense $r$-uniform $r$-partite hypergraphs, where the $i$-th coordinate of $\gamma\in\mathbb{Q}^r$ specifies the proportion of vertices from $V_i$ in the independent set. For both models, we pinpoint the size of the largest independent set and design online algorithms that achieve a multiplicative approximation factor of $r^{1/(r-1)}$ in the uniform and $(\max_i \gamma_i)^{-1/(r-1)}$ in the $r$-partite model. Furthermore, we establish matching algorithmic lower bounds, showing that these computational gaps are sharp: no online algorithms can breach these gaps.