When Does Sparsity Help for k-Independent Set in Hypergraphs and Other Boolean CSPs?

2026-05-11

AI Summary

This work investigates the impact of input sparsity—where the number of constraints $m = \Theta(n^\gamma)$ with $\gamma \geq 2$—on the computational complexity of finding size-$k$ independent sets in hypergraphs and solving binary Boolean constraint satisfaction problems (CSPs). By integrating the matrix multiplication exponent $\omega$, hypotheses on $k$-clique and 3-uniform hyperclique hardness, sparse hypergraph modeling, and fine-grained CSP classification, the authors establish conditionally tight algorithms. Their main contributions include an approximately optimal algorithm running in $O(\min\{n^{\omega k/3} + m^{k/3}, n^k\})$ time and a complete characterization of the relationship between sparsity and complexity in binary Boolean CSPs. They demonstrate that combinations of multiple constraints can yield higher complexity than individual constraints and identify broad families of constraints—such as NAND and Implication—that exhibit sharp phase-transition thresholds $\gamma_F$.

Abstract

Consider the fundamental task of finding independent sets of (constant) size $k$ in a given $n$-node hypergraph. How is the time complexity affected by the sparsity of the input, i.e., the number of hyperedges $m$? Turán's theorem implies that the problem is trivial if $m=O(n^{2-ε})$ for some $ε> 0$. Above that threshold (i.e., if $m=Θ(n^γ)$ for some $γ\ge 2$), we give a perhaps surprising algorithm with running time $O\left(\min\left\{n^{\fracω{3}k} + m^{k/3}, n^k\right\}\right)$ (for $k$ divisible by 3), which is essentially conditionally optimal for all $γ\ge 2$, assuming the $k$-clique and 3-uniform hyperclique hypotheses (here, $ω<2.372$ denotes the matrix multiplication exponent). In fact, we obtain a more detailed time complexity, sensitive to the arity distribution of the hyperedges. To study such phenomena in more generality, we study the time complexity of finding solutions of (constant) size $k$ in sparse instances of Boolean constraint satisfaction problems, where $n$ and $m$ denote the number of variables and constraints. Our results include an essentially full classification of the influence of sparsity for Boolean constraint families of binary arity. Of particular technical interest is a conditionally tight algorithm for the family consisting of the binary NAND and Implication constraints, with a running time of $Θ(m^{ωk/6 \pm c})$. Further, we identify a large class of constraint families $F$ that exhibits a sharp phase transition: there is a threshold $γ_F$ such that the problem is trivial for $m=O(n^{γ_F-ε})$, but requires essentially brute-force running time $Θ(n^{k\pm c})$ for $m=Ω(n^{γ_F})$, assuming the 3-uniform hyperclique hypothesis. Notably, in many cases the combination of constraints display higher time complexity than either constraint alone.

Problem

• sparsity
• independent set
• hypergraphs
• Boolean CSPs
• time complexity

Innovation

• sparse hypergraphs
• Boolean CSP
• conditional lower bounds
• phase transition
• matrix multiplication exponent