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

TL;DR

This paper investigates how the sparsity of hypergraphs and Boolean CSPs influences the complexity of finding fixed-size independent sets or solutions, revealing thresholds and algorithms that are conditionally optimal.

Contribution

It provides a detailed classification of the impact of sparsity on the complexity of Boolean CSPs and hypergraph problems, including new algorithms and phase transition phenomena.

Findings

An algorithm with near-optimal running time for hypergraphs with m=Θ(n^γ) edges, for γ≥2.

A tight algorithm for binary NAND and Implication constraints with specific time complexity.

Identification of a phase transition threshold γ_F for various constraint families.

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 $m$? Tur\'{a}n's theorem implies that the problem is trivial if $m=O(n^{2-\epsilon})$ for some $\epsilon> 0$. Above that threshold (i.e., if $m=\Theta(n^\gamma)$ for some $\gamma \ge 2$), we give a perhaps surprising algorithm with running time $O\left(\min\left\{n^{\frac{\omega}{3}k} + m^{k/3}, n^k\right\}\right)$ (for $k$ divisible by 3), which is essentially conditionally optimal for all $\gamma\ge 2$, assuming the $k$-clique and 3-uniform hyperclique hypotheses (here, $\omega<2.372$ 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 $\Theta(m^{\omega k/6 \pm c})$. Further, we identify a large class of constraint families $F$ that exhibits a sharp phase transition: there is a threshold $\gamma_F$ such that the problem is trivial for $m=O(n^{\gamma_F-\epsilon})$, but requires essentially brute-force running time $\Theta(n^{k\pm c})$ for $m=\Omega(n^{\gamma_F})$, assuming the 3-uniform hyperclique hypothesis. Notably, in many cases the combination of constraints display higher time complexity than either constraint alone.