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.

翻译：考虑在给定$n$节点超图中寻找（常数）大小$k$的独立集这一基本任务。输入稀疏性（即超边数量$m$）如何影响时间复杂度？Turán定理表明，若存在某个$ε>0$使得$m=O(n^{2-ε})$，则该问题平凡可解。超过该阈值后（即对于某个$γ\ge 2$有$m=Θ(n^γ)$），我们给出一个可能令人惊讶的算法，其运行时间为$O\left(\min\left\{n^{\fracω{3}k} + m^{k/3}, n^k\right\}\right)$（针对$k$可被3整除的情况），并且基于$k$-团假设与3-均匀超团假设，该算法在本质上对于所有$γ\ge 2$是条件最优的（其中$ω<2.372$表示矩阵乘法指数）。实际上，我们还获得了更精细的时间复杂度分析，该复杂度对超边的元数分布敏感。为更一般地研究此类现象，我们进一步分析了在稀疏布尔约束满足问题实例中寻找（常数）大小$k$解的时间复杂度，其中$n$和$m$分别表示变量数和约束数。我们的结果包含对二元元数布尔约束族稀疏性影响的完整分类。特别值得注意的是，针对由二元NAND与蕴含约束构成的约束族，我们提出了条件最优算法，其运行时间为$Θ(m^{ωk/6 \pm c})$。此外，我们识别出一类具有显著相变性质的约束族$F$：存在阈值$γ_F$，使得当$m=O(n^{γ_F-ε})$时问题平凡可解，而当$m=Ω(n^{γ_F})$时，基于3-均匀超团假设，问题需要基本穷举的运行时间$Θ(n^{k\pm c})$。值得注意的是，在许多情形下，约束组合呈现出的时间复杂度高于其中任意单一约束。