The Set Packing problem is, given a collection of sets $\mathcal{S}$ over a ground set $\mathcal{U}$, to find a maximum collection of sets that are pairwise disjoint. The problem is among the most fundamental NP-hard optimization problems that have been studied extensively in various computational regimes. The focus of this work is on parameterized complexity, Parameterized Set Packing (PSP): Given $r \in {\mathbb N}$, is there a collection $ \mathcal{S}' \subseteq \mathcal{S}: |\mathcal{S}'| = r$ such that the sets in $\mathcal{S}'$ are pairwise disjoint? Unfortunately, the problem is not fixed parameter tractable unless $\mathsf{W[1] = FPT}$, and, in fact, an "enumeration" running time of $|\mathcal{S}|^{\Omega(r)}$ is required unless the exponential time hypothesis (ETH) fails. This paper is a quest for tractable instances of Set Packing from parameterized complexity perspectives. We say that the input $(\mathcal{U},\mathcal{S})$ is "compact" if $|\mathcal{U}| = f(r)\cdot\Theta(\textsf{poly}( \log |\mathcal{S}|))$, for some $f(r) \ge r$. In the Compact Set Packing problem, we are given a compact instance of PSP. In this direction, we present a "dichotomy" result of PSP: When $|\mathcal{U}| = f(r)\cdot o(\log |\mathcal{S}|)$, PSP is in $\textsf{FPT}$, while for $|\mathcal{U}| = r\cdot\Theta(\log (|\mathcal{S}|))$, the problem is $W[1]$-hard; moreover, assuming ETH, Compact PSP does not even admit $|\mathcal{S}|^{o(r/\log r)}$ time algorithm. Although certain results in the literature imply hardness of compact versions of related problems such as Set $r$-Covering and Exact $r$-Covering, these constructions fail to extend to Compact PSP. A novel contribution of our work is the identification and construction of a gadget, which we call Compatible Intersecting Set System pair, that is crucial in obtaining the hardness result for Compact PSP.

翻译：集合打包问题（Set Packing）定义为：给定全域集 $\mathcal{U}$ 上的一个集合族 $\mathcal{S}$，目标是寻找一个两两不相交的集合的最大子族。该问题是最基础的 NP-难优化问题之一，已在各种计算场景中受到广泛研究。本文聚焦于其参数化复杂度，即参数化集合打包问题（Parameterized Set Packing, PSP）：给定 $r \in {\mathbb N}$，是否存在一个子族 $\mathcal{S}' \subseteq \mathcal{S}$（满足 $|\mathcal{S}'| = r$）使得 $\mathcal{S}'$ 中的集合两两不相交？遗憾的是，除非 $\mathsf{W[1] = FPT}$，该问题并非固定参数可解；事实上，除非指数时间假说（ETH）不成立，其求解需要 $|\mathcal{S}|^{\Omega(r)}$ 的"枚举"运行时间。本文从参数化复杂度视角探索集合打包的可解实例。我们将输入 $(\mathcal{U},\mathcal{S})$ 称为"紧致"的，若存在某个 $f(r) \ge r$ 使得 $|\mathcal{U}| = f(r)\cdot\Theta(\textsf{poly}( \log |\mathcal{S}|))$。紧致集合打包问题（Compact Set Packing）即给定紧致 PSP 实例。在此方向上，我们给出了 PSP 的"二分性"结果：当 $|\mathcal{U}| = f(r)\cdot o(\log |\mathcal{S}|)$ 时，PSP 属于 $\textsf{FPT}$；而当 $|\mathcal{U}| = r\cdot\Theta(\log (|\mathcal{S}|))$ 时，该问题是 $W[1]$-难的；此外，在 ETH 假设下，紧致 PSP 甚至不存在 $|\mathcal{S}|^{o(r/\log r)}$ 时间算法。尽管文献中的某些结果暗示了相关问题的紧致版本（如集合 $r$-覆盖与精确 $r$-覆盖）的困难性，但这些构造无法推广至紧致 PSP。本文的一项创新贡献在于识别并构造了一种称为"兼容相交集合系统对"（Compatible Intersecting Set System pair）的组件，该组件对于推导紧致 PSP 的困难性结果至关重要。