  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.
