On the Parameterized Complexity of Compact Set Packing

The Set Packing problem is, given a collection of sets \(\mathcal {S}\) over a ground set 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 parameter \(r \in {\mathbb N}\), 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 \(\textsf {W[1]} = \textsf {FPT} \), and, in fact, an “enumerative” 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 \(({U},\mathcal {S})\) is “compact” if \(|{U}| = f(r)\cdot \textsf {poly} ( \log |\mathcal {S}|)\), for some \(f(r) \ge r\). In the Compact PSP problem, we are given a compact instance of PSP. In this direction, we present a “dichotomy” result of PSP: When \(|{U}| = f(r)\cdot o(\log |\mathcal {S}|)\), PSP is in FPT, while for \(|{U}| = r\cdot \Theta (\log (|\mathcal {S}|))\), the problem is W[1]-hard; moreover, assuming ETH, Compact PSP does not admit \(|\mathcal {S}|^{o(r/\log r)}\) time algorithm even when \(|{U}| = r\cdot \Theta (\log (|\mathcal {S}|))\). 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. Finally, our framework can be extended to obtain improved running time lower bounds for Compact \(r\)-VectorSum.