On the Parameterized Complexity of Compact Set Packing

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