On the parameterized complexity of Compact Set Packing

11/11/2021

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by   Ameet Gadekar, et al.

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The Set Packing problem is, given a collection of sets 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 r ∈ℕ, is there a collection 𝒮' ⊆𝒮: |𝒮'| = r such that the sets in 𝒮' are pairwise disjoint? Unfortunately, the problem is not fixed parameter tractable unless 𝖶[1] = 𝖥𝖯𝖳, and, in fact, an "enumeration" running time of |𝒮|^Ω(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,𝒮) is "compact" if |𝒰| = f(r)·Θ(( log |𝒮|)), for some f(r) ≥ 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 |𝒰| = f(r)· o(log |𝒮|), PSP is in , while for |𝒰| = r·Θ(log (|𝒮|)), the problem is W[1]-hard; moreover, assuming ETH, Compact PSP does not even admit |𝒮|^o(r/log r) time algorithm.