On the parameterized complexity of Compact Set Packing

The Set Packing problem is, given a collection of sets S over a ground set U , to find 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 “enumerative” running time of |S| 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(poly(log |S|)), for some f(r) ≥ 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) ·o(log |S|), PSP is in FPT, while for |U| = r · Θ(log(|S|)), the problem is W[1]-hard; moreover, assuming ETH, Compact PSP does not even admit |S| log r) time algorithm.