Forgetfulness Can Make You Faster: An O * (8.097 k ) -time Algorithm for Weighted 3-set k -packing

Abstract

In this article, we study the classic Weighted 3-Set k -Packing problem: given a universe U , a family \( {\mathcal {S}}\) of subsets of size 3 of U , a weight function \(w : {\mathcal {S}} \rightarrow \mathbb {R}\) , \(W \in \mathbb {R}\) , and a parameter \(k \in \mathbb {N}\) , the objective is to decide if there is a subfamily \( {\mathcal {S}}\) ′ ⊆ \( {\mathcal {S}}\) of k disjoint sets and total weight at least W . We present a deterministic parameterized algorithm for this problem that runs in time O * (8.097 k ), where O * hides factors polynomial in the input size. This substantially improves upon the previously best deterministic algorithm for Weighted 3-Set k -Packing , which runs in time O * (12.155 k ) SIDMA [ 18 ], and was also the best deterministic algorithm for the unweighted version of this problem. Our algorithm is based on a novel application of the method of representative sets that might be of independent interest.