Forgetfulness Can Make You Faster: An O*(8.097k)-Time Algorithm for Weighted 3-Set k-Packing

Abstract

In this paper, 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 \({\mathcal {S}}^{\prime } \subseteq {\mathcal {S}} \) of k disjoint sets and total weight at least W. We present a deterministic parameterized algorithm for this problem that runs in time O*(8.097k), 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*(12.155k) [SIDMA 2015], 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.