Max-Distance Sparsification for Diversification and Clustering

Abstract: Let $\mathcal{D}$ be a set family that is the solution domain of some combinatorial problem. The \emph{max-min diversification problem on $\mathcal{D}$} is the problem to select $k$ sets from $\mathcal{D}$ such that the Hamming distance between any two selected sets is at least $d$. FPT algorithms parameterized by $k+\ell $, where $\ell=\max_{D\in \mathcal{D}}|D|$, and $k+d$ have been actively studied recently for several specific domains. This paper provides unified algorithmic frameworks to solve this problem. Specifically, for each parameterization $k+\ell $ and $k+d$, we provide an FPT oracle algorithm for the max-min diversification problem using oracles related to $\mathcal{D}$. We then demonstrate that our frameworks provide the first FPT algorithms on several new domains $\mathcal{D}$, including the domain of $t$-linear matroid intersection, almost $2$-SAT, minimum edge $s,t$-flows, vertex sets of $s,t$-mincut, vertex sets of edge bipartization, and Steiner trees. We also demonstrate that our frameworks generalize most of the existing domain-specific tractability results. Our main technical breakthrough is introducing the notion of \emph{max-distance sparsifier} of $\mathcal{D}$, a domain on which the max-min diversification problem is equivalent to the same problem on the original domain $\mathcal{D}$. The core of our framework is to design FPT oracle algorithms that construct a constant-size max-distance sparsifier of $\mathcal{D}$. Using max-distance sparsifiers, we provide FPT algorithms for the max-min and max-sum diversification problems on $\mathcal{D}$, as well as $k$-center and $k$-sum-of-radii clustering problems on $\mathcal{D}$, which are also natural problems in the context of diversification and have their own interests.