On the diversity problems parameterized by the size of the solutions

Abstract

Diversity optimization seeks multiple high-quality solutions that are sufficiently different from one another, providing a richer representation of the solution space than a single optimum while avoiding the prohibitive cost of complete enumeration. In this work, we introduce the notion of natural diversity, a general condition that connects a combinatorial problem $Π$ and a diversity measure $\texttt{dist}$. We show that if a pair $(Π,\texttt{dist})$ is naturally diverse and we have in hand a completion algorithm that, from a partial solution of $Π$, can complete it into a solution of $Π$, then we can produce a parameterized algorithm solving the diversity variant of $Π$ with regard to the distance $\texttt{dist}$ when parameterized by the size of the solutions and the number of expected solutions. Furthermore, we show that several widely used diversity measures, including pairwise disjointness, Hamming distance, Jaccard distance, and the Otsuka-Ochiai coefficient (under both minimum and sum aggregation), satisfy the natural diversity property for problems in which all the solutions have the same size. Finally, we demonstrate the applicability of our framework by deriving fixed-parameter algorithms for diverse variants of Minimum Vertex Cover and Minimum Steiner Tree. Our results broaden the scope of parameterized diversity algorithms by accommodating natural solution-size parameterizations and a wider class of diversity measures.