From One Solution to Many: An Oracle-Based FPT Framework for Diverse Solutions under Generalized Diversity Measures

Abstract: The problem of computing \emph{diverse} solutions has recently emerged as an important area of study, motivated by applications in fairness, robustness, and security. Instead of returning a single feasible or optimal solution, the goal is to output a \emph{collection} of meaningfully different solutions, often measured by symmetric differences. Diverse variants have been studied using sparsification, network-flow reductions, and algebraic techniques. We investigate the fixed-parameter tractability of diverse variants of an implicit set-system problem. Given parameters $k$ and $r$ and a threshold $b$, the task is to compute $r$ feasible solutions, each of size at most $k$, whose diversity under a specified objective is at least $b$. Our main contribution is an oracle-based meta-theorem. We identify a broad class of objectives, called \emph{consistently diverse}, that includes several standard measures. Assuming an \emph{exact empty-extension oracle} given a forbidden set ${\sf Forb}$, which returns a feasible solution of a prescribed size avoiding ${\sf Forb}$ or reports that none exists, we obtain a fixed-parameter tractable algorithm parameterized by $k+r$. The algorithm makes at most $(2kr)^{kr} \cdot r$ oracle calls, and in each call the oracle parameter satisfies $s+|{\sf Forb}| \leq k+2kr$. Our framework unifies and strengthens previous oracle-based approaches. Compared with Kumabe's framework (ESA 2025), which gives a doubly exponential bound on the number of oracle calls, our approach achieves the single exponential bound $2<sup>{O(kr\log(kr))}$ and directly constructs the desired tuple of solutions. We recover fixed-parameter tractable algorithms for all problems covered by that framework, with improved oracle complexity, and obtain strong bounds for diverse variants of classical graph and matroid problems.