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

2026-08-13

AI Summary

This work addresses applications requiring multiple high-diversity feasible solutions—rather than a single optimum—such as those arising in fairness and robustness. It proposes an oracle-based fixed-parameter tractable framework for implicit set systems, which efficiently constructs tuples of solutions given an upper bound \(k\) on solution size, a desired number \(r\) of solutions, and a diversity threshold \(b\). The key innovation lies in introducing a class of “consistent diversity” objective functions that unifies and strengthens existing approaches, reducing the oracle query complexity from double to single exponential, specifically to \(2^{O(kr \log(kr))}\), while directly outputting the solution set. This method yields stronger parameterized tractability results for diverse problems in graph theory and matroid theory, subsuming and significantly improving upon prior work.

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 $2^{O(kr\log(kr))}$ 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.

Problem

• diverse solutions
• fixed-parameter tractability
• set-system problem
• diversity measures
• oracle-based framework

Innovation

• diverse solutions
• fixed-parameter tractability
• oracle-based framework
• consistently diverse
• exact empty-extension oracle