# Diversity of Solutions: An Exploration Through the Lens of Fixed-Parameter Tractability Theory

## TL;DR

This paper introduces a fixed-parameter tractability approach to generating diverse collections of solutions for combinatorial problems, providing a systematic framework that automatically adapts existing algorithms with polynomial dependence on diversity.

## Contribution

It develops a novel algorithmic framework that transforms standard dynamic programming algorithms into ones capable of producing diverse solutions efficiently.

## Key findings

- Framework automatically converts algorithms for diverse solutions
- Polynomial dependence on the diversity parameter
- Applicable to a wide range of combinatorial problems

## Abstract

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. First, we consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. We then present an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.