Diversity of solutions: An exploration through the lens of fixed-parameter tractability theory - Baste, Julien; Fellows, Michael R.; Jaffke, Lars; Masařík, Tomáš; de Oliveira Oliveira, Mateus; Philip, Geevarghese; Rosamond, Frances A. (Artificial intelligence, 2022)
•Parameterized analysis of a general notion of diversity of solutions that suits a large class of combinatorial problems.•Introduction of the notion of dynamic programming core.•Efficient dynamic cores for computing one solution yield efficient dynamic cores for computing a diverse set of solutions.•The notion of diversity of solutions is also compatible with certain notions of kernel. 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.