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

Abstract

<jats:p>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. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is 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.</jats:p>

Diversity in Kemeny Rank Aggregation: A Parameterized Approach

“…The framework of diversity of solutions, under distinct notions of diversity, has found applications in several subfields of artificial intelligence, such as information search and retrieval [35,1], mixed integer programming [34,14,56], binary integer linear programming [37,60], constraint programming [41,42], SAT solving [55], recommender systems [2], routing problems [58], answer set programming [19], decision support systems [53,40], genetic algorithms [28,62], planning [6], and in many other fields. Recently, a general framework for addressing diversity of solutions from the perspective of parameterized complexity theory was developed [5]. This framework allows one to convert dynamic programming algorithms for finding an optimal solution for instances of a given problem into dynamic programming algorithms for finding a small set of diverse solutions.…”

“…A recent up-coming trend of research in artificial intelligence, called diversity of solutions [57,5,45,6,26], has focused on the development of notions of optimality that may be more appropriate in settings where subjectivity is essential. The idea is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another.…”

A Framework to Design Approximation Algorithms for Finding Diverse Solutions in Combinatorial Problems

“…As a way to solve this issue, algorithms are expected to find "diverse" solutions, and algorithms for finding "di-verse" solutions have received considerable attention in many fields such as artificial intelligence [Hanaka et al, 2021b,a;Baste et al, 2022], machine learning [Gillenwater et al, 2015], and data mining [Wang et al, 2013;Yuan et al, 2016]. There are many directions in the research on finding diverse solutions, depending on definitions of solutions and diversity measures.…”

“…From this proposition, several fixedparameter tractable (FPT) algorithms are developed. Baste et al gave algorithms for finding diverse solutions related to hitting sets [Baste et al, 2019] and those on bounded-treewidth graphs [Baste et al, 2022]. Hanaka et al [Hanaka et al, 2021b] proposed a framework to obtain FPT algorithms for finding diverse solutions in various combinatorial problems.…”

Computing Diverse Shortest Paths Efficiently: A Theoretical and Experimental Study

“…In particular, they showed that several diverse versions of combinatorial problems, such as VERTEX COVER, FEEDBACK VERTEX SET, and d-HITTING SET, are fixed-parameter tractable parameterized by the solution size plus the number of solutions (Baste et al 2019). Their follow-up research also discussed the fixed-parameter tractability of diverse versions of several combinatorial problems on bounded-treewidth graphs (Baste et al 2020).…”