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

Abstract

Finding a \emph{single} best solution is the most common objective in combinatorial optimization problems. However, such a single solution may not be applicable to real-world problems as objective functions and constraints are only ``approximately'' formulated for original real-world problems. To solve this issue, finding \emph{multiple} solutions is a natural direction, and diversity of solutions is an important concept in this context. Unfortunately, finding diverse solutions is much harder than finding a single solution. To cope with the difficulty, we investigate the approximability of finding diverse solutions. As a main result, we propose a framework to design approximation algorithms for finding diverse solutions, which yields several outcomes including constant-factor approximation algorithms for finding diverse matchings in graphs and diverse common bases in two matroids and PTASes for finding diverse minimum cuts and interval schedulings.