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Finding a \emph{single} best solution is the most common objective incombinatorial optimization problems. However, such a single solution may not beapplicable to real-world problems as objective functions and constraints areonly "approximately" formulated for original real-world problems. To solve thisissue, finding \emph{multiple} solutions is a natural direction, and diversityof solutions is an important concept in this context. Unfortunately, findingdiverse solutions is much harder than finding a single solution. To cope withdifficulty, we investigate the approximability of finding diverse solutions. Asa main result, we propose a framework to design approximation algorithms forfinding diverse solutions, which yields several outcomes includingconstant-factor approximation algorithms for finding diverse matchings ingraphs and diverse common bases in two matroids and PTASes for finding diverseminimum cuts and interval schedulings.

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