Finding Diverse Trees, Paths, and More

TL;DR

This paper studies the computational complexity of finding diverse solutions in combinatorial problems, introducing polynomial-time algorithms for certain cases like diverse spanning trees, paths, and subgraphs, expanding the scope of tractable diversity problems.

Contribution

It proves that finding diverse spanning trees maximizing pairwise Hamming distances can be solved in polynomial time, a novel result for unbounded solution sizes.

Findings

Polynomial-time algorithm for diverse spanning trees.

First known polynomial case for unbounded diverse solutions.

Extends fixed-parameter tractability to new combinatorial structures.

Abstract

Mathematical modeling is a standard approach to solve many real-world problems and {\em diversity} of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies have been devoted to finding diverse solutions. Baste et al. (Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse solutions of combinatorial problems from the perspective of fixed-parameter tractability. They considered problems of finding $r$ solutions that maximize some diversity measures (the minimum or sum of the pairwise Hamming distances among them) and gave some fixed-parameter tractable algorithms for the diverse version of several well-known problems, such as {\sc Vertex Cover}, {\sc Feedback Vertex Set}, {\sc $d$-Hitting Set}, and problems on bounded-treewidth graphs. In this work, we investigate the (fixed-parameter) tractability of problems of finding diverse spanning trees, paths, and several subgraphs. In particular, we show that, given a graph $G$ and an integer $r$, the problem of computing $r$ spanning trees of $G$ maximizing the sum of the pairwise Hamming distances among them can be solved in polynomial time. To the best of the authors' knowledge, this is the first polynomial-time solvable case for finding diverse solutions of unbounded size.