Obtaining Approximately Optimal and Diverse Solutions via Dispersion

Abstract

There has been a long-standing interest in computing diverse solutions to optimization problems. In 1995 J. Krarup posed the problem of finding $k$-edge disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer $k$, an approximation factor $c$, and an instance $I$ of an optimization problem, we aim to obtain a set of $k$ solutions to $I$ that a) are all $c$ approximately-optimal for $I$ and b) maximize the diversity of the $k$ solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. Given a metric on the space of solutions, we first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to optimized subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem. As applications of our result, we present polynomial time approximation algorithms for several problems such as diverse $c$-approximate \emph{spanning trees, maximum matchings, $s-t$ shortest paths, global min-cut, and minimum weight bases of a matroid}. In addition, we demonstrate that the \emph{$c$-approximate minimum spanning trees} can be converted into a set of $c$-approximate TSP tours where the tours are empirically diverse, advancing a step towards achieving diverse $c$-approximate TSP tours. We also explore the connection to the field of multiobjective optimization and show that the class of problems to which our result applies includes those for which the associated DUALRESTRICT problem defined by Papadimitriou and Yannakakis, and recently explored by Herzel et al. can be solved in polynomial time.