A general framework for finding diverse solutions via network flow and its applications

Abstract: In this paper, we present a general framework for efficiently computing diverse solutions to combinatorial optimization problems. Given a problem instance, the goal is to find $k$ solutions that maximize a specified diversity measure; the sum of pairwise Hamming distances or the size of the union of the $k$ solutions. Our framework applies to problems satisfying two structural properties: (i) All solutions are of equal size and (ii) the family of all solutions can be represented by a surjection from the family of ideals of some finite poset. Under these conditions, we show that the problem of computing $k$ diverse solutions can be reduced to the minimum cost flow problem and the maximum $s$-$t$ flow problem. As applications, we demonstrate that both the unweighted minimum $s$-$t$ cut problem and the stable matching problem satisfy the requirements of our framework. By utilizing the recent advances in network flows algorithms, we improve the previously known time complexities of the diverse problems, which were based on submodular function minimization.

A Framework for Efficiently Finding Diverse Solutions in Combinatorial Optimization

The paper presents a comprehensive framework for efficiently computing diverse solutions to combinatorial optimization problems, specifically focusing on the $k$-diverse problem. The goal is to identify $k$ solutions that maximize a given diversity measure, such as the sum of pairwise Hamming distances or the size of the union of the solutions. This framework innovatively reduces the problem of finding diverse solutions to classical network flow problems, enhancing computational efficiency beyond existing methods based on submodular function minimization.

Key Features of the Framework

1. Structural Properties: The framework applies to combinatorial problems satisfying two key structural properties:
   • All solutions are of equal size.
   • The family of solutions can be represented as a surjection from the family of ideals of a finite poset.
2. Network Flow Reduction: The core of the approach is reducing the $k$-diverse problem to the minimum cost flow problem and the maximum $s$-$t$ flow (minimum $s$-$t$ cut) problem in network flow.
3. Algorithmic Efficiency: By utilizing advanced network flow algorithms, notably those developed by Brand et al., the proposed framework vastly improves the time complexity associated with diverse solution computation.

Application to Specific Problems

The framework's applicability is demonstrated through classical problems like the unweighted minimum $s$-$t$ cut problem and the stable matching problem. Both satisfy the framework's structural properties, allowing the use of network flow algorithms to improve previous results both theoretically and practically.

Implications and Future Work

The implications of this research are significant for developing efficient algorithms in combinatorial optimization, particularly concerning real-world problems where diverse solutions are essential. The paper indicates that its framework could potentially be adapted further, both by exploring other diversity measures like $d_{\textup{abs}$ and by extending to other domains within optimization and beyond.

The innovative reduction of combinatorial diversity problems to network flows presents promising avenues for future investigations, including potential adaptations in artificial intelligence, where diverse solutions can lead to more robust and adaptable models.

The paper is methodologically rigorous and contributes substantial theoretical advancements, providing the basis for faster and more efficient computation of diverse solutions in combinatorial landscapes.