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

2025-04-24

AI Summary

This paper addresses the problem of computing $k$ diverse solutions in combinatorial optimization, where diversity is measured either by the sum of pairwise Hamming distances or by the cardinality of the union of solutions. Focusing on solution spaces of uniform size that can be modeled as families of ideals over a finite poset, we propose the first unified reduction of diverse solution search to network flow problems—specifically, minimum-cost flow and maximum $s$-$t$ flow. This approach departs fundamentally from the conventional paradigm relying on submodular function minimization. Our method yields the first polynomial-time algorithms for diverse solutions in classical settings including unweighted minimum $s$-$t$ cut and stable matching. The resulting algorithms exhibit significantly improved time complexity compared to existing submodular-optimization-based methods, while providing rigorous theoretical guarantees and broad applicability across combinatorial structures.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Social mining and social search on the Web

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.

Problem

• Develops a framework for computing diverse solutions in combinatorial optimization
• Maximizes diversity via Hamming distances or union size of solutions
• Applies to problems reducible to network flow with specific structural properties

Innovation

• Network flow framework for diverse solutions
• Reduces diverse solutions to flow problems
• Improves time complexity via flow algorithms