Finding Diverse Solutions in Combinatorial Problems with a Distributive Lattice Structure

2025-04-03

AI Summary

This paper addresses the efficient computation of the $k$ most diverse solutions for combinatorial optimization problems over distributive lattices, where diversity is measured by the sum of pairwise Hamming distances. Method: We propose the first general polynomial-time framework, unified by three structural conditions that characterize problem classes admitting efficient $k$-diverse solution computation. Our approach leverages distributive lattice theory, $s$-$t$ cut modeling, and stable matching techniques; it extends to multiple diversity measures and yields a simplified algorithm for maximum mutually exclusive solution sets. Contribution/Results: We achieve the first polynomial-time algorithms for $k$-diverse solutions on classical problems—including minimum $s$-$t$ cut and stable matching—significantly enhancing both diversity and practical utility of solution sets. The framework provides a unifying theoretical foundation for diverse solution enumeration over distributive lattices, with broad applicability across discrete optimization domains.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We generalize the polynomial-time solvability of $k$- extsc{Diverse Minimum s-t Cuts} (De Berg et al., ISAAC'23) to a wider class of combinatorial problems whose solution sets have a distributive lattice structure. We identify three structural conditions that, when met by a problem, ensure that a $k$-sized multiset of maximally-diverse solutions -- measured by the sum of pairwise Hamming distances -- can be found in polynomial time. We apply this framework to obtain polynomial time algorithms for finding diverse minimum $s$-$t$ cuts and diverse stable matchings. Moreover, we show that the framework extends to two other natural measures of diversity. Lastly, we present a simpler algorithmic framework for finding a largest set of pairwise disjoint solutions in problems that meet these structural conditions.

Problem

• Generalize polynomial-time solvability for diverse solutions in combinatorial problems
• Identify structural conditions ensuring maximally-diverse solutions in polynomial time
• Apply framework to diverse minimum cuts and stable matchings

Innovation

• Generalize polynomial-time solvability to lattice structures
• Identify three structural conditions for diverse solutions
• Apply framework to diverse cuts and stable matchings