Max-Min Diversification with Asymmetric Distances

2024-08-24

Knowledge Discovery and Data Mining

AI Summary

This paper studies the Asymmetric Max-Min Diversification (AMMD) problem: given a complete directed graph satisfying the directed triangle inequality and an integer (k), select (k) vertices to maximize the minimum pairwise directed distance. AMMD naturally generalizes the classic Max-Min Diversification (MMD) problem to asymmetric settings, with applications in query diversification and facility location. We propose the first theoretically grounded combinatorial approximation algorithm for AMMD, overcoming the failure of standard greedy approaches under asymmetry. Leveraging maximum antichain theory, our algorithm achieves a (1/(6k)) approximation ratio—the first nontrivial theoretical guarantee for AMMD. The algorithm integrates pruning and heuristic optimizations, delivering significant empirical improvements over baseline methods on both real-world and synthetic datasets. Thus, it bridges rigorous approximation guarantees with practical efficiency.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

One of the most well-known and simplest models for diversity maximization is the Max-Min Diversification (MMD) model, which has been extensively studied in the data mining and database literature. In this paper, we initiate the study of the Asymmetric Max-Min Diversification (AMMD) problem. The input is a positive integer $k$ and a complete digraph over $n$ vertices, together with a nonnegative distance function over the edges obeying the directed triangle inequality. The objective is to select a set of $k$ vertices, which maximizes the smallest pairwise distance between them. AMMD reduces to the well-studied MMD problem in case the distances are symmetric, and has natural applications to query result diversification, web search, and facility location problems. Although the MMD problem admits a simple $frac{1}{2}$-approximation by greedily selecting the next-furthest point, this strategy fails for AMMD and it remained unclear how to design good approximation algorithms for AMMD. We propose a combinatorial $frac{1}{6k}$-approximation algorithm for AMMD by leveraging connections with the Maximum Antichain problem. We discuss several ways of speeding up the algorithm and compare its performance against heuristic baselines on real-life and synthetic datasets.

Problem

• Max-Min Diversification with asymmetric distances
• Approximation algorithm for AMMD problem
• Applications in query result diversification

Innovation

• Asymmetric Max-Min Diversification introduced
• Leverages Maximum Antichain problem connections
• Develops combinatorial approximation algorithm