Vector Retrieval with Similarity and Diversity: How Hard Is It?

TL;DR

This paper introduces a new NP-complete optimization problem for vector retrieval balancing similarity and diversity, along with a parameter-free heuristic that outperforms existing methods in scientific QA tasks.

Contribution

It formally defines the VRSD problem, proves its NP-completeness, and proposes a novel heuristic algorithm that improves retrieval performance without manual parameter tuning.

Findings

VRSD is NP-complete, indicating high computational complexity.

The proposed heuristic outperforms MMR and DPP baselines in experiments.

Results show improved retrieval quality on scientific QA datasets.

Abstract

Dense vector retrieval is essential for semantic queries within Natural Language Processing, particularly in knowledge-intensive applications like Retrieval-Augmented Generation (RAG). The ability to retrieve vectors that satisfy both similarity and diversity substantially enhances system performance. Although the Maximal Marginal Relevance (MMR) algorithm is widely used to balance these objectives, its reliance on a manually tuned parameter leads to optimization fluctuations and unpredictable retrieval results. Furthermore, there is a lack of sufficient theoretical analysis on the joint optimization of similarity and diversity in vector retrieval. To address these challenges, this paper introduces a novel approach that characterizes both constraints simultaneously by maximizing the similarity between the query vector and the sum of the selected candidate vectors. We formally define this optimization problem, Vectors Retrieval with Similarity and Diversity (VRSD) , and prove that it is NP-complete, establishing a rigorous theoretical bound on the inherent difficulty of this dual-objective retrieval. Subsequently, we present a parameter-free heuristic algorithm to solve VRSD. Extensive evaluations on multiple scientific QA datasets , incorporating both objective geometric metrics and LLM-simulated subjective assessments, demonstrate that our VRSD heuristic consistently outperforms established baselines, including MMR and Determinantal Point Processes (k-DPP).