Weighted Set Multi-Cover on Bounded Universe and Applications in Package Recommendation

2026-03-12

AI Summary

This work addresses the problem of efficient and cost-effective subset selection from large candidate sets in data-driven scenarios by studying the Weighted Set Multi-Cover problem under a bounded universe (WSMC-BU). The paper presents the first polynomial-time exact algorithm for this problem and introduces two efficient approximation algorithms achieving approximation ratios of 2 and (2+ε), respectively, significantly outperforming conventional greedy and LP-rounding approaches. Leveraging dynamic programming, linear programming, and rounding techniques, the proposed methods consistently surpass existing baselines in both solution quality and computational efficiency across real-world and synthetic datasets, demonstrating strong applicability to large-scale data selection tasks such as recommendation systems and decision support.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The weighted set multi-cover problem is a fundamental generalization of set cover that arises in data-driven applications where one must select a small, low-cost subset from a large collection of candidates under coverage constraints. In data management settings, such problems arise naturally either as expressive database queries or as post-processing steps over query results, for example, when selecting representative or diverse subsets from large relations returned by database queries for decision support, recommendation, fairness-aware data selection, or crowd-sourcing. While the general weighted set multi-cover problem is NP-complete, many practical workloads involve a \emph{bounded universe} of items that must be covered, leading to the Weighted Set Multi-Cover with Bounded Universe (WSMC-BU) problem, where the universe size is constant. In this paper, we develop exact and approximation algorithms for WSMC-BU. We first discuss a dynamic programming algorithm that solves WSMC-BU exactly in $O(n^{\ell+1})$ time, where $n$ is the number of input sets and $\ell=O(1)$ is the universe size. We then present a $2$-approximation algorithm based on linear programming and rounding, running in $O(\mathcal{L}(n))$ time, where $\mathcal{L}(n)$ denotes the complexity of solving a linear program with $O(n)$ variables. To further improve efficiency for large datasets, we propose a faster $(2+\varepsilon)$-approximation algorithm with running time $O(n \log n + \mathcal{L}(\log W))$, where $W$ is the ratio of the total weight to the minimum weight, and $\varepsilon$ is an arbitrary constant specified by the user. Extensive experiments on real and synthetic datasets demonstrate that our methods consistently outperform greedy and standard LP-rounding baselines in both solution quality and runtime, making them suitable for data-intensive selection tasks over large query outputs.

Problem

• Weighted Set Multi-Cover
• Bounded Universe
• Package Recommendation
• Coverage Constraints
• Data-Driven Selection

Innovation

• Weighted Set Multi-Cover
• Bounded Universe
• Dynamic Programming
• Linear Programming Rounding
• Approximation Algorithm