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

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.

Figures (52)

• Figure 1: Overview of crowd-sourced data enrichment using WSMC-BU.
• Figure 2: Sample of experts with the same skill set. In this example, $H = \{Legal, Medical\}$, and the rate is considered as the weight. The figure shows the piecewise linear function $\hat{f}_H(x)$, based on the shown table.
• Figure 3: Approximation of the $\mathscr{f}(x) = x^2 + 1$ curve by a piecewise linear function $\mathscr{g}(\cdot)$, generated by the described algorithm, where $\epsilon = 3$. The function $\mathscr{g}(\cdot)$ approximates the values of the function $\mathscr{f}(\cdot)$ within a factor of $4$.
• Figure 4: The function $\hat{f}_{H_1}(x)$ is a piecewise linear function consisting of three solid segments. For $\varepsilon = 7$, the function $\hat{g}_{H_1}(x)$ is a piecewise linear function with two segments: for $x \in [0,1]$, $\hat{g}_{H_1}(x) = \hat{f}_{H_1}(x)$, while for $x \in [1,3]$, $\hat{g}_{H_1}(x)$ is given by the dashed linear segment connecting $(1,1)$ and $(3,8)$.
• Figure 5: Impact of varying number of items $\ell$ on the solution’s total weight, Census

Theorems & Definitions (11)

• Definition 1: Weighted Set Multi-Cover problem with Bounded Universe --- WSMC-BU
• Theorem 1
• Lemma 1
• Theorem 2
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 3
• Lemma 5
• Lemma 6