Bicriteria Algorithms for Submodular Cover with Partition and Fairness Constraints

TL;DR

This work studies Submodular Cover with Partition Constraints (SCP) and key variants where the ground set is partitioned and per-part budgets regulate feasibility. It introduces a unifying block-greedy framework that converts submodular maximization under partition constraints into SCP, yielding bicriteria guarantees for SCP, SCKP, and SCF via dual problems SMKP and SMF. Notably, the nonmonotone SCP case is tackled with convert-rand and a dedicated nonmono-bi routine achieving a $(1/e-\epsilon, 2/\epsilon)$-bicriteria, while the monotone knapsack-partition setting (SCKP) obtains a $(\frac{(1+\alpha)\ln(1/\epsilon)}{\ln 2}, 1-\epsilon)$-type guarantee through a block-knapsack approach and a converting theorem. For SCF, Block-Fair-Bi delivers a nearly optimal $(1-\varepsilon, \frac{\ln(1/\varepsilon)}{\ln 2})$-bicriteria, improving previous discrete methods. The paper supports these theoretical results with extensive experiments on real and synthetic data, showing improved efficiency, balanced budgets, and enhanced fairness across partitions, thereby enabling scalable, partition-aware submodular optimization in practical applications.

Abstract

In many submodular optimization applications, datasets are naturally partitioned into disjoint subsets. These scenarios give rise to submodular optimization problems with partition-based constraints, where the desired solution set should be in some sense balanced, fair, or resource-constrained across these partitions. While existing work on submodular cover largely overlooks this structure, we initiate a comprehensive study of the problem of Submodular Cover with Partition Constraints (SCP) and its key variants. Our main contributions are the development and analysis of scalable bicriteria approximation algorithms for these NP-hard optimization problems for both monotone and nonmonotone objectives. Notably, the algorithms proposed for the monotone case achieve optimal approximation guarantees while significantly reducing query complexity compared to existing methods. Finally, empirical evaluations on real-world and synthetic datasets further validate the efficiency and effectiveness of the proposed algorithms.

Figures (6)

• Figure 1: The experimental results of running the algorithms on the euall dataset, the twitch dataset, the Corel5k dataset, and the synthetic dataset. Budget: $\max_{i\in[N]}\frac{c(S\cap U_i)}{p_i}$. Cost: the size of the solution. Fairness difference: $(\max_c |S \cap U_c| - \min_c |S \cap U_c|) / |S|$.
• Figure 2: The experimental results of running the algorithms on the Corel5k dataset and the synthetic dataset. Samples: the number of queries. Budget: $\max_{i\in[N]}\frac{c(S\cap U_i)}{p_i}$.
• Figure 3: The experimental results for the SCKP problem on the Twitch dataset across different $\alpha$ values. Samples: the number of queries. Budget: $\max_{i\in[N]}\frac{c(S\cap U_i)}{p_i}$.
• Figure 4: The experimental results of running the algorithms on the Corel5k dataset and the synthetic dataset. Samples: the number of queries. Cost: the size of the returned solution. Budget: $\max_{i\in[N]}\frac{c(S\cap U_i)}{p_i}$. Fairness difference: $(\max_c |S \cap U_c| - \min_c |S \cap U_c|) / |S|$.
• Figure 5: The experimental results of running the algorithms on the ImageNet_50 dataset on the SCF problem. Samples: the number of queries. Cost: the size of the returned solution. Fairness difference: $(\max_c |S \cap U_c| - \min_c |S \cap U_c|) / |S|$.

Theorems & Definitions (42)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Corollary 2.6
• Definition 2.7
• Theorem 2.8
• proof
• proof