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Fair Set Cover

Mohsen Dehghankar, Rahul Raychaudhury, Stavros Sintos, Abolfazl Asudeh

TL;DR

The paper introduces Fair Set Cover (FSC) and its weighted variant, aiming to minimize the number of sets while enforcing demographic parity across groups. It develops naive, greedy, and fast LP-based randomized algorithms that achieve zero unfairness under certain assumptions and offer competitive approximation guarantees compared to classic Set Cover, with only modest increases in running time. The authors extend the framework to generalized and relaxed fairness notions (GFSC, GFWSC, ε-FGFSC) and demonstrate strong empirical results across real and synthetic datasets, showing fairness can be attained with minimal impact on output size and computation time. The work highlights FSC’s applicability to diverse societal problems, including team formation, licensing, and fair clustering, and situates it within the broader literature on algorithmic fairness in combinatorial optimization.

Abstract

The potential harms of algorithmic decisions have ignited algorithmic fairness as a central topic in computer science. One of the fundamental problems in computer science is Set Cover, which has numerous applications with societal impacts, such as assembling a small team of individuals that collectively satisfy a range of expertise requirements. However, despite its broad application spectrum and significant potential impact, set cover has yet to be studied through the lens of fairness. Therefore, in this paper, we introduce Fair Set Cover, which aims not only to cover with a minimum-size set but also to satisfy demographic parity in its selection of sets. To this end, we develop multiple versions of fair set cover, study their hardness, and devise efficient approximation algorithms for each variant. Notably, under certain assumptions, our algorithms always guarantee zero-unfairness, with only a small increase in the approximation ratio compared to regular set cover. Furthermore, our experiments on various data sets and across different settings confirm the negligible price of fairness, as (a) the output size increases only slightly (if any) and (b) the time to compute the output does not significantly increase.

Fair Set Cover

TL;DR

The paper introduces Fair Set Cover (FSC) and its weighted variant, aiming to minimize the number of sets while enforcing demographic parity across groups. It develops naive, greedy, and fast LP-based randomized algorithms that achieve zero unfairness under certain assumptions and offer competitive approximation guarantees compared to classic Set Cover, with only modest increases in running time. The authors extend the framework to generalized and relaxed fairness notions (GFSC, GFWSC, ε-FGFSC) and demonstrate strong empirical results across real and synthetic datasets, showing fairness can be attained with minimal impact on output size and computation time. The work highlights FSC’s applicability to diverse societal problems, including team formation, licensing, and fair clustering, and situates it within the broader literature on algorithmic fairness in combinatorial optimization.

Abstract

The potential harms of algorithmic decisions have ignited algorithmic fairness as a central topic in computer science. One of the fundamental problems in computer science is Set Cover, which has numerous applications with societal impacts, such as assembling a small team of individuals that collectively satisfy a range of expertise requirements. However, despite its broad application spectrum and significant potential impact, set cover has yet to be studied through the lens of fairness. Therefore, in this paper, we introduce Fair Set Cover, which aims not only to cover with a minimum-size set but also to satisfy demographic parity in its selection of sets. To this end, we develop multiple versions of fair set cover, study their hardness, and devise efficient approximation algorithms for each variant. Notably, under certain assumptions, our algorithms always guarantee zero-unfairness, with only a small increase in the approximation ratio compared to regular set cover. Furthermore, our experiments on various data sets and across different settings confirm the negligible price of fairness, as (a) the output size increases only slightly (if any) and (b) the time to compute the output does not significantly increase.
Paper Structure (48 sections, 20 theorems, 11 equations, 4 figures)

This paper contains 48 sections, 20 theorems, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Application Demonstration
  3. Equal Base-rate in representative data sets
  4. Business license distribution
  5. Fair Clustering
  6. Preliminaries
  7. Fairness definition
  8. Fair Set Cover
  9. Naive Algorithm
  10. Greedy Algorithm
  11. Analysis
  12. Proof:
  13. Faster algorithms
  14. $\frac{1}{2}$- approximation algorithm
  15. Proof:
  16. ...and 33 more sections

Key Result

Lemma 1

There exists a $(k(\ln n +1))$-approximation algorithm for the FSC problem that runs in $O(m\cdot k \cdot n)$ time.

Figures (4)

  • Figure 1: An illustration of a set system for the Fair Set Cover (FSC) problem, consisting of 16 points and 7 sets: 4 red sets and 3 blue sets. The Greedy Fair Set Cover algorithm (§ \ref{['subsec:greedy']}) selects $(S_1, S_5)$ as the first pair of red and blue sets, followed by the pairs $(S_3, S_6)$ and $(S_4, S_1)$. The standard greedy algorithm selects the sets $S_1$, $S_2$, $S_3$, $S_4$, and $S_7$, resulting in 4 red sets and 1 blue set, highlighting its lack of fairness.
  • Figure 2: Resume Skills: Comparing (a) the size of output cover, (b) fairness (1 is the max fairness), and (c) running time for various ground set sizes $N$. The results are averaged over 20 samples for each $N$.
  • Figure 3: Comparing (a) the size of output cover, (b) fairness based on ratio-parity, and (c) running time (averaged over 20 samples) on Adult dataset. The x-axis shows different values of $M$ (size of family of subsets or rows in the sampled dataset).
  • Figure 4: Comparing (a) the size of output cover, (b) fairness based on ratio-parity, and (c) running time (averaged over 5 samples) on POPSIM dataset. The x-axis shows different values of $N$ (size of the ground set).

Theorems & Definitions (22)

  • Definition 1: Generalized Fair Weighted Set Cover problem (GFWSC)
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Definition 2: max $k$-color cover
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Theorem 3
  • Theorem 4
  • ...and 12 more