[Paper Review] Maximizing coverage while ensuring fairness: a tale of conflicting objective

Abolfazl Asudeh, Tanya Berger‐Wolf+2 more|arXiv (Cornell University)|Jul 15, 2020

Mathematical Approximation and Integration2 citations

TL;DR

This paper introduces a combinatorial optimization framework that simultaneously maximizes coverage and ensures fairness in set selection by incorporating coloring constraints to balance the distribution of colored elements across selected sets. It presents randomized and deterministic approximation algorithms that achieve at least 63% of the optimal coverage while maintaining color proportionality within a constant factor, even under conflicting objectives.

ABSTRACT

Ensuring fairness in computational problems has emerged as a $key$ topic during recent years, buoyed by considerations for equitable resource distributions and social justice. It $is$ possible to incorporate fairness in computational problems from several perspectives, such as using optimization, game-theoretic or machine learning frameworks. In this paper we address the problem of incorporation of fairness from a $combinatorial$ $optimization$ perspective. We formulate a combinatorial optimization framework, suitable for analysis by researchers in approximation algorithms and related areas, that incorporates fairness in maximum coverage problems as an interplay between $two$ conflicting objectives. Fairness is imposed in coverage by using coloring constraints that $minimizes$ the discrepancies between number of elements of different colors covered by selected sets; this is in contrast to the usual discrepancy minimization problems studied extensively in the literature where (usually two) colors are $not$ given $a$ $priori$ but need to be selected to minimize the maximum color discrepancy of $each$ individual set. Our main results are a set of randomized and deterministic approximation algorithms that attempts to $simultaneously$ approximate both fairness and coverage in this framework.

Motivation & Objective

• To formulate a combinatorial optimization framework that balances maximum coverage and fairness in set selection.
• To address the conflict between maximizing coverage and ensuring equitable distribution of colored elements across selected sets.
• To develop approximation algorithms that simultaneously approximate both objectives under coloring constraints.
• To generalize fairness beyond equal proportions to arbitrary pre-specified color ratios in the solution.
• To provide theoretical guarantees on coverage and fairness trade-offs in the presence of conflicting objectives.

Proposed method

• Formulates the Fair Maximum Coverage (FMC) problem with χ colors, where selected sets must maintain a specified proportion of each color.
• Imposes fairness via coloring constraints that minimize discrepancies between the number of elements of different colors covered.
• Uses a randomized rounding approach based on LP relaxation with a gap of factor f in coloring constraints.
• Applies iterative rounding and dynamic programming over discretized lattice cells to approximate the solution within a (1−ε) factor of optimal coverage.
• Employs probabilistic analysis using Markov's inequality and union bounds to bound the deviation of color proportions from target ratios.
• Introduces a discretization scheme using a dilated integer lattice δ·Z^d to reduce the search space while preserving approximation guarantees.

Experimental results

Research questions

• RQ1 Can we simultaneously maximize coverage and ensure fairness in set selection when the two objectives conflict?
• RQ2 What approximation guarantees can be achieved for coverage and color proportionality in the FMC problem?
• RQ3 How can fairness be enforced via coloring constraints that balance the number of elements of each color covered?
• RQ4 Is it possible to design efficient approximation algorithms for FMC that maintain both high coverage and bounded color discrepancy?
• RQ5 Can the framework be extended to non-uniform target color proportions and general submodular objectives?

Key findings

• The proposed randomized algorithm achieves at least 63% of the optimal coverage on average while maintaining color ratios within a constant factor with high probability.
• For any constant number of colors χ, the algorithm guarantees a (1−ε)-approximation to the optimal coverage with high probability after O(log n) repetitions.
• The framework ensures that for every pair of colors, the ratio of covered elements of those colors is O(1), even under conflicting objectives.
• The LP relaxation incurs a factor-f gap in coloring constraints, which remains an open challenge to close using linear programming.
• The method achieves a (1−ε) approximation to the optimal coverage while maintaining fairness, with runtime bounded by O((∆/L)^d 2(L/δ)^d (2^{O(d)}k/ε)^d).
• The approach generalizes to arbitrary pre-specified color proportions q1, q2, ..., qχ, not just equal distributions.