Title: Improved Algorithms for Fair Matroid Submodular Maximization
Venue: NeurIPS 2025
Accepted: yes
Keywords: fairness

matroid

submodular function

approximation algorithms
Abstract: Submodular maximization subject to matroid constraints is a central problem with many applications in machine learning. As algorithms are increasingly used in decision-making over datapoints with sensitive attributes such as gender or race, it is becoming crucial to enforce fairness to avoid bias and discrimination. Recent work has addressed the challenge of developing efficient approximation algorithms for fair matroid submodular maximization. However, the best algorithms known so far are only guaranteed to satisfy a relaxed version of the fairness constraints that loses a factor 2, i.e., the problem may ask for $\ell$ elements with a given attribute, but the algorithm is only guaranteed to find $\lfloor \ell/2 \rfloor$. In particular, there is no provable guarantee when $\ell=1$, which corresponds to a key special case of perfect matching constraints.

In this work, we achieve a new trade-off via an algorithm that gets arbitrarily close to full fairness. Namely, for any constant $\varepsilon>0$, we give a constant-factor approximation to fair monotone matroid submodular maximization that in expectation loses only a factor $(1-\varepsilon)$ in the lower-bound fairness constraint. Our empirical evaluation on a standard suite of real-world datasets -- including clustering, recommendation, and coverage tasks -- demonstrates the practical effectiveness of our methods.


Peer reviews:
comments: SUMMARY:
This paper studies the problem of fair matroid monotone submodular maximization (FMMSM); the objective of this problem must take care of fairness constraints in addition to classic matroid optimization. The main results of this paper (Theorems 1.3 and 1.4) have impressive contributions in addition to earlier results, as their algorithms satisfy the fairness constraints to a tighter extent.

STRENGTHS AND WEAKNESSES:
Strengths:
1. This is a rigorous theoretical paper and the theoretical analysis is sound. It is also well-written in a clear, straight-forward style.
2. The improvements on tightness are impressive. It never violates the upper bound, and the lower bound violation factor of $1-\varepsilon$ is substantial, both for the randomized problem (Theorem 1.3) and deterministic solution (Theorem 1.4) with an additional constraint.
3. The techniques (pages 3-4) are novel to the best of my knowledge, and discussed intuitively and clearly in the section.

Weaknesses:
N/A. Although the tightness is still not optimal and constraints make the solution not universally applicable, any concrete progress on this difficult problem is meaningful.

QUESTIONS:
N/A

LIMITATIONS:
Yes, in Section 5.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies offline monotone submodular maximization subject to a matroid and a fairness constraint, described as follows:\n\nThere is a ground set $V$ partitioned into subsets $V_1, V_2, \\cdots, V_C$. \n\nFor each $c \\in [C]$, there are associated lower and upper bounds $l_c$ and $u_c$, respectively. \n\nWe are given oracle access to a monotone submodular function $f$ and a matroid $\\mathcal{I}$ defined on the ground set $V$. \n\nThe objective is to find a set $S$ that maximizes $f(S)$ subject to satisfying the following properties:\n\n$S$ is independent in the matroid ($S \\in \\mathcal{I}$).\n\nFor each  $c \\in [C]$: $l_c \\le |S \\cap V_c| \\le u_c$. \n\n\n\nIn this paper, the authors develop an algorithm that, for each $\\epsilon \\in (0, 1)$, outputs a solution that satisfies the matroid and upper bound constraints; for each $c \\in [C]$ satisfies a relaxed version of its lower bound constraint ($|S \\cap V_c| \\ge (1 - \\epsilon)l_c$) in expectation; and in expectation obtains a submodular value at least $0.499 \\cdot \\epsilon \\cdot OPT$.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nClarity and completeness in definitions and problem formulation.\n\nThe introduction is informative, it includes insightful discussions about the difficult special case of MSPM, clearly outlines existing results, their limitations and their contribution in comparison. \n\nAlthough, I could not fully read and verify the correctness of the proofs provided in the supplementary material, they seemed detailed. \n\nWeaknesses:\n\nThe approximation ratio of $0.499 \\cdot \\epsilon$ is quite weak.  \nThe fact that authors compare their solution with a stronger notion of OPT, get a weak approximation guarantee and then blame the weakness of their result on this choice is not justified for me. \n\nThe algorithms are not clearly outlined. \n\nThe running time of the algorithm is not mentioned in the paper. [I only noticed a computational complexity paragraph in the experimental section(why?) in the supplementary material. \n\nIt would have been better if you also included your final results when using the greedy algorithm instead of the local search one. (Right now, the algorithm used in experimental section mismatches the one used for theoretical results.)\n\nQUESTIONS:\n.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
The authors study the problem of maximizing a submodular function due to two types of constraints, both forming a matroid. The idea is that one type of constraints is standard, due to some structural requirements of the outcome. The second type of constraints comes from the fairness requirements: we want each protected group to be fairly represented in the selected set of elements. The problem has been already studied, and so far an algorithm was known that can violate the fairness constraints by a factor of 2, at the same time obtaining a half of the standard approximation for maximinzing a submodular function. The authors introduce a family of algorithms, parameterised by \epsilon, which trade objective function for the extent to which fairness constraints are satisfied.

STRENGTHS AND WEAKNESSES:
Strengths:
* this is an important and well motivated problem,
* the algorithm outperforms the currently know  solution with respect to both criterion on some datasets. 

Weaknesses:
* the originality is limited since some well-performing algorithm for the exact problem has been already known,
* while it is good to have a spectrum of algorithms that trade objective function for fairness constraints, it is still unclear how \epsilon should be set.

QUESTIONS:
* Does it make sense to combine your algorithm with the two-pass, and always pick the solution that achieves better objective?
* How to set the value of \epsilon in practice?
* Since two-pass and your algorithm use different ideas, is it possible to combine the two, and get an even better solution?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 2
DATE: 2025-07-03

comments: SUMMARY:
This paper presents an improved algorithm for the Fair Matroid Max-Sum Maximization (FMMSM) problem. For any \epsilon> 0, the proposed method returns an approximate solution that satisfies the matroid constraint and all fairness upper bound constraints, while ensuring that each fairness lower bound is met in expectation up to a (1 − \epsilon) factor. The paper is really about constrained submodular maximization with complicated constraints, such as partition matroid and lower fairness constraints.

STRENGTHS AND WEAKNESSES:
While the theoretical analysis is technically sound, the contribution feels fairly incremental given the existing literature, particularly [EFNTT23] and [ETNV24], which already address many of the core ideas and challenges in this setting. Additionally, I find the problem formulation difficult to justify, I am not really sure why why this setup captures a meaningful or realistic fairness scenario, and it comes across as somewhat ad hoc without a compelling motivating application.

The experimental results in Section 4, though easy to follow, also feel artificially constructed. The examples, such as exemplar-based clustering and graph coverage, are nearly identical to those commonly seen in the submodular maximization literature. This raises concerns about how well the proposed method generalizes beyond these stylized benchmarks. A broader or more practical evaluation would help establish the real-world relevance of the approach.

QUESTIONS:
I would really like to see a real application of this setup.

LIMITATIONS:
Even though I am not an expert in the fairness literature, I am less convinced that this formulation is the right one.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers fair submodular maximization subject to a matroid constraint. This is much more challenging than the simpler cardinality constraint version of the problem, where it has previously been shown that the fairness constraint plus cardinality constraint are equivalent to an instance of submodular maximization with a single matroid constraint. This problem has previously been considered before by [EFNTT23,ETNV24], but an issue with both of these results is that the proposed algorithms do not actually guarantee to meet the bottom fairness constraint (they either only meet them in expectation or by a constant factor).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- The problem is both interesting and challenging.\n- Algorithms and theoretical techniques appear novel to me.\n- Experimental results are included.\n\nWeaknesses\n- I am confused about how the results of this work improve on ETNV24, which from my understanding achieves the bounds on fairness in expectation. In particular, I have listed two questions below that would help clarify the contributions of this work relative to existing work. If this relationship can be better clairified I am willing to increase my score.\n- The experiments do not compare to ETNV24 since they say using the multilinear extension is impractical, but this comparison seems the most important from my perspective since it is the algorithm in the classic offline setting.\n\nQUESTIONS:\n- Is it possible to use concentration bounds and the algorithm of ETNV24 in order to meet the lower bounds with high probability? If so, what is the advantage of your algorithm in terms of meeting the fairness constraints?\n- Related to above question, could you describe the theoretical guarantees of your algorithm compared to ETNV24 in more depth?\n- What do you mean by that you are \"giving guarantees for every individual group rather than only in aggregate\" in contrast to related work?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-08"}