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Algorithmically Fair Maximization of Multiple Submodular Objective Functions

Georgios Amanatidis, Georgios Birmpas, Philip Lazos, Stefano Leonardi, Rebecca Reiffenhäuser

TL;DR

The paper addresses coordinating the constrained maximization of multiple submodular objectives over a shared ground set by introducing a Round-Robin protocol that enforces algorithmic fairness among $n$ agents.It demonstrates that simple greedy policies, within this protocol, yield strong approximation guarantees relative to $ ext{OPT}^-_i$ for monotone and non-monotone objectives under general $p_i$-system constraints, with improved bounds on $(\alpha,\beta)$-robust instances and via randomization.A key hardness result shows that beating greedy is NP-hard in general, underscoring the near-optimality of these simple policies, while empirical evaluation on influence maximization corroborates the practical relevance and fairness benefits of the randomized variant.The work connects to fair division by deriving EF1/FEF1-type guarantees under the proposed framework and opens several avenues for designing alternative protocols and fairness notions for broader objective classes.

Abstract

Constrained maximization of submodular functions poses a central problem in combinatorial optimization. In many realistic scenarios, a number of agents need to maximize multiple submodular objectives over the same ground set. We study such a setting, where the different solutions must be disjoint, and thus, questions of algorithmic fairness arise. Inspired from the fair division literature, we suggest a simple round-robin protocol, where agents are allowed to build their solutions one item at a time by taking turns. Unlike what is typical in fair division, however, the prime goal here is to provide a fair algorithmic environment; each agent is allowed to use any algorithm for constructing their respective solutions. We show that just by following simple greedy policies, agents have solid guarantees for both monotone and non-monotone objectives, and for combinatorial constraints as general as $p$-systems (which capture cardinality and matroid intersection constraints). In the monotone case, our results include approximate EF1-type guarantees and their implications in fair division may be of independent interest. Further, although following a greedy policy may not be optimal in general, we show that consistently performing better than that is computationally hard.

Algorithmically Fair Maximization of Multiple Submodular Objective Functions

TL;DR

The paper addresses coordinating the constrained maximization of multiple submodular objectives over a shared ground set by introducing a Round-Robin protocol that enforces algorithmic fairness among $n$ agents.It demonstrates that simple greedy policies, within this protocol, yield strong approximation guarantees relative to $ ext{OPT}^-_i$ for monotone and non-monotone objectives under general $p_i$-system constraints, with improved bounds on $(\alpha,\beta)$-robust instances and via randomization.A key hardness result shows that beating greedy is NP-hard in general, underscoring the near-optimality of these simple policies, while empirical evaluation on influence maximization corroborates the practical relevance and fairness benefits of the randomized variant.The work connects to fair division by deriving EF1/FEF1-type guarantees under the proposed framework and opens several avenues for designing alternative protocols and fairness notions for broader objective classes.

Abstract

Constrained maximization of submodular functions poses a central problem in combinatorial optimization. In many realistic scenarios, a number of agents need to maximize multiple submodular objectives over the same ground set. We study such a setting, where the different solutions must be disjoint, and thus, questions of algorithmic fairness arise. Inspired from the fair division literature, we suggest a simple round-robin protocol, where agents are allowed to build their solutions one item at a time by taking turns. Unlike what is typical in fair division, however, the prime goal here is to provide a fair algorithmic environment; each agent is allowed to use any algorithm for constructing their respective solutions. We show that just by following simple greedy policies, agents have solid guarantees for both monotone and non-monotone objectives, and for combinatorial constraints as general as -systems (which capture cardinality and matroid intersection constraints). In the monotone case, our results include approximate EF1-type guarantees and their implications in fair division may be of independent interest. Further, although following a greedy policy may not be optimal in general, we show that consistently performing better than that is computationally hard.
Paper Structure (12 sections, 20 theorems, 44 equations, 2 figures, 4 algorithms)

This paper contains 12 sections, 20 theorems, 44 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Round-Robin Framework
  4. The Effectiveness of Greedy Policies for Monotone Objectives
  5. Improved Guarantees for Robust Instances
  6. It is Hard to Optimize Over Others
  7. Dealing With Non-Monotone Objectives
  8. Improved Guarantees for Robust Instances
  9. Improving Algorithmic Fairness and Guarantees via Randomness
  10. Implications in Fair Division
  11. Empirical Evaluation
  12. Discussion and Open Questions

Key Result

Theorem 2.5

A function $f:2^M \rightarrow \mathbb{R}$ is submodular if and only if, for all $S, T \subseteq M$, Further, $f$ is monotone submodular if and only if, for all $S, T \subseteq M$,

Figures (2)

  • Figure 1: We have $500$ vertices with average degree $10$. From left to right we have the 'high', 'medium' and 'low' cases and cardinality constraint 10. While it is possible for some agents to dip below $\mathrm{OPT}_i / n$, i.e., into the $\mathrm{OPT}^-_i / n$ territory, the average is always higher and, as we progress towards less competition, typically all agents are considerably better off.
  • Figure 2: We have $200$ vertices with average degree $10$ and $5$ agents. The first set of results illustrates the 'high' case and the second one 'medium' one. The light blue colored lines correspond to each agent. The "elbow" at $40$ (which is more pronounced in the 'medium' case) is due to the items being exhausted before reaching the cardinality constraint, while the optimal does not suffer from this.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: NemhauserWF78
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['lem:mapping-monotone-general']}
  • Theorem 4.3
  • ...and 32 more