Balancing Notions of Equity: Trade-offs Between Fair Portfolio Sizes and Achievable Guarantees

TL;DR

This work studies the portfolio problem under fairness objectives, formalizing $\alpha$-approximate portfolios for classes of equity norms and quantifying the trade-off between portfolio size and approximation quality. It introduces OrderAndCount to obtain polylogarithmic portfolios for covering polyhedra and develops the primal-dual counting technique that underpins this efficiency, along with Sparsification to control dimensionality. It also presents IterativeOrdering, a framework yielding constant-factor simultaneous approximations for several symmetric monotonic norms across scheduling, routing, and clustering tasks, including new results for Completion-Times, Ordered-TSP, Ordered-Set-Cover, and $k$-Clustering, with practical polynomial-time variants. Together, these methods enable principled fairness-aware optimization by balancing the number of representative solutions against guarantees across diverse equity notions, with applications to workload distribution and critical facility planning.

Abstract

Motivated by fairness concerns, we study the `portfolio problem': given an optimization problem with set $D$ of feasible solutions, a class $\mathbf{C}$ of fairness objective functions on $D$, and an approximation factor $α\ge 1$, a set $X \subseteq D$ of feasible solutions is an $α$-approximate portfolio if for each objective $f \in \mathbf{C}$, there is an $α$-approximation for $f$ in $X$. Choosing the classes of top-$k$ norms, ordered norms, and symmetric monotonic norms as our equity objectives, we study the trade-off between the size $|X|$ of the portfolio and its approximation factor $α$ for various combinatorial problems. For the problem of scheduling identical jobs on unidentical machines, we characterize this trade-off for ordered norms and give an exponential improvement in size for symmetric monotonic norms over the general upper bound. We generalize this result as the OrderAndCount framework that obtains an exponential improvement in portfolio sizes for covering polyhedra with a constant number of constraints. Our framework is based on a novel primal-dual counting technique that may be of independent interest. We also introduce a general IterativeOrdering framework for simultaneous approximations or portfolios of size $1$ for symmetric monotonic norms, which generalizes and extends existing results for problems such as scheduling, $k$-clustering, set cover, and routing.

Figures (3)

• Figure 1: A qualitative plot to illustrate the trade-off between approximation $\alpha$ and the smallest portfolio size $|X_\alpha|$ for the Machine-Loads-Identical-Jobs problem for ordered norms. The worst-case lower bound $|X_\alpha| = \Omega\left(\frac{\log d}{\log \alpha + \log\log d}\right)$ is illustrated in red, and the upper bound $|X_\alpha| = O\left(\frac{\log d}{\log(\alpha/4)}\right)$ is illustrated in blue. The two bounds converge for $\alpha = \Omega(\log d)$.
• Figure 2: An example for makespan minimization with $2$ machines and $5$ jobs where $x^{\mathrm{OPT}}_1 < x^{\mathrm{OPT}}_2$ for optimal load vector $x^{\mathrm{OPT}}$.
• Figure 3: The vertex cover instance used in proof of Observation \ref{['thm: vertex-cover-lower-bound']}.

Theorems & Definitions (71)

• Theorem 1
• Theorem 2
• Definition 1: Portfolios
• Lemma 1: Portfolio composition
• Definition 2: Norm classes
• Lemma 2: hardy_inequalities_1952
• Lemma 3: goel_simultaneous_2006, Theorem 2.3
• Lemma 4
• Lemma 5
• Lemma 6: Dual ordered norms