Picking a Representative Set of Solutions in Multiobjective Optimization: Axioms, Algorithms, and Experiments

TL;DR

This work tackles the problem of selecting a representative fixed-size slate from the Pareto front in multiobjective optimization by adopting a social-choice perspective. It analyzes existing quality measures (uniformity and coverage), introduces a new measure, directed coverage, and provides an axiomatic framework that clarifies when each measure behaves intuitively. The authors establish computational boundaries, proving NP-hardness for three or more objectives for several measures, while showing tractable cases (notably directed coverage with few objectives) and offering algorithmic insights across cardinal, ordinal, and approval objective types. Empirical results demonstrate that the choice of measure significantly shapes the selected slate and that directed coverage often yields competitive or superior performance in practice. The work thus provides a principled basis for selecting pruning measures and highlights directions for future theoretical and algorithmic advancements.

Abstract

Many real-world decision-making problems involve optimizing multiple objectives simultaneously, rendering the selection of the most preferred solution a non-trivial problem: All Pareto optimal solutions are viable candidates, and it is typically up to a decision maker to select one for implementation based on their subjective preferences. To reduce the cognitive load on the decision maker, previous work has introduced the Pareto pruning problem, where the goal is to compute a fixed-size subset of Pareto optimal solutions that best represent the full set, as evaluated by a given quality measure. Reframing Pareto pruning as a multiwinner voting problem, we conduct an axiomatic analysis of existing quality measures, uncovering several unintuitive behaviors. Motivated by these findings, we introduce a new measure, directed coverage. We also analyze the computational complexity of optimizing various quality measures, identifying previously unknown boundaries between tractable and intractable cases depending on the number and structure of the objectives. Finally, we present an experimental evaluation, demonstrating that the choice of quality measure has a decisive impact on the characteristics of the selected set of solutions and that our proposed measure performs competitively or even favorably across a range of settings.

Figures (11)

• Figure 1: General sketch of the reduction in \ref{['thm:kcenter_hard']}. Clauses, circuits, and junctions are highlighted.
• Figure 2: Left: A circuit $C_i$ representing a variable. We draw an edge between two points when they can cover each other. This circuit has $24 = 4 \cdot 6$ points. The only way to cover all points with $6$ points is to select either all $x$ (red), or all $z$ (blue). The background grid in $\mathbb{Z}^2$ is displayed. Right: A larger circuit with two arms extending to the right.
• Figure 3: A clause gadget, representing clause $W = (i_1 \lor i_2 \lor \lnot i_3)$. The three circuits $C_{i_1}, C_{i_2}, C_{i_3}$ are combined to form the clause gadget.
• Figure 4: A junction between two circuits $C_i$ and $C_j$. Points in the central square are highlighted by which choice of points they induce for the circuits. For example: selecting $p_{x, z}$ implies that all $x$ (red) in $C_i$ and all $z$ (blue) in $C_j$ must be selected.
• Figure 5: Two circuits running in parallel. In the top circuit a crease is highlighted. Note that the cyclic sequence $x,y,z,c$ is shifted forwards by exactly one position compared to the circuit without a crease.

Theorems & Definitions (53)

• theorem 1
• proposition 1
• theorem 2
• theorem 3
• proposition 2
• proposition 3
• proposition 4
• proposition 5
• proof
• proposition 6