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Geometric Preference Elicitation for Minimax Regret Optimization in Uncertainty Matroids

Aditya Sai Ellendula, Arun K Pujari, Vikas Kumar, Venkateswara Rao Kagita

TL;DR

This work addresses robust matroid optimization under weight uncertainty using minimax regret (MMR). It introduces a geometric, preference-elicitations-based framework that treats weights as a parametric polyhedral region $U(E)$ with $W=\mathbf{Y}\boldsymbol{\lambda}$ and $\boldsymbol{\lambda}$ in a simplex, enabling iterative refinement via pairwise element preferences. By updating only newly generated extreme points and their adjacencies, and by selecting queries based on most frequent disparity pairs, the method avoids per-iteration LP solves and achieves fast convergence toward $MMR=0$. Empirical results on four standard matroids show substantial reductions in the number of queries and computation time compared to prior approaches, highlighting the practical value of the geometric, polyhedral approach for robust matroid optimization.

Abstract

This paper presents an efficient preference elicitation framework for uncertain matroid optimization, where precise weight information is unavailable, but insights into possible weight values are accessible. The core innovation of our approach lies in its ability to systematically elicit user preferences, aligning the optimization process more closely with decision-makers' objectives. By incrementally querying preferences between pairs of elements, we iteratively refine the parametric uncertainty regions, leveraging the structural properties of matroids. Our method aims to achieve the exact optimum by reducing regret with a few elicitation rounds. Additionally, our approach avoids the computation of Minimax Regret and the use of Linear programming solvers at every iteration, unlike previous methods. Experimental results on four standard matroids demonstrate that our method reaches optimality more quickly and with fewer preference queries than existing techniques.

Geometric Preference Elicitation for Minimax Regret Optimization in Uncertainty Matroids

TL;DR

This work addresses robust matroid optimization under weight uncertainty using minimax regret (MMR). It introduces a geometric, preference-elicitations-based framework that treats weights as a parametric polyhedral region with and in a simplex, enabling iterative refinement via pairwise element preferences. By updating only newly generated extreme points and their adjacencies, and by selecting queries based on most frequent disparity pairs, the method avoids per-iteration LP solves and achieves fast convergence toward . Empirical results on four standard matroids show substantial reductions in the number of queries and computation time compared to prior approaches, highlighting the practical value of the geometric, polyhedral approach for robust matroid optimization.

Abstract

This paper presents an efficient preference elicitation framework for uncertain matroid optimization, where precise weight information is unavailable, but insights into possible weight values are accessible. The core innovation of our approach lies in its ability to systematically elicit user preferences, aligning the optimization process more closely with decision-makers' objectives. By incrementally querying preferences between pairs of elements, we iteratively refine the parametric uncertainty regions, leveraging the structural properties of matroids. Our method aims to achieve the exact optimum by reducing regret with a few elicitation rounds. Additionally, our approach avoids the computation of Minimax Regret and the use of Linear programming solvers at every iteration, unlike previous methods. Experimental results on four standard matroids demonstrate that our method reaches optimality more quickly and with fewer preference queries than existing techniques.

Paper Structure

This paper contains 20 sections, 6 theorems, 13 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Classical Optimization Problem on Matroid
  4. Uncertainty Matroid
  5. Optimization in Uncertainty Matroid
  6. Preference Elicitation
  7. Minimax Regret with Preference Elicitation
  8. Problem Formulation
  9. Previous Work and Our Contribution
  10. Polyhedral Combinatorics Approach
  11. Updating $C^r$
  12. Updating $\text{Ext}(C^r)$
  13. Updating the adjacency structure of $\text{Ext}(C^r)$
  14. Selecting Strategic Preference Query
  15. Upper Bound on MMR
  16. ...and 5 more sections

Key Result

Proposition 1

For extreme points $\boldsymbol{\sigma}^u$ and $\boldsymbol{\sigma}^v$ if $\boldsymbol{\sigma}^u, \boldsymbol{\sigma}^v \in \text{Ext}(C^{r-1})$ and $\boldsymbol{\sigma}^u, \boldsymbol{\sigma}^v\in C^r$ then $\boldsymbol{\sigma}^u, \boldsymbol{\sigma}^v\in \text{Ext}(C^r)$ and the adjacency relation

Figures (7)

  • Figure 1: Rate of Convergence of Pr-A with varying values of $n$ and $p$. (a) $n =10, ~p=10$. (b) $n =20, ~p=10$. (c) $n =30, ~p=10$. (d) $n =40, ~p=10$. (e) $n =50, ~p=10$.
  • Figure 2: Performance comparison for various matroids: (a) scheduling matroids, (b) graphic matroids, (c) uniform matroids, (d) partition Matroid.
  • Figure 3: Execution time comparison for various matroids: (a) scheduling matroids, (b) graphic matroids, (c) uniform matroids, (d) partition Matroid.
  • Figure 4: Trend of MMR Variation and Execution Time
  • Figure 5: Number of Extreme Points Generated over Iterations
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 1: Uncertainty Matroid
  • Definition 2: Uniformly Optimal Basis
  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 1 more