Counterfactual Explanations for Integer Optimization Problems

TL;DR

This work investigates counterfactual explanations for integer linear optimization problems, establishing that constructing CEs is $\Sigma_2^p$-hard in general while providing tractable algorithms for several restricted mutable-parameter scenarios. It formulates weak and strong CE notions, develops constraint-generation and bilevel-based algorithms for cases where only objective parameters, only constraint parameters, RHS, or all parameters may be modified, and empirically evaluates the methods on knapsack instances up to 40 items. The computational study demonstrates that optimal CEs can be obtained in a few hours for many instances, highlights the impact of lower bounds on runtime, and offers insights into how CEs illuminate the role of individual items in optimal solutions. Overall, the paper bridges inverse/bilevel optimization with explainability, delivering practical tools to understand and steer IP-based decisions and outlining directions for heuristics and exploitation of problem structure.

Abstract

Counterfactual explanations (CEs) offer a human-understandable way to explain decisions by identifying specific changes to the input parameters of a base or present model that would lead to a desired change in the outcome. For optimization models, CEs have primarily been studied in limited contexts and little research has been done on CEs for general integer optimization problems. In this work, we address this gap. We first show that the general problem of constructing a CE is $Σ_2^p$-complete even for binary integer programs with just a single mutable constraint. Second, we propose solution algorithms for several of the most tractable special cases: (i) mutable objective parameters, (ii) a single mutable constraint, (iii) mutable right-hand-side, and (iv) all input parameters can be modified. We evaluate our approach using classical knapsack problem instances, focusing on cases with mutable constraint parameters. Our results show that our methods are capable of finding optimal CEs for small instances involving up to 40 items within a few hours.

Figures (7)

• Figure 1: Counterfactual regions for a \ref{['eq:knapsack_problem']} with right-hand side $b=3$ and three item $i_1, i_2,i_3$, the latter two of which have a mutable weight. The favored solution space consists of all all solutions that contain $i_3$, i.e., $\mathcal{D}_1 = \{ x\in \{0,1\}^3: x_3 = 1 \}$. Green and orange regions represent strong and weak CEs, respectively. Each region is labeled with the items that are part of an optimal solution and the optimum solution value. The black O marks the parameter values $(\hat{A})$ of the present problem.
• Figure 2: Progress towards optimal solution for $480$ strong CE instances of size $10,20,30$ that were solved to optimality. Red are primal and blue are dual bounds. The $x$-axis is normalized with respect to the total number of values considered in $\mathcal{C}$. The $y$-axis is normalized around each instance's optimum.
• Figure 3: Progress towards solution for $480$ weak CE instances of size $10,20,30$ that were solved to optimality. Red are primal and blue are dual bounds. The $x$-axis is normalized with respect to the total number of values considered in $\mathcal{C}$. The $y$-axis is normalized around each instance's optimum.
• Figure 4: Boxplot comparison of runtimes for $480$ strong CE instances with sizes $10$, $20$, $30$, and $40$. Instances not solved to optimality in time are assigned $36000$s/$10$h.
• Figure 5: Boxplot comparison of runtimes for $480$ weak CE instances with sizes $10$, $20$, $30$, and $40$. Instances not solved to optimality in time are assigned $36000$s/$10$h.

Theorems & Definitions (33)

• Definition 1
• Definition 2
• Example 1
• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Theorem 3
• proof