Original: Counterfactual Explanations for Optimization-Based Decisions in the Context of the GDPR
Revised: Counterfactual Explanations for Optimization-Based Decisions in the Context of the GDPR

Original: Abstract
Revised: Abstract

Original: The General Data Protection Regulations (GDPR) entitle individuals to explanations for automated decisions.
Revised: The General Data Protection Regulations (GDPR) entitle individuals to explanations for automated decisions.

Original: The form, comprehensibility, and even existence of such explanations remain open problems, investigated as part of explainable AI.
Revised: The form, comprehensibility, and even existence of such explanations remain open problems, investigated as part of explainable AI.

Original: We adopt the approach of counterfactual explanations and apply it to decisions made by declarative optimization models.
Revised: We adopt the approach of counterfactual explanations and apply it to decisions made by declarative optimization models.

Original: We argue that inverse combinatorial optimization is particularly suited for counterfactual explanations but that the computational difﬁculties and relatively nascent literature make its application a challenge.
Revised: We argue that inverse combinatorial optimization is particularly suited for counterfactual explanations but that the computational difﬁculties and relatively nascent literature make its application a challenge.

Original: To make progress, we address the case of counterfactual explanations that isolate the minimal differences for an individual.
Revised: To make progress, we address the case of counterfactual explanations that isolate the minimal differences for an individual.

Original: We show that under two common optimization functions, full inverse optimization is unnecessary.
Revised: We show that under two common optimization functions, full inverse optimization is unnecessary.

Original: In particular, we show that for functions of the form of the sum of weighted binary variables, which includes frameworks such as weighted MaxSAT, a solution can be found by solving a slightly modiﬁed version of the original optimization model.
Revised: In particular, we show that for functions of the form of the sum of weighted binary variables, which includes frameworks such as weighted MaxSAT, a solution can be found by solving a slightly modiﬁed version of the original optimization model.

Original: In contrast, the sum of weighted integer variables can be solved with a binary search over a series of modiﬁcations to the original model.
Revised: In contrast, the sum of weighted integer variables can be solved with a binary search over a series of modiﬁcations to the original model.

Original:  Introduction
Revised:  Introduction

Original: There has been substantial development of algorithmic systems that make recommendations or decisions about highimpact issues such as healthcare, homelessness, recidivism and parole, and social services [Eubanks, 2018; Azizi et al. , 2018].
Revised: There has been substantial development of algorithmic systems that make recommendations or decisions about highimpact issues such as healthcare, homelessness, recidivism and parole, and social services [Eubanks, 2018; Azizi et al. , 2018].

Original: The General Data Protection Regulations (GDPR)
Revised: The General Data Protection Regulations (GDPR)

Original: [Parliament and Council of the European Union, 2016] declare that individuals subjected to an automated decision have a right to an explanation of the decision.
Revised: [Parliament and Council of the European Union, 2016] declare that individuals subjected to an automated decision have a right to an explanation of the decision.

Original: While the legal scope of such a right is an active topic of debate, there is widespread consensus on the importance of implementing such a concept [Selbst and Powles, 2018; Wachter et al. , 2017a ] and research on explanations is a central direction in the ﬁeld of explainable AI (XAI)
Revised: While the legal scope of such a right is an active topic of debate, there is widespread consensus on the importance of implementing such a concept [Selbst and Powles, 2018; Wachter et al. , 2017a ] and research on explanations is a central direction in the ﬁeld of explainable AI (XAI)

Original: [Guidotti et al. , 2018].
Revised: [Guidotti et al. , 2018].

Original: For example, consider an optimization-based automated medical scheduling system that assigns an individual to an appointment three months in the future.
Revised: For example, consider an optimization-based automated medical scheduling system that assigns an individual to an appointment three months in the future.

Original: The individual may ask a question such as “Why can’t I be seen next month?”
Revised: The individual may ask a question such as “Why can’t I be seen next month?”

Original: In this paper, we adopt the approach of counterfactual explanations [Wachter et al. , 2017b] by deriving the minimal change to the world that would have resulted in the alternative indicated by the questioner.
Revised: In this paper, we adopt the approach of counterfactual explanations [Wachter et al. , 2017b] by deriving the minimal change to the world that would have resulted in the alternative indicated by the questioner.

Original: We develop an approach to generate counterfactual explanations for optimization-based decisions using a generalization of inverse combinatorial optimization [ Heuberger, 2004 ] .
Revised: We develop an approach to generate counterfactual explanations for optimization-based decisions using a generalization of inverse combinatorial optimization [ Heuberger, 2004 ] .

Original: Under some restrictions, we show how to formulate the inverse problem and, further, how to solve it under two common objective functions: the sum of weighted binary variables and the sum of weighted integer variables.
Revised: Under some restrictions, we show how to formulate the inverse problem and, further, how to solve it under two common objective functions: the sum of weighted binary variables and the sum of weighted integer variables.

Original: Our solution techniques demonstrate that the computational effort to produce an explanation is equivalent to that of the original optimization problem in the former case and a log-factor more expensive than that in the latter case.
Revised: Our solution techniques demonstrate that the computational effort to produce an explanation is equivalent to that of the original optimization problem in the former case and a log-factor more expensive than that in the latter case.

Original: In the next section, we provide background on counterfactual explanations and inverse optimization before turning to our formal problem deﬁnition in Section 3.
Revised: In the next section, we provide background on counterfactual explanations and inverse optimization before turning to our formal problem deﬁnition in Section 3.

Original: Sections 4 and 5 present our contributions w.r.t.
Revised: Sections 4 and 5 present our contributions w.r.t.

Original: the two objective functions and we then discuss the limitation of our work and relate it to existing work in the ﬁnal sections.
Revised: the two objective functions and we then discuss the limitation of our work and relate it to existing work in the ﬁnal sections.

Original:  Background 2. Counterfactual Explanations in XAI
Revised:  Background 2. Counterfactual Explanations in XAI

Original: A large part of XAI, especially in machine learning (ML), has focused on explaining decision by making the internal logic of an algorithm more understandable
Revised: A large part of XAI, especially in machine learning (ML), has focused on explaining decision by making the internal logic of an algorithm more understandable

Original: [ Burrell, 2016 ] .
Revised: [ Burrell, 2016 ] .

Original: A common approach is to generate local approximations of opaque models (e.g., neural networks) with interpretable models (e.g., linear models or decision trees) from which the impact of features on decisions is easier to comprehend [Ribeiro et al. , 2016; Simonyan et al. , 2013].
Revised: A common approach is to generate local approximations of opaque models (e.g., neural networks) with interpretable models (e.g., linear models or decision trees) from which the impact of features on decisions is easier to comprehend [Ribeiro et al. , 2016; Simonyan et al. , 2013].

Original: This approach has been criticized as insufﬁcient in generating the explanations envisioned by the GDPR, since it does not explicitly describe the circumstances under which a different decision would have been reached.
Revised: This approach has been criticized as insufﬁcient in generating the explanations envisioned by the GDPR, since it does not explicitly describe the circumstances under which a different decision would have been reached.

Original: Such a lack makes it difﬁcult to contest the decision and to act to change it, two key goals for user-centered explanations [Wachter et al. , 2017b].
Revised: Such a lack makes it difﬁcult to contest the decision and to act to change it, two key goals for user-centered explanations [Wachter et al. , 2017b].

Original: A different line of research in XAI addresses these issues with counterfactual explanations [Wachter et al. , 2017b] which have the following form: given that a set of facts resulted in decision x , decision ¯ x would have been made had these facts been changed in a particular way and the rest of the world had remained unchanged.
Revised: A different line of research in XAI addresses these issues with counterfactual explanations [Wachter et al. , 2017b] which have the following form: given that a set of facts resulted in decision x , decision ¯ x would have been made had these facts been changed in a particular way and the rest of the world had remained unchanged.

Original: Using our appointment scheduling example: the individual’s medical acuity level was 3 and the appointment was scheduled in three months; had the acuity level been 5 (i.e., more acute), the appointment would have been scheduled next month.
Revised: Using our appointment scheduling example: the individual’s medical acuity level was 3 and the appointment was scheduled in three months; had the acuity level been 5 (i.e., more acute), the appointment would have been scheduled next month.

Original: More precisely, a counterfactual is situation that is “contrary to the facts”: an imagined alternative to a past event.
Revised: More precisely, a counterfactual is situation that is “contrary to the facts”: an imagined alternative to a past event.

Original: Counterfactual reasoning invokes a comparison between the imagined event and the factual event [Epstude and Roese, 2008].
Revised: Counterfactual reasoning invokes a comparison between the imagined event and the factual event [Epstude and Roese, 2008].

Original: By this deﬁnition, a counterfactual explanation involves two counterfactuals.
Revised: By this deﬁnition, a counterfactual explanation involves two counterfactuals.

Original: First there is a counterfactual question , “Why decision x and not ¯ x ?”, with the imagined decision ¯ x being the ﬁrst counterfactual.
Revised: First there is a counterfactual question , “Why decision x and not ¯ x ?”, with the imagined decision ¯ x being the ﬁrst counterfactual.

Original: The explanation then provides a set of hypothetical facts, different from the original facts, as the second counterfactual.
Revised: The explanation then provides a set of hypothetical facts, different from the original facts, as the second counterfactual.

Original: To distinguish between the two counterfactuals, we refer to decision ¯ x as the foil and the hypothetical facts as the counterfactual values , using similar terminology to Miller [2019].
Revised: To distinguish between the two counterfactuals, we refer to decision ¯ x as the foil and the hypothetical facts as the counterfactual values , using similar terminology to Miller [2019].

Original: Further, it is typically preferable for such an explanation to minimally perturb the facts that led to the original decision.
Revised: Further, it is typically preferable for such an explanation to minimally perturb the facts that led to the original decision.

Original: As a counterfactual explanation explicitly addresses what needs to happen for a decision to be different and avoids the internal logic of the algorithm, such an explanation is better suited for explanations for users [ Wachter et al. , 2017b ] .
Revised: As a counterfactual explanation explicitly addresses what needs to happen for a decision to be different and avoids the internal logic of the algorithm, such an explanation is better suited for explanations for users [ Wachter et al. , 2017b ] .

Original: 2. Inverse Combinatorial Optimization
Revised: 2. Inverse Combinatorial Optimization

Original: Standard (or, forward ) optimization searches for an optimal assignment of variables given a model consisting of an objective function, constraints, and parameters.
Revised: Standard (or, forward ) optimization searches for an optimal assignment of variables given a model consisting of an objective function, constraints, and parameters.

Original: Inverse optimization, in contrast, starts with a model, initial parameter values, and a known solution, and ﬁnds the minimal change to the parameter values that would make the solution optimal [ Demange and Monnot, 2014; Heuberger, 2004 ] .
Revised: Inverse optimization, in contrast, starts with a model, initial parameter values, and a known solution, and ﬁnds the minimal change to the parameter values that would make the solution optimal [ Demange and Monnot, 2014; Heuberger, 2004 ] .

Original: To formulate the forward and inverse problems, let x ∈
Revised: To formulate the forward and inverse problems, let x ∈

Original: X ⊆ R n be a decision vector subject to a feasible set X , c ∈ C ⊆ R n be an initial parameter vector subject to feasible set C , and f : C ×
Revised: X ⊆ R n be a decision vector subject to a feasible set X , c ∈ C ⊆ R n be an initial parameter vector subject to feasible set C , and f : C ×

Original: X
Revised: X

Original: → R be the objective function.
Revised: → R be the objective function.

Original: It is assumed that the components of c are present only in the objective, not in any constraints.
Revised: It is assumed that the components of c are present only in the objective, not in any constraints.

Original: The minimization variant of the forward problem (cid:104) X, c, f (cid:105) is to ﬁnd an optimal x ∗ such that f ( c, x ∗ )
Revised: The minimization variant of the forward problem (cid:104) X, c, f (cid:105) is to ﬁnd an optimal x ∗ such that f ( c, x ∗ )

Original: =
Revised: =

Original: min { f ( c, x ) : x ∈ X } .
Revised: min { f ( c, x ) : x ∈ X } .

Original: In the inverse problem, deﬁned w.r.t.
Revised: In the inverse problem, deﬁned w.r.t.

Original: a norm (cid:107)·(cid:107) , we are given a solution ˆ x ∈ X that we wish to make optimal by minimally modifying the parameter vector.
Revised: a norm (cid:107)·(cid:107) , we are given a solution ˆ x ∈ X that we wish to make optimal by minimally modifying the parameter vector.

Original: Let ¯ c ∈ C be the modiﬁed parameter vector.
Revised: Let ¯ c ∈ C be the modiﬁed parameter vector.

Original: The inverse optimization problem is then
Revised: The inverse optimization problem is then

Original: [Equation]
Revised: [Equation]

Original: In partial inverse optimization [Demange and Monnot, 2014], we are no longer given a full solution ˆ x , but instead a partial solution ¯ x pi ∈ R for i ∈ I ⊆ { 1 , . . . , n } and, similarly, must ﬁnd minimally perturbed parameters.
Revised: In partial inverse optimization [Demange and Monnot, 2014], we are no longer given a full solution ˆ x , but instead a partial solution ¯ x pi ∈ R for i ∈ I ⊆ { 1 , . . . , n } and, similarly, must ﬁnd minimally perturbed parameters.

Original: The partial inverse optimization problem is identical to the above model with the additional constraint: ˆ x i = ¯ x pi , ∀ i ∈ I .
Revised: The partial inverse optimization problem is identical to the above model with the additional constraint: ˆ x i = ¯ x pi , ∀ i ∈ I .

Original:  Problem Deﬁnition
Revised:  Problem Deﬁnition

Original: Let x ∗ be an optimal solution to an instance of a (forward) optimization problem, (cid:104) X, c, f (cid:105) .
Revised: Let x ∗ be an optimal solution to an instance of a (forward) optimization problem, (cid:104) X, c, f (cid:105) .

Original: Assume we are asked a counterfactual question “Why is x ∗ not different?”
Revised: Assume we are asked a counterfactual question “Why is x ∗ not different?”

Original: such that this difference can be expressed by a foil constraint set , ψ , that x ∗ does not satisfy.
Revised: such that this difference can be expressed by a foil constraint set , ψ , that x ∗ does not satisfy.

Original: In our medical scheduling example, this question might be “Why isn’t my appointment scheduled next month?”, and ψ would represent the constraint that the patient’s procedure is in the next month.
Revised: In our medical scheduling example, this question might be “Why isn’t my appointment scheduled next month?”, and ψ would represent the constraint that the patient’s procedure is in the next month.

Original: Let X ψ ⊆
Revised: Let X ψ ⊆

Original: X represent all points in X that satisfy ψ .
Revised: X represent all points in X that satisfy ψ .

Original: Any point in X ψ is a foil (as per Section 2.1), because it represents a counterfactual decision the user has asked about.
Revised: Any point in X ψ is a foil (as per Section 2.1), because it represents a counterfactual decision the user has asked about.

Original: We assume X ψ (cid:54) = ∅ .
Revised: We assume X ψ (cid:54) = ∅ .

Original: Let X cψ =
Revised: Let X cψ =

Original: X \ X
Revised: X \ X

Original: ψ be the complement of X ψ , and any point in X cψ be called a factual solution (by deﬁnition,
Revised: ψ be the complement of X ψ , and any point in X cψ be called a factual solution (by deﬁnition,

Original: x ∗ ∈ X cψ ).
Revised: x ∗ ∈ X cψ ).

Original: An explanation is then generated by ﬁnding the counterfactual parameters ¯ c ∈ C that cause a foil to become optimal while minimally perturbing the initial parameters c .
Revised: An explanation is then generated by ﬁnding the counterfactual parameters ¯ c ∈ C that cause a foil to become optimal while minimally perturbing the initial parameters c .

Original: We therefore deﬁne the Nearest Counterfactual
Revised: We therefore deﬁne the Nearest Counterfactual

Original: Explanation (NCE) problem for some norm (cid:107)·(cid:107) as:
Revised: Explanation (NCE) problem for some norm (cid:107)·(cid:107) as:

Original: [Equation]
Revised: [Equation]

Original: The explanation, then, is: “If the parameters had been ¯ c instead of c , the alternative decision would have been reached.”
Revised: The explanation, then, is: “If the parameters had been ¯ c instead of c , the alternative decision would have been reached.”

Original: This formulation is a novel generalization of both inverse optimization and partial inverse optimization, since it allows a foil to be described with a constraint set, not only a (partial) set of value assignments.
Revised: This formulation is a novel generalization of both inverse optimization and partial inverse optimization, since it allows a foil to be described with a constraint set, not only a (partial) set of value assignments.

Original: For example, a patient can ask why they are not seen in a given time interval, as opposed to only asking why they are not seen on a given day.
Revised: For example, a patient can ask why they are not seen in a given time interval, as opposed to only asking why they are not seen on a given day.

Original: 3. Restrictions
Revised: 3. Restrictions

Original: General inverse optimization is computationally difﬁcult, especially with discrete variables [ Demange and Monnot, 2014 ] , and the literature is still developing robust solution approaches.
Revised: General inverse optimization is computationally difﬁcult, especially with discrete variables [ Demange and Monnot, 2014 ] , and the literature is still developing robust solution approaches.

Original: We make the following restrictions here to make progress on a subspace that nonetheless includes a number of interesting problems, including our medical example.
Revised: We make the following restrictions here to make progress on a subspace that nonetheless includes a number of interesting problems, including our medical example.

Original: We limit ψ to one constraint on one component of x with the form ψ : x j ≤ m 0 or ψ : x j ≥ m 0 , j ∈ [ n ] , m 0 ∈ N 0 .
Revised: We limit ψ to one constraint on one component of x with the form ψ : x j ≤ m 0 or ψ : x j ≥ m 0 , j ∈ [ n ] , m 0 ∈ N 0 .

Original: 1 We are interested in explanations for one individual, often modeled with a single variable and parameter, x j and c j , and our explanations consider only changes to c j , and not the other components of c .
Revised: 1 We are interested in explanations for one individual, often modeled with a single variable and parameter, x j and c j , and our explanations consider only changes to c j , and not the other components of c .

Original: This restriction means that an explanation is about a characteristic of the questioner as opposed to others that the questioner may have no knowledge of.
Revised: This restriction means that an explanation is about a characteristic of the questioner as opposed to others that the questioner may have no knowledge of.

Original: It also protects the privacy of others as their real and counterfactual parameter values are not revealed in an explanation.
Revised: It also protects the privacy of others as their real and counterfactual parameter values are not revealed in an explanation.

Original: For the original parameters, c , all foils are strictly suboptimal solutions, i.e. (cid:64) x
Revised: For the original parameters, c , all foils are strictly suboptimal solutions, i.e. (cid:64) x

Original: ∈ X ψ such that min { f ( c, x ) : x ∈ X } =
Revised: ∈ X ψ such that min { f ( c, x ) : x ∈ X } =

Original: min
Revised: min

Original: { f ( c, x ) : x ∈ X ψ } .
Revised: { f ( c, x ) : x ∈ X ψ } .

Original: We do not address cases where the foil is an alternative optimal solution to the original problem because such a situation produces
Revised: We do not address cases where the foil is an alternative optimal solution to the original problem because such a situation produces

Original: the unsatisfying explanation: “The foil is just as suitable as the initial decision, but was not selected due to arbitrary details of the algorithm (e.g., tie breaking).”
Revised: the unsatisfying explanation: “The foil is just as suitable as the initial decision, but was not selected due to arbitrary details of the algorithm (e.g., tie breaking).”

Original: The components of c and ¯ c appear only in the objective function, not in any constraints.
Revised: The components of c and ¯ c appear only in the objective function, not in any constraints.

Original: Therefore, any x ∈ X remains feasible for all values of c and ¯ c . 4.
Revised: Therefore, any x ∈ X remains feasible for all values of c and ¯ c . 4.

Original: There exists at least one feasible foil: X ψ (cid:54) = ∅ .
Revised: There exists at least one feasible foil: X ψ (cid:54) = ∅ .

Original: We return to these restrictions in more depth in Section 6.
Revised: We return to these restrictions in more depth in Section 6.

Original: We now address the NCE problem for two common objective functions: the weighed sum of binary variables and the weighted sum of general integer variables.
Revised: We now address the NCE problem for two common objective functions: the weighed sum of binary variables and the weighted sum of general integer variables.

Original:  Weighted Sum of Binary Variables
Revised:  Weighted Sum of Binary Variables

Original: Consider a problem where a subset of individuals are chosen from a group, for example, to receive some service.
Revised: Consider a problem where a subset of individuals are chosen from a group, for example, to receive some service.

Original: For such a problem, a natural model is to represent each individual as a binary variable where the 1 represents provision of the service and where some individual measure of the importance of each individual is represented by a scalar value.
Revised: For such a problem, a natural model is to represent each individual as a binary variable where the 1 represents provision of the service and where some individual measure of the importance of each individual is represented by a scalar value.

Original: The appropriate objective function is to maximize the sum of the quality of the chosen individuals and the obvious foil question from an individual is “Why wasn’t I selected for the service?”
Revised: The appropriate objective function is to maximize the sum of the quality of the chosen individuals and the obvious foil question from an individual is “Why wasn’t I selected for the service?”

Original: Deﬁnition 4.1 (Weighted Sum of Binary Variables (WSBV)) .
Revised: Deﬁnition 4.1 (Weighted Sum of Binary Variables (WSBV)) .

Original: Let x ∈ X ⊆ { 0 , 1 } n be a binary decision vector, c ∈ N n 0 a vector of integer parameters such that c is not present in any constraints that deﬁne X , and c · x = (cid:80) ni
Revised: Let x ∈ X ⊆ { 0 , 1 } n be a binary decision vector, c ∈ N n 0 a vector of integer parameters such that c is not present in any constraints that deﬁne X , and c · x = (cid:80) ni

Original: =1 c i x
Revised: =1 c i x

Original: i be a scalar product of two vectors in R n . 

Revised: i be a scalar product of two vectors in R n .

Original: The WSBV maximization problem is max { c · x : x ∈ X } .
Revised: The WSBV maximization problem is max { c · x : x ∈ X } .

Original: Let NCE 0-1 be the NCE problem for WSBV.
Revised: Let NCE 0-1 be the NCE problem for WSBV.

Original: We are given a WSBV instance (cid:104) X, c (cid:105) and a foil constraint ψ .
Revised: We are given a WSBV instance (cid:104) X, c (cid:105) and a foil constraint ψ .

Original: Because x is a binary vector and due to restriction 1, ψ must have the form ψ : x j = 1 or ψ : x j = 0 .
Revised: Because x is a binary vector and due to restriction 1, ψ must have the form ψ : x j = 1 or ψ : x j = 0 .

Original: Let ¯ c ∈ N n 0 be the modiﬁed parameter vector, which is identical to c with the exception of component ¯ c
Revised: Let ¯ c ∈ N n 0 be the modiﬁed parameter vector, which is identical to c with the exception of component ¯ c

Original: j replacing c j , since we are only interested in that one parameter.
Revised: j replacing c j , since we are only interested in that one parameter.

Original: Thus, using the L 1 norm, we deﬁne the
Revised: Thus, using the L 1 norm, we deﬁne the

Original: NCE 0-1 problem as
Revised: NCE 0-1 problem as

Original: [Equation]
Revised: [Equation]

Original: Rather than solving a general inverse optimization problem, we can solve this problem in closed-form given optimal solutions to the original problem and to the original problem plus ψ .
Revised: Rather than solving a general inverse optimization problem, we can solve this problem in closed-form given optimal solutions to the original problem and to the original problem plus ψ .

Original: Intuitively, any reduction in the objective function value incurred by the foil must be exactly compensated by the increase in ¯ c j in order for the foil to be optimal.
Revised: Intuitively, any reduction in the objective function value incurred by the foil must be exactly compensated by the increase in ¯ c j in order for the foil to be optimal.

Original: Let the following function denote the difference between the optimal foil and optimal factual solution
Revised: Let the following function denote the difference between the optimal foil and optimal factual solution

Original: [Equation]
Revised: [Equation]

Original: Since X ψ ⊆ X , it holds that ∆ X ψ ( y ) ≥ 0 for any ψ , X ⊆ { 0 , 1 } n , and y ∈ N n 0 .
Revised: Since X ψ ⊆ X , it holds that ∆ X ψ ( y ) ≥ 0 for any ψ , X ⊆ { 0 , 1 } n , and y ∈ N n 0 .

Original: We prove the validity of our closed form approach for the case when ψ : x j = 1 , but it can easily be modiﬁed for the case of ψ : x j = 0 , as well as the minimization variant of a WSBV problem.
Revised: We prove the validity of our closed form approach for the case when ψ : x j = 1 , but it can easily be modiﬁed for the case of ψ : x j = 0 , as well as the minimization variant of a WSBV problem.

Original: Theorem 1.
Revised: Theorem 1.

Original: The (cid:104) X, c, ψ (cid:105) NCE 0-1 problem, with ψ : x j =and a non-empty foil set X ψ (cid:54) = 0 has an optimal solution ¯
Revised: The (cid:104) X, c, ψ (cid:105) NCE 0-1 problem, with ψ : x j =and a non-empty foil set X ψ (cid:54) = 0 has an optimal solution ¯

Original: [Equation]
Revised: [Equation]

Original: j =
Revised: j =

Original: [Equation]
Revised: [Equation]

Original: ∆
Revised: ∆

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: By restriction 2, none of the optimal solutions of (cid:104) X, c (cid:105) satisfy ψ .
Revised: By restriction 2, none of the optimal solutions of (cid:104) X, c (cid:105) satisfy ψ .

Original: In this case we can write
Revised: In this case we can write

Original: [Equation]
Revised: [Equation]

Original: Since the foil constraint has the form ψ : x j = 1 , for each
Revised: Since the foil constraint has the form ψ : x j = 1 , for each

Original: [Equation]
Revised: [Equation]

Original: cψ we have x j = 0 .
Revised: cψ we have x j = 0 .

Original: Recalling that ¯ c is identical to c everywhere except the coordinate j , for every ¯ c we have that
Revised: Recalling that ¯ c is identical to c everywhere except the coordinate j , for every ¯ c we have that

Original: [Equation]
Revised: [Equation]

Original: Similarly, for each x ∈ X ψ we must have x j = 1 and that the contribution of the non- j components must be identical for every ¯ c , such that
Revised: Similarly, for each x ∈ X ψ we must have x j = 1 and that the contribution of the non- j components must be identical for every ¯ c , such that

Original: [Equation]
Revised: [Equation]

Original: Next, note that by (7) we have that for every solution ¯ c of an
Revised: Next, note that by (7) we have that for every solution ¯ c of an

Original: NCE (cid:104) X, c, ψ (cid:105) problem, it holds that ∆ Xψ (¯ c ) = 0 .
Revised: NCE (cid:104) X, c, ψ (cid:105) problem, it holds that ∆ Xψ (¯ c ) = 0 .

Original: Thus,
Revised: Thus,

Original: [Equation]
Revised: [Equation]

Original: Therefore, we have ¯ c j = c j +∆ Xψ ( c ) .
Revised: Therefore, we have ¯ c j = c j +∆ Xψ ( c ) .

Original: It is clear that ¯ c j ∈ N 0 , because both c ∈ N 0 and x ∈ N 0 , and therefore ∆ Xψ ( c ) ∈ N 0 .
Revised: It is clear that ¯ c j ∈ N 0 , because both c ∈ N 0 and x ∈ N 0 , and therefore ∆ Xψ ( c ) ∈ N 0 .

Original: Optimality follows from uniqueness of ¯ c .
Revised: Optimality follows from uniqueness of ¯ c .

Original: Theorem 1 shows that the computational effort required to form an explanation arises solely from re-solving the original problem with the added constraint x j = 1 .
Revised: Theorem 1 shows that the computational effort required to form an explanation arises solely from re-solving the original problem with the added constraint x j = 1 .

Original: As mentioned, the proof can easily be modiﬁed when the foil constraint is ψ : x j = 0 , to give the optimal solution ¯ c ∗ j = c j − ∆ Xψ ( c ) .
Revised: As mentioned, the proof can easily be modiﬁed when the foil constraint is ψ : x j = 0 , to give the optimal solution ¯ c ∗ j = c j − ∆ Xψ ( c ) .

Original: The Scope of Explanations for WSBV We now show two examples of explanations that can be generated as a result of solving NCE 0-1 .
Revised: The Scope of Explanations for WSBV We now show two examples of explanations that can be generated as a result of solving NCE 0-1 .

Original: In the 0-1 knapsack problem (KP) we are given a knapsack capacity W ∈ N and a set of items [ n ] ⊆ N , with each item i ∈ [ n ] associated with a weight w i ∈ N , a proﬁt c i ∈ N , and a binary variable x i ∈ { 0 , 1 } set to 1 iff the item is included in the knapsack.
Revised: In the 0-1 knapsack problem (KP) we are given a knapsack capacity W ∈ N and a set of items [ n ] ⊆ N , with each item i ∈ [ n ] associated with a weight w i ∈ N , a proﬁt c i ∈ N , and a binary variable x i ∈ { 0 , 1 } set to 1 iff the item is included in the knapsack.

Original: The 0-1 KP is max { (cid:80) ni
Revised: The 0-1 KP is max { (cid:80) ni

Original: =1 x i c i : (cid:80) ni
Revised: =1 x i c i : (cid:80) ni

Original: =1 x i w i ≤ W } , that is, to select a subset of items such that the sum of the proﬁts of the selected items is maximized and the sum of their weights does not exceed the knapsack capacity.
Revised: =1 x i w i ≤ W } , that is, to select a subset of items such that the sum of the proﬁts of the selected items is maximized and the sum of their weights does not exceed the knapsack capacity.

Original: KP problems are used for many real-life applications, for instance for capital budgeting decisions and our example of providing a limited service [Kellerer et al. , 2013].
Revised: KP problems are used for many real-life applications, for instance for capital budgeting decisions and our example of providing a limited service [Kellerer et al. , 2013].

Original: Using the
Revised: Using the

Original: NCE 0-1 , the explanation for a person not receiving a service can be “Because the beneﬁt of offering you this service was only c j .
Revised: NCE 0-1 , the explanation for a person not receiving a service can be “Because the beneﬁt of offering you this service was only c j .

Original: If it had increased to ¯ c ∗ j , you would have been offered the service.”
Revised: If it had increased to ¯ c ∗ j , you would have been offered the service.”

Original: Conversely, a person who was offered the service can assess how much their beneﬁt c j can decrease before the service is assigned to someone else.
Revised: Conversely, a person who was offered the service can assess how much their beneﬁt c j can decrease before the service is assigned to someone else.

Original: Another important class of WSBV problems is the weighted MaxSAT problem [Li and Manya, 2009].
Revised: Another important class of WSBV problems is the weighted MaxSAT problem [Li and Manya, 2009].

Original: We are given a conjunction of n disjunctive clauses, with each clause i ∈
Revised: We are given a conjunction of n disjunctive clauses, with each clause i ∈

Original: [ n ] associated with a weight c i ∈ N and a binary variable x i ∈ { 0 , 1 } that is assigned 1 iff the clause is satisﬁed.
Revised: [ n ] associated with a weight c i ∈ N and a binary variable x i ∈ { 0 , 1 } that is assigned 1 iff the clause is satisﬁed.

Original: The objective is to maximize the total weight of the satisﬁed clauses, max (cid:80) ni
Revised: The objective is to maximize the total weight of the satisﬁed clauses, max (cid:80) ni

Original: =1 x i c i .
Revised: =1 x i c i .

Original: Using our explanation technique, we can ask “Why is clause j not satisﬁed?”, and receive an explanation “Because the weight of this clause was only c j .
Revised: Using our explanation technique, we can ask “Why is clause j not satisﬁed?”, and receive an explanation “Because the weight of this clause was only c j .

Original: If it increased to ¯ c ∗ j , it would be satisﬁed.”
Revised: If it increased to ¯ c ∗ j , it would be satisﬁed.”

Original:  Weighted Sum of Integer Variables
Revised:  Weighted Sum of Integer Variables

Original: We now consider a problem class that includes the medical appointment scheduling example introduced above.
Revised: We now consider a problem class that includes the medical appointment scheduling example introduced above.

Original: In such a problem, the goal is to schedule appointments to maximize some form of health outcome which is often approximated by the minimization of the acuity-weighted sum of appointment times.
Revised: In such a problem, the goal is to schedule appointments to maximize some form of health outcome which is often approximated by the minimization of the acuity-weighted sum of appointment times.

Original: A foil question from an individual would have the form of asking why their appointment was not scheduled at some other time (e.g., earlier).
Revised: A foil question from an individual would have the form of asking why their appointment was not scheduled at some other time (e.g., earlier).

Original: Deﬁnition 5.1 (Weighted Sum of Integer Variables (WSIV)) .
Revised: Deﬁnition 5.1 (Weighted Sum of Integer Variables (WSIV)) .

Original: Let x ∈ X ⊆ N n 0 be an integer decision vector and c ∈ N nan integer parameter vector, such that c is not present in the constraints that deﬁne X .
Revised: Let x ∈ X ⊆ N n 0 be an integer decision vector and c ∈ N nan integer parameter vector, such that c is not present in the constraints that deﬁne X .

Original: A WSIV minimization problem is of the form min { c · x : x ∈ X } .
Revised: A WSIV minimization problem is of the form min { c · x : x ∈ X } .

Original: We deﬁne the explanation problem NCE N 0 .
Revised: We deﬁne the explanation problem NCE N 0 .

Original: Given is a
Revised: Given is a

Original: WSIV instance (cid:104) X, c (cid:105) and a foil constraint ψ , which by restriction 2 takes the form ψ : x j ≤ m 0 or ψ : x j ≥ m 0 , m 0 ∈ N 0 , over the coordinate j ∈ [ n ] of the variables x .
Revised: WSIV instance (cid:104) X, c (cid:105) and a foil constraint ψ , which by restriction 2 takes the form ψ : x j ≤ m 0 or ψ : x j ≥ m 0 , m 0 ∈ N 0 , over the coordinate j ∈ [ n ] of the variables x .

Original: As before, let ¯ c be the modiﬁed parameter vector, identical to c for all coordinates other than j .
Revised: As before, let ¯ c be the modiﬁed parameter vector, identical to c for all coordinates other than j .

Original: The NCE N 0 using the Lnorm is then
Revised: The NCE N 0 using the Lnorm is then

Original: [Equation]
Revised: [Equation]

Original: In our appointment scheduling example, we are given n patients (jobs) and m medical resources.
Revised: In our appointment scheduling example, we are given n patients (jobs) and m medical resources.

Original: Let c ∈ N n 0 represent patient acuity (a higher value means higher acuity), p ∈ N nrepresent job processing times, and x ∈ X ⊆ N n 0 represent job completion times, where X contains any constraints on the schedule (e.g., resource capacity, task precedence, etc.).
Revised: Let c ∈ N n 0 represent patient acuity (a higher value means higher acuity), p ∈ N nrepresent job processing times, and x ∈ X ⊆ N n 0 represent job completion times, where X contains any constraints on the schedule (e.g., resource capacity, task precedence, etc.).

Original: The goal is to ﬁnd a schedule that minimizes the weighted sum of completion times, min { c · x : x ∈ X } .
Revised: The goal is to ﬁnd a schedule that minimizes the weighted sum of completion times, min { c · x : x ∈ X } .

Original: The question “Why can’t I be seen before time m 0 ?” is exactly represented by the NCE N 0 problem with ψ : x j ≤ m 0 .
Revised: The question “Why can’t I be seen before time m 0 ?” is exactly represented by the NCE N 0 problem with ψ : x j ≤ m 0 .

Original: 5. Theoretical Results
Revised: 5. Theoretical Results

Original: We present theoretical results for the ψ : x j ≤ m 0 case.
Revised: We present theoretical results for the ψ : x j ≤ m 0 case.

Original: The proof can be modiﬁed to show that Theorem 2 holds for the case of ψ : x j ≥ m 0 as well.
Revised: The proof can be modiﬁed to show that Theorem 2 holds for the case of ψ : x j ≥ m 0 as well.

Original: The NCE N 0 is more involved than NCE 0-1 because we cannot rely on the fact that x j = 0 in the optimal factual solution.
Revised: The NCE N 0 is more involved than NCE 0-1 because we cannot rely on the fact that x j = 0 in the optimal factual solution.

Original: As a result, modiﬁcations to the c j parameter change the objective value of both the optimal counterfactual and factual solutions and we must reason about the change in their relative values as c j varies.
Revised: As a result, modiﬁcations to the c j parameter change the objective value of both the optimal counterfactual and factual solutions and we must reason about the change in their relative values as c j varies.

Original: We aim to prove that there is a constant c ∗ j ∈ R 0+ such that [ c ∗ j , ∞ ) ∩ N 0 is the set of all feasible solutions to the NCE, i.e. all values ¯ c j ∈ N 0 result in feasible solutions iff ¯
Revised: We aim to prove that there is a constant c ∗ j ∈ R 0+ such that [ c ∗ j , ∞ ) ∩ N 0 is the set of all feasible solutions to the NCE, i.e. all values ¯ c j ∈ N 0 result in feasible solutions iff ¯

Original: [Equation]
Revised: [Equation]

Original: j ≥
Revised: j ≥

Original: [Equation]
Revised: [Equation]

Original: This result will allow the use of binary search over a number of instances of the forward optimization problem to solve the
Revised: This result will allow the use of binary search over a number of instances of the forward optimization problem to solve the

Original: NCE.
Revised: NCE.

Original: To this end, as above, we deﬁne an auxiliary function
Revised: To this end, as above, we deﬁne an auxiliary function

Original: [Equation]
Revised: [Equation]

Original: To establish the observation above it is enough to prove the following.
Revised: To establish the observation above it is enough to prove the following.

Original: Theorem 2.
Revised: Theorem 2.

Original: Let (cid:104) X, c, ψ (cid:105) be an NCE problem with ψ : x j ≤ m 0 .
Revised: Let (cid:104) X, c, ψ (cid:105) be an NCE problem with ψ : x j ≤ m 0 .

Original: Then,
Revised: Then,

Original: ∆ Xψ ( y ) is a monotonically non-increasing function in y j . 2.
Revised: ∆ Xψ ( y ) is a monotonically non-increasing function in y j . 2.

Original: There is a c 0 j ∈ R 0+ such that ∆ Xψ ( c 0 ) = 0 and ccoincides with c on every entry other than j .
Revised: There is a c 0 j ∈ R 0+ such that ∆ Xψ ( c 0 ) = 0 and ccoincides with c on every entry other than j .

Original: In what follows, we assume y and ¯ y are two vectors in R n 

Revised: In what follows, we assume y and ¯ y are two vectors in R n

Original: + that are identical over all coordinates except j where y j ≤ ¯ y j , and denote their positive difference by δ y .
Revised: + that are identical over all coordinates except j where y j ≤ ¯ y j , and denote their positive difference by δ y .

Original: Intuitively, we show that if y j increases by positive δ y , both components of ∆ Xψ (¯ y ) will increase, but the negative component min x ∈ X ¯ y · x will increase faster, and the function will eventually vanish.
Revised: Intuitively, we show that if y j increases by positive δ y , both components of ∆ Xψ (¯ y ) will increase, but the negative component min x ∈ X ¯ y · x will increase faster, and the function will eventually vanish.

Original: Let us start with an observation that uses the form of the foil constraint to provide bounds on how fast the components of ∆ Xψ (¯ y ) can increase w.r.t.
Revised: Let us start with an observation that uses the form of the foil constraint to provide bounds on how fast the components of ∆ Xψ (¯ y ) can increase w.r.t.

Original: ¯ y j .
Revised: ¯ y j .

Original: Intuitively, part 1 below uses the minimum value of z j to derive a lower bound on the increase of the optimal factual solution w.r.t.
Revised: Intuitively, part 1 below uses the minimum value of z j to derive a lower bound on the increase of the optimal factual solution w.r.t.

Original: ¯ y j .
Revised: ¯ y j .

Original: Similarly, part 2 uses the maximum value of z j to give an upper bound on the increase of the optimal foil solution w.r.t.
Revised: Similarly, part 2 uses the maximum value of z j to give an upper bound on the increase of the optimal foil solution w.r.t.

Original: y j .
Revised: y j .

Original: Lemma 1.
Revised: Lemma 1.

Original: Let Z be a ﬁnite feasible set.
Revised: Let Z be a ﬁnite feasible set.

Original: If for each z ∈ Z it holds that k 0 ≤ z j , we have δ y k 0 ≤ min z ∈ Z ¯ y · z − min z ∈ Z y · z . 2.
Revised: If for each z ∈ Z it holds that k 0 ≤ z j , we have δ y k 0 ≤ min z ∈ Z ¯ y · z − min z ∈ Z y · z . 2.

Original: If for each z ∈ Z it holds that z j ≤ K 0 , we have min z ∈ Z ¯
Revised: If for each z ∈ Z it holds that z j ≤ K 0 , we have min z ∈ Z ¯

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Let z ∗
Revised: Let z ∗

Original: ∈
Revised: ∈

Original: Z be an optimal solution for the minimization problem (cid:104) Z, y (cid:105) , and let ¯ z ∗ ∈
Revised: Z be an optimal solution for the minimization problem (cid:104) Z, y (cid:105) , and let ¯ z ∗ ∈

Original: Z be an optimal solution for the minimization problem (cid:104) Z, ¯ y (cid:105) .
Revised: Z be an optimal solution for the minimization problem (cid:104) Z, ¯ y (cid:105) .

Original: Note also that by deﬁnition of ¯ y , for each vector x ∈ R n we have
Revised: Note also that by deﬁnition of ¯ y , for each vector x ∈ R n we have

Original: [Equation]
Revised: [Equation]

Original: Statement (1): since z ∗ is optimal given y , we have y · z ∗ ≤ y · ¯ z ∗ .
Revised: Statement (1): since z ∗ is optimal given y , we have y · z ∗ ≤ y · ¯ z ∗ .

Original: Thus, using (12) we deduce
Revised: Thus, using (12) we deduce

Original: [Equation]
Revised: [Equation]

Original: Statement (2) is proved similarly.
Revised: Statement (2) is proved similarly.

Original: As ¯ z ∗ is optimal for ¯ y ,
Revised: As ¯ z ∗ is optimal for ¯ y ,

Original: [Equation]
Revised: [Equation]

Original: Then, again using (12) we can write,
Revised: Then, again using (12) we can write,

Original: [Equation]
Revised: [Equation]

Original: We can now proceed to the proof of Theorem 2.
Revised: We can now proceed to the proof of Theorem 2.

Original: Proof.
Revised: Proof.

Original: Assume that X cψ (cid:54) = ∅ , since otherwise both claims hold trivially.
Revised: Assume that X cψ (cid:54) = ∅ , since otherwise both claims hold trivially.

Original: First, we aim at proving that ∆ Xψ is monotonic non-increasing within the scope of one coordinate.
Revised: First, we aim at proving that ∆ Xψ is monotonic non-increasing within the scope of one coordinate.

Original: Recall that y and ¯ y are two vectors in R n 0+ that are identical over all coordinates except j where y j ≤ ¯ y j , with their positive difference denoted δ y .
Revised: Recall that y and ¯ y are two vectors in R n 0+ that are identical over all coordinates except j where y j ≤ ¯ y j , with their positive difference denoted δ y .

Original: We shall prove that
Revised: We shall prove that

Original: [Equation]
Revised: [Equation]

Original: Let us look at the three following cases:
Revised: Let us look at the three following cases:

Original: a) The case ∆ Xψ (¯ y )
Revised: a) The case ∆ Xψ (¯ y )

Original: = 0 is trivial, since by deﬁnition ∆ Xψ ( y ) ≥ 0 for every y ∈ R n 0+ .
Revised: = 0 is trivial, since by deﬁnition ∆ Xψ ( y ) ≥ 0 for every y ∈ R n 0+ .

Original: b) Let ∆ Xψ ( y )
Revised: b) Let ∆ Xψ ( y )

Original: = 0 .
Revised: = 0 .

Original: Let x ∗ ,
Revised: Let x ∗ ,

Original: ψ ∈ X ψ be an optimal solution for the minimization problem (cid:104) X, y (cid:105) .
Revised: ψ ∈ X ψ be an optimal solution for the minimization problem (cid:104) X, y (cid:105) .

Original: By deﬁnition of ψ , this means x ∗ ,ψ j ≤ m 0 .
Revised: By deﬁnition of ψ , this means x ∗ ,ψ j ≤ m 0 .

Original: Let x c ∈
Revised: Let x c ∈

Original: X
Revised: X

Original: cψ be a solution in the complement of X ψ , meaning that, x ∗ ,ψj
Revised: cψ be a solution in the complement of X ψ , meaning that, x ∗ ,ψj

Original: [Equation]
Revised: [Equation]

Original: In order for ∆ Xψ ( y ) =
Revised: In order for ∆ Xψ ( y ) =

Original: 0 , we must have y · x ∗ ,ψ ≤ y · x
Revised: 0 , we must have y · x ∗ ,ψ ≤ y · x

Original: [Equation]
Revised: [Equation]

Original: = ⇒
Revised: = ⇒

Original: [Equation]
Revised: [Equation]

Original: By this, we have that min x ∈ X ψ ¯ y · x ≤ min x ∈ X
Revised: By this, we have that min x ∈ X ψ ¯ y · x ≤ min x ∈ X

Original: cψ ¯ y · x .
Revised: cψ ¯ y · x .

Original: Which implies that ∆ Xψ (¯ y ) = 0 .
Revised: Which implies that ∆ Xψ (¯ y ) = 0 .

Original: c) Lastly, assume
Revised: c) Lastly, assume

Original: [Equation]
Revised: [Equation]

Original: For ∆ Xψ to be monotonically non-increasing, the following difference of deltas must be non-positive:
Revised: For ∆ Xψ to be monotonically non-increasing, the following difference of deltas must be non-positive:

Original: [Equation]
Revised: [Equation]

Original: We can observe that (13) will hold iff all foils are suboptimal, meaning that all optimal solutions lie in X cψ .
Revised: We can observe that (13) will hold iff all foils are suboptimal, meaning that all optimal solutions lie in X cψ .

Original: Using this condition and rearranging (14),
Revised: Using this condition and rearranging (14),

Original: [Equation]
Revised: [Equation]

Original: Next, we insert bounds on these terms from Lemma 1.
Revised: Next, we insert bounds on these terms from Lemma 1.

Original: Note that for each x ∈ X ψ it holds that x j ≤ m 0 .
Revised: Note that for each x ∈ X ψ it holds that x j ≤ m 0 .

Original: Thus, by Lemma 1 part 2 we have
Revised: Thus, by Lemma 1 part 2 we have

Original: [Equation]
Revised: [Equation]

Original: Similarly, for each x ∈ X cψ it holds that x j ≥ m 

Revised: Similarly, for each x ∈ X cψ it holds that x j ≥ m

Original: + 1 .
Revised: + 1 .

Original: Thus, by Lemma 1 part 1 we have
Revised: Thus, by Lemma 1 part 1 we have

Original: [Equation]
Revised: [Equation]

Original: Applying these two inequalities to 14 we have
Revised: Applying these two inequalities to 14 we have

Original: [Equation]
Revised: [Equation]

Original: Next, we prove the second part of Theorem 2, that there is a c 0 j ∈ R 0+ such
Revised: Next, we prove the second part of Theorem 2, that there is a c 0 j ∈ R 0+ such

Original: that ∆ Xψ ( c 0 ) = 

Revised: that ∆ Xψ ( c 0 ) =

Original: and c 0 coincides with the unmodiﬁed parameter vector c on every entry other than j .
Revised: and c 0 coincides with the unmodiﬁed parameter vector c on every entry other than j .

Original: We must show that
Revised: We must show that

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Let us deﬁne a vector x min ∈ N n 0 , where every entry other than j is zero and x min
Revised: Let us deﬁne a vector x min ∈ N n 0 , where every entry other than j is zero and x min

Original: j = m 

Revised: j = m

Original: + 1 .
Revised: + 1 .

Original: For every x ∈ X cψ , we observe that x min i ≤ x i for each coordinate i .
Revised: For every x ∈ X cψ , we observe that x min i ≤ x i for each coordinate i .

Original: Thus, for every c (cid:48) ∈ R n 

Revised: Thus, for every c (cid:48) ∈ R n

Original: + it holds that c (cid:48) ·
Revised: + it holds that c (cid:48) ·

Original: [Equation]
Revised: [Equation]

Original: Therefore, (15) will hold if we can satisfy
Revised: Therefore, (15) will hold if we can satisfy

Original: [Equation]
Revised: [Equation]

Original: To this end, we pick an arbitrary x ψ ∈ X ψ and recall that x ψj ≤ m 0 .
Revised: To this end, we pick an arbitrary x ψ ∈ X ψ and recall that x ψj ≤ m 0 .

Original: We will satisfy (16) if we can ﬁnd a c 0 j ∈
Revised: We will satisfy (16) if we can ﬁnd a c 0 j ∈

Original: [Equation]
Revised: [Equation]

Original: 0+ such that
Revised: 0+ such that

Original: [Equation]
Revised: [Equation]

Original: Thus, we can pick c
Revised: Thus, we can pick c

Original: [Equation]
Revised: [Equation]

Original: Note that selecting this value of c 

Revised: Note that selecting this value of c

Original: j will also provide an upper bound for the binary search, as discussed next.
Revised: j will also provide an upper bound for the binary search, as discussed next.

Original: 5. Solving NCE N 0 Problems
Revised: 5. Solving NCE N 0 Problems

Original: We use the results of Theorem 2 to show that NCE N 0 can be solved with binary search.
Revised: We use the results of Theorem 2 to show that NCE N 0 can be solved with binary search.

Original: Theorem 2 assures us that feasible solutions for NCE N 0 (cid:104) X, c, ψ (cid:105) , with ψ : x j ≤ m 0 , are the integers that lie within the interval [ c ∗ j , ∞ ) ⊆
Revised: Theorem 2 assures us that feasible solutions for NCE N 0 (cid:104) X, c, ψ (cid:105) , with ψ : x j ≤ m 0 , are the integers that lie within the interval [ c ∗ j , ∞ ) ⊆

Original: R 

Revised: R

Original: + (i.e.
Revised: + (i.e.

Original: ∆ Xψ vanishes for each integer in this interval).
Revised: ∆ Xψ vanishes for each integer in this interval).

Original: Since we are looking for the minimal solution, to solve the problem it is sufﬁcient to ﬁnd (cid:100)
Revised: Since we are looking for the minimal solution, to solve the problem it is sufﬁcient to ﬁnd (cid:100)

Original: c ∗ j (cid:101) .
Revised: c ∗ j (cid:101) .

Original: The last part of the proof of Theorem 2 provides us with an upper bound on the optimal solution, c 0 j ∈ R 0+ such that c ∗ j ≤ c 0 j .
Revised: The last part of the proof of Theorem 2 provides us with an upper bound on the optimal solution, c 0 j ∈ R 0+ such that c ∗ j ≤ c 0 j .

Original: To compute c 0 j , we must ﬁnd a feasible foil, x ψ ∈
Revised: To compute c 0 j , we must ﬁnd a feasible foil, x ψ ∈

Original: X ψ .
Revised: X ψ .

Original: The smallest possible value of c 

Revised: The smallest possible value of c

Original: j will be given by x ψ ∈ argmin { c · x − c j x j : x ∈ X ψ } .
Revised: j will be given by x ψ ∈ argmin { c · x − c j x j : x ∈ X ψ } .

Original: Alternatively, it may be more convenient to use the original forward objective function, so that x ψ ∈ argmin { c · x : x ∈ X ψ } , though this may result in a higher value for c 0 j .
Revised: Alternatively, it may be more convenient to use the original forward objective function, so that x ψ ∈ argmin { c · x : x ∈ X ψ } , though this may result in a higher value for c 0 j .

Original: The lower bound on (cid:100) c ∗ j (cid:101) is c j + 1 , because c j < (cid:100) c ∗ j (cid:101) since all foils are suboptimal at c j (by restriction 2).
Revised: The lower bound on (cid:100) c ∗ j (cid:101) is c j + 1 , because c j < (cid:100) c ∗ j (cid:101) since all foils are suboptimal at c j (by restriction 2).

Original: Since ∆ Xψ is a monotonically non-increasing function, it is enough to run binary search on the integer set { c j + 1 , . . .
Revised: Since ∆ Xψ is a monotonically non-increasing function, it is enough to run binary search on the integer set { c j + 1 , . . .

Original: , (cid:100) c 0 j (cid:101)} to solve the problem.
Revised: , (cid:100) c 0 j (cid:101)} to solve the problem.

Original: This, together with the identity (cid:100) log 2 (cid:100) x (cid:101)(cid:101)
Revised: This, together with the identity (cid:100) log 2 (cid:100) x (cid:101)(cid:101)

Original: = (cid:100) log 2 x (cid:101) for each x ≥ 1 , results in the following statement:
Revised: = (cid:100) log 2 x (cid:101) for each x ≥ 1 , results in the following statement:

Original: Proposition 1.
Revised: Proposition 1.

Original: Let (cid:104) X, c, ψ (cid:105) be an NCE problem.
Revised: Let (cid:104) X, c, ψ (cid:105) be an NCE problem.

Original: Given a vector c 0 ∈ N n , where c 0 j replaces the j ’th entry in the vector c , such that ∆ Xψ ( c 0 ) = 0 , the (cid:104) X, c, ψ (cid:105) problem can be solved within at most (cid:100) log 2 ( c 0 j − c j ) (cid:101) solutions of the forward WSIV minimization problems of the form (cid:104)
Revised: Given a vector c 0 ∈ N n , where c 0 j replaces the j ’th entry in the vector c , such that ∆ Xψ ( c 0 ) = 0 , the (cid:104) X, c, ψ (cid:105) problem can be solved within at most (cid:100) log 2 ( c 0 j − c j ) (cid:101) solutions of the forward WSIV minimization problems of the form (cid:104)

Original: ¯ c, X ψ (cid:105) and (cid:104) ¯ c, X cψ (cid:105) .
Revised: ¯ c, X ψ (cid:105) and (cid:104) ¯ c, X cψ (cid:105) .

Original:  Discussion and Limitations
Revised:  Discussion and Limitations

Original: We demonstrated that we can generate counterfactual explanations for general optimization models with two common objective functions.
Revised: We demonstrated that we can generate counterfactual explanations for general optimization models with two common objective functions.

Original: The additional computational effort is solving either a single or logarithmic number of forward optimization problems, depending on the objective function.
Revised: The additional computational effort is solving either a single or logarithmic number of forward optimization problems, depending on the objective function.

Original: Our contributions were made within a restricted context.
Revised: Our contributions were made within a restricted context.

Original: Our ﬁrst limitation (Section 3.1) restricted the scope for explanation to a single model parameter.
Revised: Our ﬁrst limitation (Section 3.1) restricted the scope for explanation to a single model parameter.

Original: In the context of the GDPR, the restriction to characteristics of the questioner is reasonable both for the need to protect private information and from the perspective of generating a meaningful explanation: a counterfactual that changes characteristics of other individuals has limited value.
Revised: In the context of the GDPR, the restriction to characteristics of the questioner is reasonable both for the need to protect private information and from the perspective of generating a meaningful explanation: a counterfactual that changes characteristics of other individuals has limited value.

Original: Lifting the restriction to address a set of parameters concerning the questioner would require general partial inverse optimization.
Revised: Lifting the restriction to address a set of parameters concerning the questioner would require general partial inverse optimization.

Original: While most inverse optimization literature deals with multiple parameters [Heuberger, 2004], especially in the case of discrete variables, such problems are still challenging.
Revised: While most inverse optimization literature deals with multiple parameters [Heuberger, 2004], especially in the case of discrete variables, such problems are still challenging.

Original: We see this extension as a prime area for near-term future research.
Revised: We see this extension as a prime area for near-term future research.

Original: Our second and fourth restrictions require that a foil solution exists and is not optimal for the factual problem.
Revised: Our second and fourth restrictions require that a foil solution exists and is not optimal for the factual problem.

Original: Both restrictions were made because it is unclear what kind of counterfactual explanation is appropriate if they are lifted.
Revised: Both restrictions were made because it is unclear what kind of counterfactual explanation is appropriate if they are lifted.

Original: If the foil solution does not exist, then the Nearest Counterfactual
Revised: If the foil solution does not exist, then the Nearest Counterfactual

Original: Explanation problem is infeasible and no possible modiﬁcation can result in the questioner’s counterfactual.
Revised: Explanation problem is infeasible and no possible modiﬁcation can result in the questioner’s counterfactual.

Original: If the foil solution is optimal for the original problem, then there is no reason why it was not chosen as the factual solution.
Revised: If the foil solution is optimal for the original problem, then there is no reason why it was not chosen as the factual solution.

Original: In either case, there does not appear to be a meaningful explanation.
Revised: In either case, there does not appear to be a meaningful explanation.

Original: Finally, we required that none of the counterfactual parameters appear in the constraints.
Revised: Finally, we required that none of the counterfactual parameters appear in the constraints.

Original: Without such a restriction, not all solutions x ∈ X would remain feasible as the parameters change, substantially complicating the mathematical structure used in our results.
Revised: Without such a restriction, not all solutions x ∈ X would remain feasible as the parameters change, substantially complicating the mathematical structure used in our results.

Original: While there exists some work in inverse optimization under such assumptions (e.g., [ Yang et al. , 1997 ] ), approaches to the problem are not well-developed and form another important research direction.
Revised: While there exists some work in inverse optimization under such assumptions (e.g., [ Yang et al. , 1997 ] ), approaches to the problem are not well-developed and form another important research direction.

Original: In addition to the above restrictions, our work relies on the assumption that the users know and understand their weight.
Revised: In addition to the above restrictions, our work relies on the assumption that the users know and understand their weight.

Original: For a medical scheduling example, we assume patients know or can access their acuity scores.
Revised: For a medical scheduling example, we assume patients know or can access their acuity scores.

Original: For the example of solving a knapsack problem to select clients, we assume clients know how much they are ready to pay for a product.
Revised: For the example of solving a knapsack problem to select clients, we assume clients know how much they are ready to pay for a product.

Original: In our framework, a user has several choices after an explanation: they can accept the decision, act to change it (e.g., get a second opinion of their acuity), or contest the decision on the grounds that it is unfair to use this weight in this way.
Revised: In our framework, a user has several choices after an explanation: they can accept the decision, act to change it (e.g., get a second opinion of their acuity), or contest the decision on the grounds that it is unfair to use this weight in this way.

Original: In the latter case, they may ask why the system makes distinctions based on the weight, leading to ethical discussions which are important but beyond the scope of explaining the decision.
Revised: In the latter case, they may ask why the system makes distinctions based on the weight, leading to ethical discussions which are important but beyond the scope of explaining the decision.

Original: Though, we note that our approach provides an explanation that can enable a user to pose such metaexplanation questions.
Revised: Though, we note that our approach provides an explanation that can enable a user to pose such metaexplanation questions.

Original: If our assumption does not hold and the user does not understand how their weight was assigned, then they no longer need an explanation of the optimization-based decision, but rather an explanation of how the inputs to the model are determined.
Revised: If our assumption does not hold and the user does not understand how their weight was assigned, then they no longer need an explanation of the optimization-based decision, but rather an explanation of how the inputs to the model are determined.

Original: This is again beyond our scope as it now demands an explanation from the automated or human system that assigned the weights.
Revised: This is again beyond our scope as it now demands an explanation from the automated or human system that assigned the weights.

Original: But again, by explaining that a change in weight would change the outcome, our approach allows the user to question another part of the decision making system.
Revised: But again, by explaining that a change in weight would change the outcome, our approach allows the user to question another part of the decision making system.

Original:  Related Work
Revised:  Related Work

Original: Much recent work has looked at counterfactual explanations for classiﬁers, with seminal work by Wachter et.
Revised: Much recent work has looked at counterfactual explanations for classiﬁers, with seminal work by Wachter et.

Original: al [2017b].
Revised: al [2017b].

Original: Most relevant is work that generates explanations with a binary search over a sequence of combinatorial models [Karimi et al. , 2020; Russell, 2019].
Revised: Most relevant is work that generates explanations with a binary search over a sequence of combinatorial models [Karimi et al. , 2020; Russell, 2019].

Original: To our knowledge, none of the work addresses decisions made by explicit optimization models nor makes the connection to inverse optimization.
Revised: To our knowledge, none of the work addresses decisions made by explicit optimization models nor makes the connection to inverse optimization.

Original: Explainable constraint programming has mainly focused on explaining infeasibility [Freuder, 2017], often through the identiﬁcation of a minimal set of unsatisﬁable constraints [Junker, 2004].
Revised: Explainable constraint programming has mainly focused on explaining infeasibility [Freuder, 2017], often through the identiﬁcation of a minimal set of unsatisﬁable constraints [Junker, 2004].

Original: A parallel literature exists in mathematical programming [Chinneck, 2008].
Revised: A parallel literature exists in mathematical programming [Chinneck, 2008].

Original: While such sets could be viewed as counterfactuals (i.e., one of these constraints must be different), this connection has not been clearly developed and the use of inverse optimization is absent.
Revised: While such sets could be viewed as counterfactuals (i.e., one of these constraints must be different), this connection has not been clearly developed and the use of inverse optimization is absent.

Original: Other research has looked at counterfactual explanations in
Revised: Other research has looked at counterfactual explanations in

Original: AI Planning, highlighting the difference between speciﬁc factual and counterfactual plans [ Fox et al. , 2017 ] and describing properties of all counterfactual plans [ Eiffer et al. , 2020 ] .
Revised: AI Planning, highlighting the difference between speciﬁc factual and counterfactual plans [ Fox et al. , 2017 ] and describing properties of all counterfactual plans [ Eiffer et al. , 2020 ] .

Original: The only previous work that we are aware of that relates counterfactual explanations with inverse optimization is that of Brandao and Magazzeni [ 2020 ] which generates counterfactual explanations for path planning using an inverse shortest path formulation.
Revised: The only previous work that we are aware of that relates counterfactual explanations with inverse optimization is that of Brandao and Magazzeni [ 2020 ] which generates counterfactual explanations for path planning using an inverse shortest path formulation.

Original: While clearly making the connection with inverse optimization, the work focuses on the polynomially solvable inverse shortest path problem and has a substantially narrower scope than ours.
Revised: While clearly making the connection with inverse optimization, the work focuses on the polynomially solvable inverse shortest path problem and has a substantially narrower scope than ours.

Original:  Conclusion
Revised:  Conclusion

Original: We have shown that counterfactual explanations for decisions made by an optimization model can be generated through a generalization of inverse combinatorial optimization, a problem that has been studied in mathematical programming.
Revised: We have shown that counterfactual explanations for decisions made by an optimization model can be generated through a generalization of inverse combinatorial optimization, a problem that has been studied in mathematical programming.

Original: Within the context of legislation such as the General Data
Revised: Within the context of legislation such as the General Data

Original: Protection Regulations that mandates the right to an explanation, our approach can produce user-centered explanations that allow an individual to contest and act to change the decision to which they were subject
Revised: Protection Regulations that mandates the right to an explanation, our approach can produce user-centered explanations that allow an individual to contest and act to change the decision to which they were subject

Original: [Wachter et al. , 2017b].
Revised: [Wachter et al. , 2017b].

Original: Our formulation allows an individual to inquire about any change to a decision that can be represented with a constraint set on the original formulation.
Revised: Our formulation allows an individual to inquire about any change to a decision that can be represented with a constraint set on the original formulation.

Original: With some restrictions, we show that counterfactual explanations for models with two common objective functions can be generated with limited additional computational effort beyond solving the original model.
Revised: With some restrictions, we show that counterfactual explanations for models with two common objective functions can be generated with limited additional computational effort beyond solving the original model.

Original: We believe this general approach to counterfactual explanations for optimization-based decisions is a fruitful yet nascent direction that deserves further development.
Revised: We believe this general approach to counterfactual explanations for optimization-based decisions is a fruitful yet nascent direction that deserves further development.

Revised: Acknowledgements This research was supported by the

Revised: Natural Sciences and Engineering Research Council of

Revised: Canada.