Title: Parsimonious Predictions for Strategyproof Scheduling
Venue: NeurIPS 2025
Accepted: yes
Keywords: algorithmic game theory

mechanism design

scheduling

learning with predictions
Abstract: We consider the problem of scheduling $m$ jobs on $n$ unrelated
  strategic machines to minimize the maximum load of any machine, but
  the machines are strategic and may misreport processing times to
  minimize their own load. The pioneering work of Nisan and Ronen gave
  an $n$-approximate deterministic strategyproof mechanism for this
  setting, and this was recently shown to be best possible by the
  breakthrough results of Christodoulou et al. This large approxation
  guarantee begs the question: how can we avoid these large worst-case
  results. In this work, we use the powerful framework of algorithms
  with (machine-learned) predictions to bypass these strong
  impossibility results. We show how we can predict $O(m+n)$ values to
  obtain a deterministic strategyproof algorithm whose makespan is
  within a constant factor of the optimal makespan when the
  predictions are correct, and $O(n)$ times the optimum no matter how
  poor the predictions are.

Parsimonious Predictions for Strategyproof Scheduling

Abstract

We consider the problem of scheduling m jobs on n unrelated strategic machines to minimize the maximum load of any machine. As the machines are strategic they may misreport processing times to minimize their own load. The pioneering work of Nisan and Ronen gave an n -approximate deterministic strategyproof mechanism for this setting, and this was recently shown to be best possible by the breakthrough results of Christodoulou et al. This large approxation guarantee begs the question: how can we avoid these large worst-case results. In this work, we use the powerful framework of algorithms with (machine-learned) predictions to bypass these strong impossibility results. We show how we can predict O ( m + n ) values to obtain a deterministic strategyproof algorithm whose makespan is within a constant factor of the optimal makespan when the predictions are correct, and O ( n ) times the optimum no matter how poor the predictions are.

1 Introduction

Weconsider the NP-hard problem of scheduling jobs on unrelated machines to minimize the makespan, i.e., the load of the most loaded machine, in the setting where the machines are agents which can behave strategically. In this game-theoretic setting, introduced by Nisan and Ronen (2001), only the machines know the processing times for the jobs. Since they suffer a cost for processing jobs (i.e., the processing time/size of the job), they have an incentive to misreport sizes (i.e., to overstate or understate them) to the mechanism in order to get a more desirable bundle of jobs. Our goal is to design a mechanism that achieves (a) a good approximation ratio, so that we always find a schedule with makespan comparable to that of the optimal solution on the true processing sizes, and (b) strategyproofness, so that no machine has any incentive to misstate their processing times.

In the non-strategic setting of classical approximation algorithms, the unrelated machines scheduling problem can be approximated to a factor of 2 (due to works of Lenstra et al. (1990) and Shmoys and Tardos (1993)), and it is NP-hard to approximate it better than a factor of 3 / 2 , leaving a small gap between the constants in the upper and lower bounds. But when the machines can behave strategically, it is now known that the optimal approximation factor of any strategyproof mechanism is not a constant, but exactly n . The paper of Nisan and Ronen (2001) gave a mechanism achieving the n -approximation, and now a matching lower bound of n for any strategyproof mechanism is known due to Christodoulou et al. (2022, 2023). This leaves us with a large gap between the results that are achievable in the non-strategic and strategic settings in the worst-case.

Since worst-case instances may be pathological and the analysis pessimistic, we can ask: how can we give mechanisms that perform better in non-worst-case settings? An important approach of algorithms with predictions or learning-augmented framework , has gained momentum, giving a balance between the robustness guarantees of the worst-case model, and better results when given

machine-learned information. Concretely, the model asks: can we perform better if we have some (machine-learned) untrusted predictions about the instance? We seek the following two outcomes:

1. Robustness: this requires that the mechanism's performance is not much worse than the performance without predictions, regardless of whether the predictions are correct or not.

2. Consistency: this asks that if the predictions are indeed correct (i.e., the predictions indeed conform to the parameters/features of the true instance they are supposed to be predicting), then the mechanism's performance is close to that achievable if it truly believed the predictions. We may also ask for an error-tolerant version of consistency, where the algorithm's performance relates to the 'error' in the predictions.

This line of work was initiated in the setting of online algorithms by Mahdian et al. (2012), Purohit et al. (2018) and Lykouris and Vassilvitskii (2021) (see also the survey article by Mitzenmacher and Vassilvitskii (2022)); it was used in strategic settings by Agrawal et al. (2024), Xu and Lu (2022) and Gkatzelis et al. (2022). (See the annotated bibliography compiled by Balkanski et al. (2024)).

The unrelated machine scheduling problem was studied in this learning-augmented setting by Xu and Lu (2022) and Agrawal et al. (2024), who observed that following the predictions blindly gives an excellent consistency of 1 , but has arbitrarily poor robustness if the predictions are faulty; to remedy this, they gave algorithms with bounded robustness and consistency. The current best result is due to Balkanski et al. (2023), who gave the SCALED GREEDY algorithm that is (1 + c ) n -robust and (4 + 2 /c ) -consistent, for any c > 0 . Their algorithm uses the entire matrix ̂ p ∈ R n × m + of processing times, where ̂ p ij is the prediction for the true processing time p ij of the job j on machine i .

Christodoulou et al. (2024) explore whether we can use a smaller set of predictions, namely output predictions , where the mechanism is merely given the predicted machine for each job. They give an algorithm that uses just these m many predictions, and that is O (1) -robust and ( n 2 ) -consistent. Their algorithm-and indeed, all the above algorithms in this domain-compute machine 'biases'. They use a greedy-like algorithm using the reported processing times weighted by these biases to assign the jobs. They also show that, given just assignment predictions, no such weighted VGC mechanism can obtain much better trade-offs.

In light of these results, we ask: can we get a different set of linearly many predictions, i.e., O ( n + m ) quantities, and achieve the same guarantees?

1.1 Our Results

Our main result answers this question in the affirmative, and gives nearly optimal robustness/consistency tradeoffs with only a linear number of predictions.

Theorem 1.1 (Informal Main Theorem) . There exists a strategyproof mechanism that takes predictions for the value of the optimal makespan, a 'bias' for each machine, and a 'good' machine for each job, and produces an assignment which is (1 + c ) n -robust, and (2 + 1 c -1 ) -consistent for any choice of c > 1 .

This achieves essentially the same guarantees as the result of Balkanski et al. (2023) (our results replace a 1 / c term in the consistency guarantee by 1 / ( c -1) ), while using far fewer predictions-only linearly many instead of quadratically many. And using these few extra predictions (corresponding to the 'biases' for each machines) to avoid the lower bounds of Christodoulou et al. (2024).

Our approach is based on an essentially simple idea: instead of trying to predict the optimal machines for each job, we predict the machines which would be output by an approximation algorithm. This algorithm follows the 'relax-and-round' approach: it solves a linear programming (LP) relaxation for the unrelated machine scheduling problem, and then converts the fractional solution into an integral one. Crucially, the assignments given by this rounding algorithm are supported by linear programming duality. We use this in two ways: firstly, we use the machine variables of the dual program as the machine biases. Secondly, we use the structure of the dual solutions (e.g., complementary slackness) along with the output of the rounding process to prove robustness guaranteees.

While this approach leads to an O ( n 2 ) -robust algorithm, the dual structure of the classical LP relaxation is not rigid enough. The next idea is to write a new LP relaxation for unrelated machine scheduling problem, which allows us to ensure that the duals are bounded, which gives us more

control over the robustness. The final ingredient is showing a good approximation guarantee for this new relaxation.

Finally, like Balkanski et al. (2023), we can extend our work to an error-tolerant version. In this setting, we are given a parameter η > 1 , and predictions are considered ' η -good' if they correspond to some processing times that are within a factor of η of the true processing times. The goal now is to be consistent when the predictions are ' η -good', otherwise the algorithm is required to be robust. The formal definitions and details of the following result appear in Appendix B.

Theorem 1.2 (Informal Error-Tolerant Theorem) . There exists a strategyproof mechanism that takes the above predictions for the value of the optimal makespan, the optimal dual solution and the predicted rounded solution, and produces an assignment which is (1 + c ) η 2 n -robust (when the predictions are not η -good), and ̂ η 2 γ (2 + 1 c -1 ) -consistent (when they are ̂ η -good for some ̂ η ≤ η ) for any choice of c > 1 .

Comparing the Prediction Models. An advantage of parsimonious predictions is that they require only linear space to manipulate and store, compared to the quadratic storage used in the works ofXu and Lu (2022), Agrawal et al. (2024), Balkanski et al. (2023). It is an open problem to prove that learning these smaller number predictions from data has a smaller sample complexity. (We briefly discuss the learnability of the processing times in a PAC-like model in Appendix A.) That said, we can use the predictions given to Balkanski et al. (2023) (i.e., predictions for all the processing times) to obtain our results by first computing the predictions that our mechanism expects (which are all functions of the processing times), and using these 1 + n + m values for our mechanism.

1.2 Other Related Work

Strategyproof mechanisms are increasingly important in open and decentralized systems, since the resources belong to different agents who may act in their own interest and may not cooperate unless they are given the right incentives. In such settings, it is often important to ensure notions of nonmanipulability and fairness, either for reasons of social welfare, or because of legal requirements. Strategyproof mechanisms can help ensure that no single agent can manipulate the outcome to their advantage. Specifically for settings of scheduling jobs to machines, note that machines may misreport processing times to reduce their workload, which could lead to poor scheduling decisions and delays for jobs. The first work for the unrelated machine scheduling problem in the strategyproof setting was due to Nisan and Ronen (2001). The lower bounds were given by Nisan and Ronen (2001), Christodoulou et al. (2009), Koutsoupias and Vidali (2013), Giannakopoulos et al. (2021), Dobzinski and Shaulker (2020), Ashlagi et al. (2012), Christodoulou et al. (2007), culminating in a matching lower bound of n in the recent works of Christodoulou et al. (2022, 2023).

The work of Christodoulou et al. (2024) considers the output predictions model, where the mechanism is merely given a predicted output for each request: this corresponds to a predicted machine for each job in our setting. They give an algorithm that is ( β +1) -robust and ( n 2 /β ) -consistent for any fixed β ∈ [1 , n ] ; it assigns a bias of r ij = 1 for the predicted machine i , and r i ′ j = n/β for every other machine i ′ , and chooses the weighted minimizer of r ij p ij . They also show that, given just assignment predictions, no weighted VGC mechanism can give a significantly better trade-off than β vs. n 2 /β . In contrast, our work shows that if we are also given a prediction for OPT and the dual variables (one for each machine), we can achieve much stronger results.

The use of linear programs and duality as predictions is widespread: they have been used, e.g., the works of Lavastida et al. (2021), Lattanzi et al. (2020), Li and Xian (2021) uses linear/convex programs and their duals to perform load balancing in the online setting. Moreover, linear programming duals are used in the work of Dinitz et al. (2021); it uses learned duals as a 'warm start' to speed up algorithms for finding maximum weight bipartite matchings; these ideas are further explored by Davies et al. (2023), Polak and Zub (2024), Sakaue and Oki (2022). However, our use in mechanism design, and in particular, our approach of controlling the variation in the duals to make the mechanism more 'robust', seems to not have been used before.

2 Definitions

An instance I of the unrelated machine scheduling problem consists of (i) a set J of m jobs, (ii) a collection M of n machines, and (iii) a processing time p ij for each job j ∈ J and machine i ∈ M .

Let p denote the vector of all processing times. Given an instance I , an assignment of jobs to machines is a vector x ∈ { 0 , 1 } n × m , where x ij = 1 if and only if job j is assigned to machine i . Such an assignment x satisfies ∑ i x ij = 1 , and induces a load of ∑ j p ij x ij on machine i ∈ M .

The makespan of assignment x is MS ( p , x ) := max i ∈ M ∑ j p ij x ij .

Define the optimal (integer) makespan to be OPT ( p ) := min assignments x MS ( p , x ) , where the minimum is taken over all possible assignments x of jobs to machines, and the optimal (integer) assignment be denoted by x ⋆ = x ⋆ ( p ) .

Given an assignment x , we can also associate an allocation function φ : J → M with it, where φ ( j ) = i ⇐⇒ x ij = 1 . We frequently move between allocation functions φ and assignments x , as needed. (In some cases, we consider x ∈ { 0 , 1 } n × m vectors where ∑ i x ij ≥ 1 , but we can always reduce some of the x ij values to zero and get ∑ i x ij = 1 , without increasing the machine loads and the makespans.)

2.1 Strategyproofness

Consider an instance of the unrelated machine scheduling problem, where the machines report their processing times for all the jobs. Let p ∈ R n × m + be the true processing times for all machine-job pairs. (As is standard, we use p i to denote the i th row of matrix p , and p -i to denote the rest of the rows.) Now suppose some machine i ∈ M reports p ′ i ∈ R m + as its vector of processing times (where p ′ i may or may not equal p i ), and the other machines truthfully report p -i ∈ R m + :

let x ( p ′ i , p -i ) ∈ { 0 , 1 } n × m + be the assignment of jobs to machines, and

let π ( p ′ i , p -i ) ∈ R n × m + be the payments made by the mechanism to machines.

Define the utility of machine i to be

Observe that while both the map x and the payments π are functions of the reports ( p ′ i , p -i ) , the second term in the definition of utility uses the true processing times.

Definition 2.1 (Strategyproofness) . An allocation function φ is strategyproof if there exists payments π such that for any machine i and for any misreport of the processing times vector p ′ i , we have U i ( p i , p -i ) ≥ U i ( p ′ i , p -i ) .

Definition 2.2 (Monotonicity) . Let x = x ( p i , p -i ) and x ′ = ( p ′ i , p -i ) denote the allocations of the mechanism when machine i reports p i and p ′ i respectively. An allocation mechanism is monotone if for every machine i and for every two reports of the machine p i and p ′ i , the associated allocations satisfy ∑ j ∈ J ( x ij -x ′ ij )( p ij -p ′ ij ) ≤ 0 . Moreover, call the mechanism item-wise monotone if the allocations satisfy ( x ij -x ′ ij )( p ij -p ′ ij ) ≤ 0 ∀ j ∈ J .

Theorem 2.3 (Nisan and Ronen (2001), Saks and Yu (2005)) . For an allocation algorithm, there exists a payment function π such that ( φ, π ) is strategyproof if and only if the allocation algorithm satisfies the monotonicity property.

Since any item-wise monotone mechanism is also monotone, the following corollary holds:

Corollary 2.4. An allocation algorithm is strategyproof if it satisfies the item-wise monotonicity property. In other words, if for every machine i , suppose x = x ( p i , p -i ) and x ′ = ( p ′ i , p -i ) , and suppose these allocations satisfy ( x ij -x ′ ij )( p ij -p ′ ij ) ≤ 0 for every job j . Then the allocation algorithm is monotone, and hence there exists a payment function that gives strategyproofness.

3 A New Linear Program Relaxation

We now give a new linear programming relaxation for the scheduling problem, such that the dual variables corresponding to the machines lie in some bounded range. It is this control on the dual variables allows us to show strong robustness guarantees without losing much on the consistency.

The standard linear programming relaxation for the unrelated machine scheduling problem takes an instance I and a target makespan T , and defines the set of permissible edges E ( T, p ) := { ( i, j ) |

p ij ≤ T } by removing all the edges in M × J which correspond to processing times larger than the target value T . The feasibility linear program P ( T, p ) is then defined as follows:

Again, the first constraints allow the fractional assignment to allocate more than 1 unit of job j , but any such solution can be transformed into one that satisfies these constraints with equality. Using inequalities simplifies the linear programming dual D ( T, p ) :

Our new linear programming relaxation for the unrelated machine scheduling problem takes an instance I and a target makespan T , and defines the set of permissible edges E ( T, p ) := { ( i, j ) | p ij ≤ T } by removing all the edges in M × J which correspond to processing times larger than the target value T . Then for any a scalar c ≥ 1 , the linear program relaxation P c ( T, p ) is the following:

In other words, the primal constraint for machine i ∈ M says:

The first constraints assigns each job to some machine, and is a natural one. But the second constraint is mysterious due to the variables Y i and Z : to demystify this, let us write its dual program D c ( T, p ) :

Note that the dual program has variables α j for each job j ∈ J , and variables β i for each machine i ∈ M ; the latter should be thought of as machine 'weights'. Observe that this LP is very similar to the dual for the natural relaxation, where constraints (4) and (5) are the new constraints added to control the range of the β i values. Moreover, complementary slackness says that x ij ( α j -β i p ij ) = 0 .

Proposition 3.1. If T ≥ OPT ( p ) , then both the primal and dual linear programs above are feasible.

Proof. Indeed, consider the assignment x ⋆ ( p ) that achieves the optimal makespan for input p ; it is a feasible solution to the primal program. Moreover, the all-zero solution is feasible for the dual program.

3.1 Properties of the New Programs

The optimal solution for the new dual program D c ( T, p ) has a value no larger than that of the standard dual, since it is a maximization problem with more constraints. Of course, the added constraints could conceivably make the dual infeasible, but we now show that this is not possible.

Lemma 3.2 (LPs are Bounded) . For any c > 1 and any T ≥ OPT ( p ) , the primal P c ( T, p ) is feasible, and any (fractional) solution ( x , Y , Z ) to it has value at least Z (1 -1 / c ) -T / c ≥ -T / c . Moreover, its dual D c ( T, p ) is feasible and bounded as well.

Proof of Lemma 3.2. The original primal P ( T, p ) is feasible because of Proposition 3.1; moreover, any solution for the original primal LP is also feasible for the new primal P c ( T, p ) by just setting Z = Y i = 0 . To show the bound on the primal objective value, consider the constraint (3). The non-negativity of the sum ∑ p ij x ij implies that Y i ≤ T + Z . Consequently, the primal objective value is

where we used the non-negativity of the Z variable in the final step. Since the primal P c ( T, p ) is feasible and bounded from below, we can use strong duality to infer that the dual D c ( T, p ) is also feasible and bounded, which completes the proof.

While the objective function for the primal P c ( T, p ) is not meaningful per se , we can also show that the optimal fractional solution has small makespan.

Lemma 3.3 (Fractional Makespan) . For c > 1 and target T ≥ OPT ( p ) , an optimal fractional solution ( ˜ x , ˜ Y , ˜ Z ) for the linear program P c ( T, p ) has (fractional) makespan

Proof. Since any solution to the classical primal is also a solution to the new primal (by setting Z = Y i = 0 ), the optimal solution to the primal P c ( T, p ) has objective value at most zero. Moreover, Lemma 3.2 implies that the optimal solution to P c ( T, p ) has objective value at least ˜ Z (1 -1 /c ) -T/c . Combining these facts,

Using constraint (3) again, but this time ignoring the non-negative ˜ Y i terms, we get that

for each machine i ∈ M , as claimed.

In other words, any optimal fractional solution ˜ x for P c ( T, p ) is also a solution to the following relaxed version of the original linear program P ( T, p ) :

as long as we set λ = c c -1 . (Note that when the relaxation parameter λ = 1 , then Q 1 ( T, p ) = P ( T, p ) , i.e., we get back the original primal linear program.)

3.2 Rounding Algorithm

Assume that T ≥ OPT ( p ) . In this case, the rounding algorithms of Lenstra et al. (1990) and Shmoys and Tardos (1993) take a fractional solution ˜ x ≥ 0 satisfying the relaxed LP Q λ ( T, p ) and output an integer assignment x † ∈ { 0 , 1 } n × m with the following properties:

∑ i x † ij = 1 so that each job is assigned to exactly one machine,

x † is also a solution to Q λ ′ ( T, p ) but now with λ ′ = λ +1 .

each job j is assigned to some machine i to which it was originally fractionally assigned in ˜ x (i.e., x † ij = 1 = ⇒ ˜ x ij > 0 ), and moreover

In other words, this map satisfies

This analysis can be tightened a bit to make the additive loss depend on the largest processing time of any job in the optimal solution, but we stick to the analytical form (7) for simplicity.

4 The Mechanism and the Predictions

Our mechanism takes the following predictions: (i) a predicted threshold ̂ T , (ii) predicted dual variables ̂ β i ∈ [1 /cn, 1] for each machine i ∈ M , and (iii) a predicted assignment ̂ φ : J → M . In the ideal solution, the threshold ̂ T should equal OPT ( p ) , the variables ̂ β should be an optimal solution to the modified dual program D c ( T, p ) , and the predicted assignment ̂ x should be the assignment x † arising from optimally solving the modified primal P c ( T, p ) and then rounding it as in Section 3.2. The predicted assignment ̂ φ corresponds to a vector ̂ x ∈ { 0 , 1 } n × m , and henceforth we will use the two objects interchangeably. Finally, observe that if we are instead given a prediction ̂ p of the processing times, we can compute the values above-as described in Section 3-and use them instead.

4.1 The Mechanism

To begin, the DUAL-PREDICTOR mechanism checks whether the predicted dual variables satisfy ∑ i ̂ β i ≤ 1 and min i ̂ β i ≥ 1 /cn . If not, it rejects the predictions and just runs the greedy algorithm, which is n -robust and truthful Nisan and Ronen (2001). If the ̂ β values are in the correct range, the mechanism considers the reported processing times p ∈ R n × m along with the predictions and does the following:

Mechanism 1: DUAL-PREDICTOR

In words, the DUAL-PREDICTOR mechanism builds a set of machines small ( j ) := { j | p ij ≤ ̂ T } on which the job is 'small' compared to our prediction for the optimal makespan, based on the reported sizes p and the prediction ̂ T for the optimal makespan. If this set is empty (a sure indicator of the predictions being incorrect), it assigns the job to the machine on which it has the smallest reported size. Else it assigns it to a machine in small ( j ) that minimizes ̂ β i p ij , breaking ties in favor of the predicted machine ̂ φ ( j ) .

Atechnical observation: our mechanism uses the reported p ij values to define the set small ( j ) (which can be thought of as modifying the biases β i ). This is a departure from previous algorithms, which define the biases r ij based on the predictions and the reported processing times.

5 The Analysis

We now analyze the quality of the assignment φ returned by the DUAL-PREDICTOR mechanism: our proof proceeds in three natural steps.

We first show (1 + c ) n -robustness of the mechanism by proving that no matter what the predictions are, the makespan of the assignment produced by the mechanism is at most (1 + c ) n · OPT ( p ) .

Next, we show (2 + 1 / ( c -1) ) -consistency. We say that when the predictions are correct (i.e., when ̂ T ≈ OPT ( p ) , and the ̂ β i values correspond to an optimal solution to the dual D c ( ̂ T, p ) , and also the predicted assignment ̂ φ corresponds to the integer allocation x † obtained taking an optimal solution ˜ x to the primal P c ( ̂ T, p ) and rounding it as in Section 3.2), then our algorithm achieves a makespan of at most (2 + 1 / c -1 ) · ̂ T ; i.e., a nearly optimal solution.

Finally we show that the mechanism is strategyproof, by showing that the mechanism is monotone.

5.1 Robustness

For robustness, we show that no matter what the predictions are, the mechanism's assignment x has makespan MS ( p , x ) that is at most O ( n ) times the optimal makespan. Let x ⋆ be the optimal assignment, so that OPT ( p ) = MS ( p , x ⋆ ) .

Theorem 5.1. Given any predictions, MS ( p , x ) ≤ (1 + c ) n · MS ( p , x ⋆ ) .

Proof. Let φ and φ ⋆ be maps from J to M that correspond to the assignments x and x ⋆ in the natural way. Define r j := max(1 , cn · ̂ β φ ⋆ ( j ) ) . To begin, we claim that for any job j ,

Indeed, consider the various cases:

If small ( j ) = ∅ , then p φ ( j ) ,j ≤ p i,j for all machines i , and hence also for machine φ ⋆ ( j ) .

Else if | small ( j ) | ≥ 1 , we consider two subcases.

(a) If φ ⋆ ( j ) ̸∈ small ( j ) , then ̂ T < p φ ⋆ ( j ) ,j . Moreover, we assign job j to some job in small ( j ) , and therefore p φ ( j ) ,j ≤ ̂ T . Chaining the two implies p φ ( j ) ,j < p φ ⋆ ( j ) ,j .

(b) Else both φ ( j ) and φ ⋆ ( j ) belong to small ( j ) . The mechanism's selection criterion ensures that ̂ β φ ( j ) p φ ( j ) ,j ≤ ̂ β φ ⋆ ( j ) p φ ⋆ ( j ) ,j . Since the ̂ β values range between 1 / cn and 1 , the claim immediately follows.

This proves the claim (8). Now consider any machine i , and let J i := { j | φ ( j ) = i } be the jobs assigned to machine i by the mechanism. The load of machine i is

where we use ∑ i ̂ β i ≤ 1 in the final step.

5.2 Consistency

We now show the consistency guarantees: when the predictions are correct, the makespan of the mechanism's assignment is at most (2+ 1 c -1 ) times the optimal makespan. We consider the following slightly relaxed definition:

Definition 5.2 ( γ -Correct Predictions) . For γ ≥ 1 , the predictions ( ̂ T, ̂ β, ̂ φ ) are γ -correct for processing times p if the following conditions hold:

̂ T ∈ [ OPT ( p ) , γ OPT ( p )] ,

̂ β = ˜ β for an optimal solution ( ˜ α , ˜ β ) to the dual program D c ( ̂ T, p ) , and

the assignment ̂ x corresponding to the prediction ̂ φ is the assignment x † obtained by rounding an optimal solution ˜ x to the primal P c ( ̂ T, p ) , as in Section 3.2.

When predictions are 1 -correct, we refer to them as correct predictions. We show that γ -correct predictions imply a good makespan.

Lemma 5.3 (Consistency) . If the predictions ( ̂ T, ̂ β, ̂ φ ) are γ -correct for processing times p , then

Proof. Observe that the set small ( j ) is exactly the set of permissible edges from E ( ̂ T, p ) := { ( i, j ) | p ij ≤ ̂ T } which are incident to job j . Since ̂ T ≥ OPT ( p ) , the set small ( j ) is non-empty since it contains the machine that j is assigned to in the optimal integer solution x ⋆ . The mechanism now

assigns job j to the machine φ ( j ) ∈ small ( j ) minimizing ̂ β i p ij = ˜ β i p ij . We claim that this machine is in fact ̂ φ ( j ) . The claim implies that x (the output of the mechanism) equals ̂ x (the assignment corresponding to the predicted allocation), which in turn equals x † (the rounded solution). Hence (7) gives us MS ( p , x ) = MS ( p , ̂ x ) ≤ (2 + 1 / ( c -1) ) · ̂ T , which completes the proof.

We now prove the claim that φ ( j ) = ̂ φ ( j ) . By the property of the rounding algorithm, we know that ̂ x i,j = 1 only if ˜ x i,j > 0 . Moreover, we have ˜ x i,j > 0 only for ( i, j ) ∈ E ( ̂ T, p ) and hence ̂ φ ( j ) belongs to small ( j ) . We now claim that

for all i ∈ small ( j ) . Indeed, observe that ˜ α j ≤ ˜ β i p ij for all ( i, j ) ∈ small ( j ) because of dual feasibility. Moreover, complementary slackness implies that ˜ x ij ( ˜ α j -˜ β i p ij ) for all i ∈ small ( j ) . Since ˜ x ̂ φ ( j ) ,j > 0 we have that that ˜ α j = ˜ β ̂ φ ( j ) p ̂ φ ( j ) ,j . Moreover, using that ̂ β = ˜ β , we get that ̂ φ ( j ) is one of the minimizers of ̂ β i p ij , proving (9). Finally, our tie-breaking strategy ensures that the mechanism assigns the job j to ̂ φ ( j ) .

As an aside, note that while we used the fact that ̂ φ is obtained from the rounding algorithm of Section 3.2, any ̂ φ satisfying the properties above (in particular, that ˜ x ̂ φ ( j ) ,j > 0 ), and having low load, could be used instead.

5.3 Strategyproofness

To show that the the mechanism above is strategyproof, recall the results from Section 2.1, and specifically Corollary 2.4 which shows that an item-size monotone mechanism is strategyproof.

Theorem 5.4 (Strategyproofness) . The Dual-Predictor mechanism is item-size monotone and hence strategyproof.

Proof. Let the true processing times be given by p , and consider any other report p i made by machine i . If x = x ( p i , p -i ) and x ′ = x ( p ′ i , p -i ) are the allocations under the correct reports and the misreport respectively, we want to show that

Let small ( j ) and small ′ ( j ) denote the sets corresponding to the runs of the mechanism when given reports ( p i , p -i ) and ( p ′ i , p -i ) respectively.

1. Case I : Suppose x ij = 1 , i.e., the job j is assigned to machine i under the correct reports. Since x ′ ij ∈ { 0 , 1 } , (10) follows immediately in case p ij ≤ p ′ ij , so it suffices to show that p ′ ij < p ij = ⇒ x ′ ij = 1 . We consider two cases based on the size of small ( j ) .

Suppose small ( j ) = ∅ , then for machine i to be assigned job j , it must be that i = arg min i ′ p i ′ ,j . Since p ′ ij ≤ ̂ T , it must be the case that | small ′ ( j ) | ≤ 1 . If small ′ ( j ) = ∅ , then p ′ ij < p ij means we still have i = arg min i ′ p ′ i ′ ,j . Else small ′ ( j ) = { i } , and then the mechanism always allocates job j to the unique machine i in small ′ ( j ) . This means x ′ ij = 1 , proving the claim for the case when small ( j ) = ∅ .

̸

Else | small ( j ) | ≥ 1 . Since j is assigned to machine i , we infer that i ∈ small ( j ) and ̂ β i p ij ≤ ̂ β i ′ p i ′ ,j for all i ′ ∈ small ( j ) . (This is vacuously true when | small ( j ) | = 1 .) Since p ′ ij < p ij and the other sizes p i ′ ,j remain unchanged for i ′ = i , we get small ′ ( j ) = small ( j ) . Moreover, since ̂ β i ≥ 0 , the job j will continue to be assigned to machine i , and hence x ′ ij = 1 .

2. Case II : Suppose x ij = 0 . Since x ′ ij ∈ { 0 , 1 } , (10) follows immediately in case p ′ ij ≤ p ij , so it suffices to show that p ′ ij > p ij = ⇒ x ′ ij = 0 .

Let i j be the machine that is assigned the job j in the assignment x : i.e., x i j ,j = 1 . If small ( j ) = ∅ , then p ij > p i j j since it is not assigned the job j . Now, p ′ ij is even higher, and hence x ′ ij = 0 .

̸

and therefore, x ′ ij = 0 .

This case analysis implies that any machine i misreporting its processing time for job j leads to an allocation satisfying the item-wise monotonicity property (10). Now Corollary 2.4 implies that the mechanism is strategyproof.

6 Closing Remarks

This work extends the investigation of learning-augmented algorithms for optimization problems for which the pessimistic lower bounds can be circumvented using machine-learned predictions. We considered the unrelated machine scheduling problem in the strategic settings, where the agents (machines) can lie about the sizes of jobs to get an allocation with lower load. We show how to use only a linear number of predictions (essentially one value for each job and for each machine) to get constant consistency and optimal O ( n ) robustness, improving on previous work that either achieved suboptimal robustness with linearly many predictions, or used quadratically many predictions for the optimal results. Our approach used a new LP relaxation with small integrality gap that also controlled the range of the dual values.

Several tantalizing directions remain open: can we improve the bounds even further? Since a typical case is when the number of machines is much smaller than the number of jobs, can we use only O ( n ) predictions and get similar algorithmic guarantees? Can we predict these predictions much faster than predicting job sizes? Finally, what other algorithmic problems can use this 'range-controlled dual' idea to get improvements?

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A Learnability of Predictions

Let us show how we can learn predictions of the processing time, say in a PAC setting, where each instance p is obtained by first drawing a context/feature a from a distribution D over some set A , and getting the matrix n × m of processing times using an unknown map f ( a ) ∈ P := R n × m + . Our goal is to learn some map ̂ f : A → P such that

Then, at the time the mechanism is run on the true input, we suppose that the predictor sees the actual contexts/features a , and gives the prediction ̂ f ( a ) to the algorithm. The guarantee above ensures that Pr [ D ( f ( a ) , ̂ f ( a )) ≥ e ε ] ≤ δ , where the error parameter e ε ≈ (1 + ε ) for small ε .

Let us sketch one way to solve this learning problem. First, we can focus on each coordinate f ij separately and learn it with confidence 1 -δ ′ := 1 -δ/ ( mn ) ; a trivial union bound then gives the claim. The goal now becomes that of learning a real-valued univariate function g : A → R + , where g ( a ) = ln f ij ( a ) , to an additive accuracy of ε with high confidence 1 -δ ′ . At this point, we can use standard tools from machine learning theory to give bounds on the number of samples that suffice to learn g to an additive error of ε . Of course, one needs to make some assumptions on the complexity of the map ˜ g , which would depend on the applications at hand.

As a concrete example of these results, let us assume that the processing times lie in the range [1 , . . . , e K ] , and define ˜ g ( a ) := ⌊ g ( a ) ⌋ , then the function ˜ g becomes { 0 , 1 , . . . , K } -valued. Furthermore, the gap due to this approximation | ˜ g -g | ≤ 1 ; translating back via the definitions, the true processing time f ij ( a ) and its estimate ˆ f ij ( a ) = exp(˜ g ( a )) , would differ by at most a constant factor. Finally, given this { 0 , 1 , . . . , K } -valued function ˜ g , we can use the results of Ben-David et al. (1995) on learning such functions exactly with high confidence; this work reduces the learning of this function to showing the finiteness of VC-dimension of some related binary classification problems. In all these cases, the sample complexity behaves like O (( d Ψ /ε 2 ) log 1 /δ ) , where the parameter d Ψ is some notion of the dimension of associated concept classes.

B Error tolerant Dual-Predictor Mechanism

It can be shown that the DUAL-PREDICTOR mechanism is somewhat brittle - even in the presence of minor errors, the mechanism achieves the robustness guarantee. In this section, we design an error tolerant mechanism that builds on the previous mechanism and achieves a constant approximation factor not only when the predictions are accurate, but also when the predictions are approximately accurate.

Specifically, our strategyproof mechanism ERROR-TOLERANT DUAL-PREDICTOR will take as input an error threshold η ≥ 1 in addition to the predictions used in Mechanism 1: it will output an allocation with consistency guarantees if the error of the prediction is at most η , and it will always ensures the robustness guarantees. (Both the guarantees now depend on η .)

B.1 Approximately Correct Predictions

To begin, define the following 'distortion' between two vectors p , q of processing times:

Observe that distortion is symmetric, so that D ( p , q ) = D ( q , p ) .

Definition B.1 ( ( γ, η ) -Approximate Correctness) . We now say that predictions ( ̂ T, ̂ β, ̂ φ ) are ( γ, η ) -approximately correct with respect to the (true) processing times p if there exist

(a) processing times q such that the distortion D ( q , p ) ≤ η , and

(b) predictions ( η ̂ T, ̂ β, ̂ φ ) are γ -correct for processing times q , where γ -correctness is defined as in Definition 5.2.

We say that the processing times q and estimate ̂ T certify ( γ, η ) -approximate correctness.

B.2 Error-Tolerant Mechanism

The error-tolerant mechanism ERROR-TOLERANT DUAL-PREDICTOR is given a parameter η ≥ 1 , along with predictions that are claimed to be ( η, γ ) -approximately correct, and it proceeds very similarly to the original DUAL-PREDICTOR mechanism: building a set of 'small' machines for each job and then using a weighted greedy mechanism to assign to one of them (or to the machine with smallest reported processing time) if the set is empty.

Mechanism 2: Error-Tolerant Dual-Predictor

Input: Reported values p ij for all ( i, j ) ∈ M × J , predictions ̂ β ∈ R n + , map ̂ φ : J → M , and R

̸

return the vector x corresponding to this allocation φ .

There are two differences between the DUAL-PREDICTOR mechanism and this error-tolerant version: first, we now add in the predicted machine ̂ φ ( j ) to the set small ( j ) even if the reported size p ̂ φ ( j ) ,j is larger than the threshold ̂ T , as long as it is not too large. Secondly, we modify the multipliers β to give the predicted machine an advantage, before choosing the machine for job j . Observe that for the setting of η = 1 , the two mechanisms coincide.

B.3 Robustness

The proof of robustness is quite similar to that of the original mechanism.

Theorem B.2. Given any predictions, MS ( p , x ) ≤ (1 + c ) n · η 2 · MS ( p , x ⋆ ) .

Proof. Let φ and φ ⋆ be maps from J to M that correspond to the assignments x and x ⋆ in the natural way. Define

To begin, we claim that for any job j ,

Indeed, consider the various cases:

If small ( j ) = ∅ , then p φ ( j ) ,j ≤ p i,j for all machines i , and hence also for machine φ ⋆ ( j ) .

Else if | small ( j ) | ≥ 1 , we consider two subcases.

(a) If φ ⋆ ( j ) ̸∈ small ( j ) , then ̂ T < p φ ⋆ ( j ) ,j . Moreover, we assign job j to some job in small ( j ) , and therefore p φ ( j ) ,j ≤ η 2 ̂ T . Chaining the two implies p φ ( j ) ,j < η 2 · p φ ⋆ ( j ) ,j .

(b) Else both φ ( j ) and φ ⋆ ( j ) belong to small ( j ) . The mechanism's selection criterion ensures that

Since the ̂ β values range between 1 / cn and 1 , the claim immediately follows.

This proves the claim (13). Finally a calculation similar to the proof from Theorem 5.1 shows that the load of machine i is at most η 2 · ( n + cn ) · MS ( p , x ⋆ ) , completing the proof.

B.4 Consistency

Now suppose that the predictions ( ̂ T, ̂ β, ̂ φ ) are ( γ, η ) -approximately correct, and let q be the processing times that certify this ( γ, η ) -approximate correctness. (Note that q is not known to the mechanism.) Unwrapping the definition, we get that

D ( p , q ) ≤ η , so that p /η ≤ q ≤ η p ,

OPT ( q ) ≤ η ̂ T ≤ γ OPT ( q ) ,

̂ β = ˜ β corresponds to an optimal dual solution ( ˜ α , ˜ β ) to the dual D c ( η ̂ T, q ) , and

̂ φ ( j ) corresponds to the solution x † obtained by rounding an optimal fractional solution ˜ x to the primal P c ( η ̂ T, q ) .

Lemma B.3 (Consistency) . If the predictions are ( γ, η ) -approximately correct, then

where D ( p , q ) = η ≤ η

Proof. We first argue that the set small ( j ) is non-empty. Indeed, the machine ̂ φ ( j ) lies in the support of some primal solution to the linear program P c ( η ̂ T, q ) and hence q ̂ φ ( j ) ,j ≤ η ̂ T . Since p i,j ≤ η q i,j for all ( i, j ) , we have p ̂ φ ( j ) ,j ≤ η 2 ̂ T . By the definition of the mechanism, the machine ̂ φ ( j ) is added to the set small ( j ) if it satisfies this last inequality, thereby ensuring that small ( j ) is non-empty. Hence, the mechanism assigns to the machine in small ( j ) minimizing β i p ij .

Now consider the machines in small ( j ) = { i | p ij ≤ ̂ T } ∪ { ̂ φ ( j ) } . Since p ≥ ̂ p /η , the former set is contained in { i | q ij ≤ η ̂ T } . Morever, as argued in the previous paragraph, ̂ φ ( j ) also belongs to this set. In other words, the pairs { ( i, j ) | i ∈ small ( j ) } are contained in the permissible edges E ( η ̂ T, q ) .

Next, the feasibility for the dual program D c ( η ̂ T, q ) means that ˜ α j ≤ ˜ β i q ij for all ( i, j ) ∈ E ( η ̂ T, q ) ; moreover, complementary slackness implies that ˜ α j = ˜ β i q ij for machines i ∈ small ( j ) with ˜ x ij > 0 , i.e., in the support of the optimal fractional solution. Hence, the machine ̂ φ ( j ) is a minimizer of ˜ β i q i,j = ̂ β i q i,j over machines i ∈ small ( j ) . However, we do not have access to the processing times q ij but only to p ij . To fix this issue, we use (i) the fact that q ij /η ≤ p ij ≤ η p ij and (ii) the definition of β (as in in the mechanism) to infer that ̂ φ ( j ) is the minimizer of β i p ij over machines i ∈ small ( j ) . This means x = ̂ x = x † .

Now suppose that D ( p , q ) = ̂ η ≤ η . Then the resulting makespan is

which completes the proof.

B.5 Strategyproofness

The proof is essentially identical to the proof of Theorem 5.4, so we only sketch it here.

Proof. (Sketch) For a job j , if a machine i was not in small ( j ) , the only way for machine i to become part of small ′ ( j ) is to reduce its processing time for the job j . Likewise, the only way to be removed from small ( j ) is to increase its processing time. Therefore, if machine i already had the job j ( x ij = 1 ), reducing its processing time ensures that it still receives the job, irrespective of whether small ( j ) is empty or not.

̸

On the other hand, if machine i did not receive the job, the case when small ( j ) = ∅ is trivial. If small ( j ) = ∅ , increasing its processing time can only increase its scaled processing time until it eventually does not belong to small ′ ( j ) ; this shows x ij = x ′ ij = 0 as required.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: As the abstract and introduction claims, we designed a learning-augmented job scheduling mechanism for unrelated machines that requires only Θ( n + m ) predictions and achieves nearly optimal consistency and robustness. Both consistency and robustness are proved thoroughly.

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The answer NA means that the abstract and introduction do not include the claims made in the paper.

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Question: Does the paper discuss the limitations of the work performed by the authors?

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of scheduling m jobs on n strategic, unrelated machines to minimize makespan, a setting where worst-case analysis shows that any strategyproof mechanism has a pessimistic guarantee. To bypass this limitation, the authors use the learning-augmented framework, designing a mechanism that uses parsimonious (i.e., simple) predictions. The core of their method is an LP relaxation designed to control the range of its dual variables, which are then used as predicted machine \"biases”.  The proposed mechanism takes as input a predicted optimal makespan, a predicted bias for each machine, and a predicted \"good\" machine for each job. It achieves a makespan that is (1+c)n times the optimum regardless of prediction quality (robustness) and (2+1/(c−1)) times the optimum when predictions are correct (consistency), essentially matching the performance of previous work that required complex predictions.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**  \nA very well-written paper on a nice problem.  Analysis appears to all be sound, and the results are surprising in that parsimonious (i.e., succinct/simple) predictions can achieve the same (nearly optimal) consistency/robustness trade off for this problem.  The authors achieve this result via a new linear programming relaxation and dualization for the problem that may be of independent interest.\n\n**Weaknesses**  \nMy sense is that learning-augmented papers at NeurIPS typically include some small numerical studies to show how the newly proposed algorithms compare against baselines — such a study would be a valuable addition to this paper.  I would guess that those results would strictly increase the paper’s strength, since algorithms that use parsimonious predictions often compare quite favorably against those that use more complicated predictions models.\n\nQUESTIONS:\nFor NeurIPS, I think it would be valuable to give a little bit of practical context about the unrelated machine scheduling problem in strategic settings — i.e., what are some applications that this problem models (or approximately models)?\n\nMinor comment: on line 180, I believe “assims” should be “assigns”.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers learning-augmented truthful mechanisms for makespan scheduling on unrelated machines. In makespan scheduling on unrelated machines, we are given $m$ jobs to be scheduled on $n$ machines, and the goal is to assign the jobs to the machines with the objective to minimize the maximum processing load assigned to a single machine. The processing time $p_{ij}$ of a job $j$ depends on the machine $i$ to which it is assigned. In this paper, the authors consider a strategic variant of this problems, where the machines are selfish agents that might misreport their processing times for the jobs in order to receive a smaller processing load. The goal is to design a strategyproof scheduling mechanism in which no agent has an incentive to misreport their processing times, while still minimizing the objective of the underlying scheduling problem. In this setting, the best-possible approximation ratio of any strategyproof mechanism is $n$. In the learning-augmented setting, the mechanism has access to untrusted predictions on the problem instance. If the meachnism has access to predicted processing times for all job-machine-pairs, then there is a $\\mathcal{O}(1)$-consistent and $\\mathcal{O}(n)$ -robust algorithm (Balkanski et al. (2023)), where the consistency is the approximation ratio for perfect predictions and the robustness is the approximation ratio for arbitrary predictions. If the number of predicted values is linear, then the previously best-known learning-augmented algorithm is $\\mathcal{O}(1)$-consistent and $\\mathcal{O}(n^2)$-robust (Christodoulou et al. 2024). The main contribution of this paper is a $\\mathcal{O}(1)$-consistent and $\\mathcal{O}(n)$-robust algorithm using a linear number of predicted values. \n\nTo achieve this result, the authors assume access to a predicted optimal job assignment, a predicted optimal makespan, and predicted optimal variables for the dual of a certain LP-relaxation. At its core, the presented mechanism greedily matches a job $j$ to the machine $i$ of minimum $p_{ij}\\beta_i$, where $p_{ij}$ is the reported processing time of job $j$ on machine $i$ and $\\beta_i$ is the predicted dual variable, provided that the reported time is smaller then the prediceted optimal makespan. The predicted assignment is only used for tiebreaking. The key idea of the analysis is the usage of a new LP-relaxation that guarantees the dual variables to be within a certain range. Assuming access to predictions on these dual variables, the authors can now use that the predicted values are within this range, which is crucial for proving the improved guarantees. In the appendix, the authors give an error-tolerant version of this mechanism.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n* The presented algorithm essentially matches the previously best-known consistency and robustness trade-off, while using significantly smaller predictions ($\\mathcal{O}(n+m)$ instead of $\\mathcal{O}(n \\cdot m)$ ), and significantly improves upon the previously best-known guarantees for predictions of a linear size. This seems to be significant progress towards the best-possible trade-off for small predictions.\n* The main technical contribution to achieve this result is a new LP-relaxation. This new LP-relaxation allows to bound the dual variables within a certain range, which is crucial for the proof of the consistency and robustness guarantees. In my opinion, this is a new and very elegant idea to achieve improved results. \n* Since many learning-augmented algorithms in the literature are based on predicted dual variables, the used techniques could potentially also be applicable in different contexts, which raises intriguing follow-up questions.\n\nWeaknesses:\n* In the appendix, the authors show how to design an error-tolerant version of their mechanism, which is also less brittle (more robust against small perturbations in the predicted values). I agree that this approach gives an error-dependency and addresses the brittleness with respect to the predicted makespan and the predicted assignment. However, I am not sure about error-tolerance with respect to the predicted $\\beta$-values. The original algorithm is defined to immediately discard infeasible predicted $\\beta$-values, even if they are only outside the feasibility range by an $\\epsilon$. The error-tolerant version seems to do the same (at least the proof of Theorem C.2 still assumes that the predicted values are feasible), which would mean that error-tolerant mechanism remains very brittle against slightly infeasible $\\beta$-values.\n* While the presented approach uses smaller predictions than the algorithm by Balkanski et al. (2023), it is completely unclear to me what the benefit of these smaller predictions is. As stated by the authors, it remains unclear whether it is easier to learn these smaller predictions than the $m \\cdot n$ processing times. Since the presented learnability result is with respect to the processing times, it remains unknown whether it is possible to learn the smaller predictions without also learning the processing times. Even the error-threshold $\\eta$ is with respect to the distortion of processing times and not a function of the actual predictions.\n* Apart from the new LP-relaxation, most of the used techniques seem to be already established in the area of learning-augmented algorithm design, which somewhat limits the technical contribution beyond the new relaxation. Admittedly, you could also see this as a strength of the new relaxation rather than a weakness.\n\nQUESTIONS:\n* How does the error-tolerant algorithm behave if the predicted $\\beta$-values are slightly infeasible, i.e., outside the range $[1/cn,1]$? Does it immediately fall back into the robustness case, or does it still achieve improved guarantees?\n* Apart from the new LP-relaxation, what would you consider the main technical contribution of your paper?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the unrelated machine scheduling to minimize the makespan. The processing time matrix is not known a priori. The machines are strategic agents, meaning that they may misreport processing times to benefit themselves, i.e., lower their processing load. The problem is to design a strategyproof mechanism where misreporting can only lower each agent’s utility and minimize the makespan. The problem is considered under the algorithms with predictions setting, where the mechanism is given predictions about makespan, machine loads, and the job-to-machine mapping. The problem of strategyproof scheduling has been studied where the predicted processing times are accessible and a constant consistent and O(n) robust algorithm exists. The author proposes a strategyproof mechanism based on linear program relaxation and rounding, with the number of predictions required being linear in the number of machines and jobs. The consistency is constant and robustness is in O(n), i.e., the machine is approximately optimal if the predictions are correct and O(n) times the optimum under arbitrarily bad predictions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n1. The problem is well-presented. \n2. The contribution of reducing the number of predictions is critical. \n3. The key metrics of algorithms with predictions are analyzed, e.g., consistency and robustness. \n\nWeaknesses: \nAlthough the attempt to reduce the number of predictions is meaningful, the total contribution does not seem to pass the bar of NeurIPS. It did not at least convince the reviewer that it was the case. I will elaborate below. \n\nThe definition of prediction error is too tailored to the analysis and the algorithm and it is very strict. There are two weaknesses: 1) the prediction for the makespan is at least the optimum and 2) the predictions for machine load and assignment need to be exact so that the mechanism can output a solution (after rounding) that matches the optimal solution to the modified primal. First of all, it seems unclear how the bound of the makespan predictions can be achieved. Second, it is hard to examine the hardness of the second condition; it is unclear if the machine load and assignment can easily end up getting the exact outcome in practice. These two weaknesses make the error metric unintuitive and hard to examine its effectiveness in measuring the prediction quality. It seems possible that some set of predictions does not have correctness. It seems hard for any learning model to produce predictions that are gamma-correct for any gamma. \n\nThe weakness in the prediction error makes the consistency hard to interpret. It is still the case that the algorithm is constant consistent when all predictions are perfect, but it does not make meaningful sense when the predictions are not perfect and some predictions do not correspond to any gamma-correctness. Therefore, the reviewer has to assess the consistency not as a function of the prediction error, but as a constant for the case of perfect predictions. This drawback significantly lowers the technical contribution of the paper. \n\nIn my opinion, the assessment of prediction error definition is inherently discretionary. I assess it as indecent due to 1) being too strict and tailored to the algorithm and 2) non-coverage for all predictions. This makes the analysis for an error-dependent consistency less contributive — I have to interpret the consistency as being meaningful for the case of the perfect prediction but not the general case. This lowers the contribution of the paper to below the bar of acceptance. \n\nMinor issues: \n1. It is unclear when reading section 3 “a new linear program relaxation”, what the old is and what the contributions of the new one are. The authors include the old linear program in the appendix, but a clear discussion is needed for the readers to appreciate the contributions. \n2. Incomplete sentence in Line 300, “complementary slackness implies that …”.\n\nQUESTIONS:\n1. It seems that the predicted job-to-machine map phi is not as important, which is used just for tie-breaking. Is it possible to drop this via still achieving the same robustness and consistency? \n\n2. Can the authors justify strong sensibility and rationale for the prediction error metric — gamma-correct predictions? If the authors can show that a gamma-correct prediction for some gamma is always (or nearly always) achievable via some learning techniques, the reviewer may consider a re-evaluation.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies an important problem from the algorithmic game theory literature, strategy-proof scheduling. The authors show how it is possible to use a linear number of predictions (O(n+m) where m if the number of jobs and n is the number of machines) to improve on the approximations known when taking a worst-case analysis approach. They do so by using a novel framing of the problem, and match or improve on known results in the space, using less predictions/predicted information.\n\nSTRENGTHS AND WEAKNESSES:\nOverall this is a strong, technically interesting paper, that advances the state-of-the art in strategy-proof scheduling, a classic problem from algorithmic game theory. \n\nStrengths:\n- Studying a well-motivated problem (strategic scheduling) from the algorithmic game theory literature.\n- Novel framing of the problem (i.e. a new framing of the underlying linear programming relaxation) which allows them to provide a nearly optimal tradeoffs with respect to consistency and robustness using a linear number of predictions.\n- Appears to be technically sound and full proofs are provided in the appendix (with good proof sketches in the main paper).\n\nWeaknesses:\n- The paper is dense which limits its accessibility. While reading the work, I was trying to figure out how I might use the results in practice. While I understood the role of the beta's in the dual formulation, I was less confident when it come to understanding how I would \"predict\" them if I was considering them as features of the machine. More clarity here would have been useful.\n\nQUESTIONS:\n1. I understood mathematically the role of the  predicted \\beta’s in the dual formulation. Furthermore, the paper does state that they can be viewed as “machine weights” (line 183) or machine biases (line 245).  However, if I was to actually deploy the mechanism in practice, is there an intuitive way to obtain these weights/biases? \n\n2. Theorem 5.1 provides a guarantee on the robustness of the mechanism, no matter what the prediction quality is. Theorem 5.2 looks at consistency and provides a more nuanced insight since the result depends on the quality of the prediction. Is it possible to revisit Theorem 5.1 and consider prediction quality? If so, would this require a different analytical approach?\n\n3. Can you provide examples of other, specific domains where the proposed approach might be useful?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}