Paper: Learning Node Selection via Tripartite Graph Representation in Mixed Integer Linear Programming (ICLR-2024)

Reviewer comment (Experiments):
There is no ablation study results to support the advantages of the proposed method. For example, how about using the traditional bipartite graph representation instead of the newly proposed tripartite scheme? What are the contributions of the three reward componenets on the final performance?

Author response:
Thank you for your insightful comments on the evaluation aspects of our paper. We recognize the importance of conducting extensive experiments and ablation studies to validate the effectiveness of our proposed method. We are actively working on supplementing our experiments to address these gaps. However, due to time constraints, we were not able to complete these additional experiments in time for this submission. We are committed to continuing this work and will prioritize completing these evaluations in the near future.

Reviewer comment (Writing):
The tripartite graph representation in Section 3.2 could be explained a lot more clearly. As is, it is just defined as an abstract object but the authors never define clearly how it is built from the underlying MIP. It is more or less possible to infer how they do this from Figure 1, but all of this ought to be clearly defined in the body of the paper.

Author response:
We appreciate your constructive feedback on Section 3.2 regarding the tripartite graph representation. In response to your comments, we have revised the manuscript to include a detailed explanation of how the tripartite graph is built step-by-step.

Reviewer comment (Theory):
I’m not really sure what Theorem 3.2 is saying. Branch-and-bound is a deterministic algorithm, so indeed all the information needed for its execution is contained at the root node for a fixed node selection policy. A brief glance at the proofs in the appendix makes me skeptical that Theorems 3.1 and 3.2 are saying anything rigorous/meaningful about branch-and-bound.

Author response:
Thank you for your insightful observation regarding Theorem 3.2. It appears there may be a misunderstanding regarding the focus and implications of the theorem, which I would like to clarify.

Your comment rightly points out that branch-and-bound (B&B) is a deterministic algorithm and, with a fixed node selection policy, the outcome is determined by the information at the root node. However, the scenario we address in our theorem is different. Theorem 3.2 asserts that, regardless of the node selection strategy employed at any specific round of the B&B algorithm, all information obtained through the exploration of nodes up to that point can be effectively represented using only the information from the root node and the current leaf nodes. It means that for the purpose of representing the state of the B&B tree at any given round, it is not necessary to record detailed information about all the intermediate nodes that have been explored. Instead, it suffices to focus on the root node and the leaf nodes, which carry the most recent branching constraints and updated bounds. This theorem significantly streamlines the representation of the B&B process. By focusing on the root and leaf nodes, we avoid the redundancy of recording extensive information about every intermediate node, thereby reducing computational overhead and improving efficiency.

Reviewer comment (Theory):
I do not understand Theorem 4.2 and its proof. The authors claim here that node selection and variable selection are “independent”. The proof seems to show that given a fixed tree state, the optimal node does not depend on the variable selection policy. But the assumption that there is a given fixed tree state means that the variable selection decisions have already been made at the point of node selection. Varying the variable selection policy would change the current subtree, and the premise of the theorem wouldn’t hold. I don’t think it is scientifically valid to claim from this theorem that node selection and variable selection are “independent”, and my guess is that most viewpoints in MIP solving would take issue to this claim.

Author response:
Thank you for your insightful comments and for pointing out the need for clarity regarding Theorem 4.2. Your expertise in this area is greatly appreciated. 

We apologize for any confusion caused by our use of the term "independent" in our initial draft. In the revised version of the paper, we have adjusted our terminology to more accurately reflect the relationship between node selection and variable selection within the branch-and-bound (B&B) process.

Regarding Theorem 4.2, our intention was to highlight that while the variable selection process shapes the tree, the node selection strategy can independently assess the tree's current state to determine the next course of action. As you correctly point out, variable selection decisions indeed influence the formation of the current subtree, and varying variable selection policies would result in different subtrees.

The core idea we wish to convey is that once a specific subtree is formed, the node selection process focuses solely on this existing state of the tree to identify the most promising node for exploration. This decision-making process within the node selection phase does not require direct knowledge of the variable selection policies that led to the current state of the tree. It is this aspect of the process that we sought to characterize as being 'independent' of the variable selection strategy.
The contribution of this theorem lies in addressing a common concern in prior research: the interdependence of node and variable selection in B&B algorithms. While it is true that variable selection influences the shaping of the subtree and, consequently, the options available for node selection, our theorem clarifies that the node selection strategy itself does not need to factor in the specifics of the variable selection decisions. Instead, it can effectively operate based on the current state of the tree, regardless of the variable selection path that led to it.

Reviewer comment (Writing):
The writing in the paper needs a lot of improvement. There are grammatical errors and issues with sentence structure in nearly every paragraph that make the paper difficult to read.

Author response:
Thank you for pointing out the writing issues. We have thoroughly reviewed and revised the paper to enhance its readability and clarity.

Reviewer comment (Reproducibility):
Doesn’t the set NC in the tripartite graph representation grow very large since every leaf node is stored? Are there any limitations due to this?

Author response:
Thank you for your astute observation regarding the potential size of the set NC in our tripartite graph representation. Indeed, as the number of leaf nodes in the branch-and-bound tree increases, the corresponding set of node constraints (NC) could potentially grow large.

However, it's important to note that we only need to store a limited amount of information for each leaf node—specifically, the unique constraints added due to branching and the upper and lower bounds associated with these nodes. This quantity of information, while seemingly extensive, does not significantly exceed the storage requirements compared to previous approaches where each node is treated as an independent MILP problem.

Moreover, it is crucial to understand that these leaf nodes represent the current options available for node selection in the branch-and-bound process. Prior to making a selection, it is indeed necessary to represent the characteristics of each option comprehensively. Therefore, despite the potential size of NC, representing this information is essential for effective decision-making within our proposed framework.

Additionally, we are actively exploring methods to further reduce and optimize the storage of this information without compromising the effectiveness of node selection in our framework.

Reviewer comment (Evaluation):
Since the performance is inconsistent on larger cases, can author provide any insights on what type of MILP problem would be suitable for the proposed methods?

Author response:
Thank you for your insightful question, which has prompted valuable reflection on our part regarding the performance of our methods across different types of MILP problems.
Our approach is designed to be generally applicable to a broad range of MILP problems. However, as you rightly pointed out, we observe varying performance depending on the specific characteristics of each problem. We appreciate the inspiration your question has provided and, it reflects a common phenomenon where a general-purpose model exhibits varying performances across different tasks. This variation could be linked to deeper issues of interpretability within GNN models. While a definitive answer might be complex, we can, based on our experience and observations, offer some conjectural insights.

We hypothesize that our method may be more suitable for MILP problems where the relationship between decision variables and constraints is more straightforward and less obscured by complex interrelations. An example of where our method might show more promise is in problems like the Weighted Partial MaxSAT (WPMS). In such problems, the structure may allow for clearer constraints, which our method could potentially leverage more effectively.

Reviewer comment (Experiments):
What is the rule of setting hyperparameter n (# of nodes for selection)? What if I choose some other values (e.g., n=30,50,100)

Author response:
Thank you for raising this excellent question. The primary reason for setting a parameter $n$ is to manage the dynamic variation in the number of leaf nodes within the branch and bound tree. By imposing a limit on $n$, we constrain the action space of our reinforcement learning model to a manageable subset of nodes, facilitating more efficient learning and decision-making processes. In our experiments, we opted for a relatively large value of $n$, ensuring that it was generally greater than or equal to the actual number of leaf nodes. Our observations suggest that when $n$ is adjusted around the number of leaf nodes, the results remain largely consistent, indicating a degree of robustness in this approach.

However, it is important to acknowledge that we did not extensively explore scenarios where $n$ is significantly smaller. The impact of a reduced $n$ on our results might depend on the heuristics used for pre-selecting nodes to be included in this subset.

Reviewer comment (Experiments):
Do you also restrict the number of nodes to be selected for SCIP?

Author response:
Thank you for your question！In our approach, we do not impose a restriction on the number of nodes to be selected specifically for SCIP. The rationale behind this decision is that our method for pre-selecting the top $n$ nodes, based on their scores, leverages SCIP's default node selection strategies. In such a setup, if we were to pre-select the top $n$ nodes using the same rules and then choose the highest-scoring node from this subset, the outcome would essentially be identical to directly selecting the top node according to SCIP's default process. This is because the node that emerges as the highest-scoring in both scenarios is the same. Therefore, introducing a step to limit the selection to $n$ nodes within SCIP would be redundant.

Reviewer comment (Experiments):
The empirical study was done on pretty easy instances and the slowest approaches took only less than half a minute to solve them. It is more common for this kind of study to test on harder and/or larger instances.

Author response:
For weakness 1 and 2:

We sincerely appreciate the valuable comment made by the reviewers that have helped enhance our paper. We acknowledge that our initial experiments were conducted on relatively easier instances, primarily due to time and computational resource constraints. We are currently in the process of expanding our experiments to include more challenging datasets, specifically following the approaches used in He et al. (2014) and Nair et al. (2020). The medium datasets being incorporated include MIK (Atamtürk, 2003), which consists of a set of MILP problems with knapsack constraints, and CORLAT (Gomes et al., 2008), a real dataset utilized for constructing a wildlife corridor for grizzly bears in the Northern Rockies region. This will include not only runtime and the number of nodes but also more detailed metrics such as win/loss rates.
Due to time constraints, these additional experiments are still underway and have not been completed for inclusion in this revision. However, we are committed to providing a more comprehensive evaluation in our final version, should the paper be accepted.

Reviewer comment (Evaluation):
Only runtime and number of nodes were reported in evaluations. It will be important to include more metrics such as win/loss rate and scatter plots to understand the per-instance performance.

Author response:
For weakness 1 and 2:

We sincerely appreciate the valuable comment made by the reviewers that have helped enhance our paper. We acknowledge that our initial experiments were conducted on relatively easier instances, primarily due to time and computational resource constraints. We are currently in the process of expanding our experiments to include more challenging datasets, specifically following the approaches used in He et al. (2014) and Nair et al. (2020). The medium datasets being incorporated include MIK (Atamtürk, 2003), which consists of a set of MILP problems with knapsack constraints, and CORLAT (Gomes et al., 2008), a real dataset utilized for constructing a wildlife corridor for grizzly bears in the Northern Rockies region. This will include not only runtime and the number of nodes but also more detailed metrics such as win/loss rates.
Due to time constraints, these additional experiments are still underway and have not been completed for inclusion in this revision. However, we are committed to providing a more comprehensive evaluation in our final version, should the paper be accepted.

Reviewer comment (Writing):
Writing can be improved. (1) The edges between V and NC are never defined. I think they are defined similarly to the edges between V and C; (2) Gap reward in Theorem 4.1 is undefined (though defined later); (3) It would be important to describe the intuitive idea for proving each theorem. (4) Typo: Section 3.2: V \cup C = V\cup NC = C\cup NC should be taking the intersection instead. Section 4.2 mordern -> modern.

Author response:
For weakness 4:

Thank you for pointing this out. We made corrections and clarifications in the revision.

Reviewer comment (Theory):
The node constraints vertices represent constraints added in addition to the root problem. They seem to be only one variable if they are constraints added by branching, as mentioned in 3.2 (the 4th line in the paragraph). In that case, each NC vertex will have only one edge connecting to one variable vertex. Is the tripartite representation necessary? This would be more useful if you consider cuts added to the nodes.

Author response:
For Question 1:

Specifically, concerning the necessity of the tripartite representation: although each leaf node indeed adds only one constraint relative to its parent node due to branching, these constraints, relative to the original problem, accumulate over several rounds of branching. Therefore, representing these subproblems (leaf nodes) in contrast to the original problem's constraint space requires multiple edges connecting to variable nodes, not just one.