Paper: The Central Spanning Tree Problem (ICLR-2024)

Reviewer comment (Novelty):
The introduced problems, CST and BCST problems, are currently less meaningful in both theoretical and practical senses than the existing problems. To introduce a new problem, it is desired that at least one of the following points are realized: (I) The newly introduced problem has a strength or offers a trade-off compared to the existing problems. (II) The newly introduced problem introduces a new better solution/insight on the existing problems.

Author response:
- **Advantages of CST over MRCT**: The consistent performance of mSTreg across diverse $\alpha$ values stands out as a significant strength, aligning with our goal of creating a useful heuristic. Therefore, the point that the CST is as difficult to solve as the MRCT is secondary, as the CST has still valuable advantages over the MRCT. While MRCT's higher star-shape tendency enhances stability, it risks information loss, especially in higher dimensions, as depicted in Figure 12. The parameterized nature of $\alpha$ in CST mitigates this, allowing for a tailored balance between robustness and data fidelity. This flexibility is particularly crucial in higher dimensions, where the CST's interpolation surpasses the limitations of MRCT. These observations underscore the superiority of the more general CST and its branched version, emphasizing their real-world applicability, specifically in higher dimensions.

Reviewer comment (Novelty):
Regarding (I), although it is empirically confirmed that the CST is robust against data perturbations, the MRCT is also robust against data noise. The expectation was that the CST problem could be solved more easily than the MRCT problem. However, CST is NP-hard for $\alpha>0$ (Appendix C), and the performance of the proposed heuristics (mSTreg) has no clear correspondence to the $\alpha$ value (Appendix N). This means that the MRCT problem can be solved to the same extent as the CST/BCST problem, providing no significant advantage of CST/BCST over MRCT.

Author response:
Concerning the level of difficulty in solving the more general CST/BCST compared to the MRCT, it is important to clarify that we are not asserting that our proposal is inherently easier to solve. Rather, our argument centers on the observation that the parameterized CST/BCST exhibits more appealing properties than the MRCT. As a result, opting for the CST/BCST might be justified even if both are equally challenging to solve.

Reviewer comment (Experiments):
Regarding (II), the proposed heuristics for solving CST/MRCT problems (mSTreg) is applied to existing problems, namely the Steiner tree problem and MRCT problem. For the Steiner tree problem, only instances with known optimal solutions are used as benchmarks. For the MRCT problem, mSTreg falls short compared to the existing algorithm GRASP_PR. Although the authors claim that GRASP_PR is time-consuming due to heavy local search and can be improved by using mSTreg for preprocessing, this claim is not experimentally verified. Thus, currently the proposed heuristics, mSTreg, only outperforms existing methods for the newly proposed CST/BCST problems.

Author response:
Regarding the 5-minute time limit, we aimed to demonstrate that mSTreg can outperform the MRCT in terms of efficiency, while yielding satisfactory solutions. Notably, mSTreg, executed with just 20 iterations, produced a solution within seconds. In contrast, GRASP_PR did not consistently enhance this solution within the specified 5-minute timeframe. Extending the runtime of GRASP_PR would likely result in further improvements, as it would explore the solution space more extensively. It is important to emphasize that GRASP_PR, initialized with mSTreg, also adhered to the 5-minute time limit, surpassing the other two alternatives. This demonstrates the clear benefit of the combined approach and highlights how initializing with mSTreg can enhance GRASP_PR.

Reviewer comment (Presentation):
The main weakness of this article is its presentation. From my point of view, it is not in the right format for a conference like ICLR. The main paper refers extensively to the appendices, which are 30 pages long. Some parts of the main paper are difficult to read without calling up the long version, particularly the optimization algorithm (section 4). Mention should also be made of the link with the minimum concave cost network flow problem, which helps to show that the problem is NP-hard: some elements are given in the main body, but the link can only be understood by carefully reading the appendix.

Author response:
- **Presentation**: While we acknowledge the length of our Appendix, we believe it serves the purpose of providing comprehensive details for readers interested in a deeper understanding of our work. Our intention is to strike a balance between clarity in the main paper and the availability of detailed information in the Appendix. We contend that our manuscript does its best at conveying the most relevant information in the main part. Regarding the comprehension of our heuristic, we would value specific feedback on the aspects that may have been challenging, enabling us to make targeted improvements in the clarity of the section.

Reviewer comment (Experiments):
The main advantage of the proposed model is its robustness to noise. However, it is only compared with the minimum spanning tree (and the Steiner tree in the case where nodes can be added), which is known not to be robust to noise. However, competing algorithms are mentioned: some are even used as benchmarks to measure the accuracy of the heuristic. The paper's main motivation (robustness) should be more thoroughly documented.

Author response:
- **Robustness discussion**: To the best of our knowledge, we are the first ones to delve into the robustness of the geometrical structure of a spanning tree in response to data perturbations, such as noise. We thank the reviewer for providing additional references. However, it's crucial to highlight that the robustness treated by these papers differs from ours. Bezrukov et al.'s paper, although related by name, focuses on computing a Frechet mean among all spanning trees of a graph using the symmetric difference of edge sets as a metric. The paper by Kasperski and Zielinski studies the mST with uncertain edge costs. Thus, they aim to find a tree whose cost is robust under different realizations of the edge costs. Thus, both papers neglect the geometrical stability of the tree. Consequently, our comparison only included the mST and the MRCT, as these are natural limits of our CST proposal. Note that, though the MRCT already existed previously in the literature, its robustness, as explained here, had not been mentioned before. Moreover, as discussed in  "Advantages of CST over MRCT"  (see response to H47r), the MRCT's robustness may cause an information loss in contrast to the parameterized CST. In light of this, we contend that our study effectively addresses the specific robustness considerations within our scope. Simultaneously, we acknowledge that our related work did not mention alternative robustness criteria, and we will amend this in the final version.

Reviewer comment (Writing):
If the understanding of the definition of Steiner points on page 3 is correct, the data are only used for the terminal nodes of the tree, while all the internal nodes are added (they are not part of the data) and their position is to be optimized. Confirmation is asked whether this is really the case.

Author response:
- **Role of Steiner points (SPs)**: Indeed, your understanding of the SPs is right. Regarding the formulation of the optimization problem in (2), we posit that it is appropriately formulated. While (1) only minimizes over the set of spanning trees, (2) jointly minimizes both the trees and the SPs' positions, what from our perspective does not pose a conflict. (2) is equivalent to minimizing the SPs' positions given a fixed topology and subsequently selecting the topology that achieves the minimum cost. If there are specific aspects of (2) that are unclear or concerning, we would appreciate additional details to address them comprehensively. 

  Regarding the relevance of the Steiner points to model the structure, we acknowledge your valid point. Although SPs are not inherent to the original data, there are scenarios, where abstracting the data requires points that are not necessarily part of the raw data. For example, in many single-cell trajectory inference methods, it is common to define cluster centers (which are not part of the actual data) to describe the trajectory. Also in skeletonization the SPs become relevant as argued below in response "CST and Skeleton". We will emphasize these aspects in the final manuscript to provide a more comprehensive understanding of the applicability of the SPs.

Reviewer comment (Theory):
If the previous point holds, it raises two questions: (i) Is optimization problem (2) well formulated compared to equation (1)?

Author response:
- **Role of Steiner points (SPs)**: Indeed, your understanding of the SPs is right. Regarding the formulation of the optimization problem in (2), we posit that it is appropriately formulated. While (1) only minimizes over the set of spanning trees, (2) jointly minimizes both the trees and the SPs' positions, what from our perspective does not pose a conflict. (2) is equivalent to minimizing the SPs' positions given a fixed topology and subsequently selecting the topology that achieves the minimum cost. If there are specific aspects of (2) that are unclear or concerning, we would appreciate additional details to address them comprehensively. 

  Regarding the relevance of the Steiner points to model the structure, we acknowledge your valid point. Although SPs are not inherent to the original data, there are scenarios, where abstracting the data requires points that are not necessarily part of the raw data. For example, in many single-cell trajectory inference methods, it is common to define cluster centers (which are not part of the actual data) to describe the trajectory. Also in skeletonization the SPs become relevant as argued below in response "CST and Skeleton". We will emphasize these aspects in the final manuscript to provide a more comprehensive understanding of the applicability of the SPs.

Reviewer comment (Evaluation):
If the previous point holds, it also raises the question: (ii) Is this model really relevant to extract the tree structure of a point cloud? The reviewer thinks this point should be better discussed throughout the paper.

Author response:
- **Role of Steiner points (SPs)**: Indeed, your understanding of the SPs is right. Regarding the formulation of the optimization problem in (2), we posit that it is appropriately formulated. While (1) only minimizes over the set of spanning trees, (2) jointly minimizes both the trees and the SPs' positions, what from our perspective does not pose a conflict. (2) is equivalent to minimizing the SPs' positions given a fixed topology and subsequently selecting the topology that achieves the minimum cost. If there are specific aspects of (2) that are unclear or concerning, we would appreciate additional details to address them comprehensively. 

  Regarding the relevance of the Steiner points to model the structure, we acknowledge your valid point. Although SPs are not inherent to the original data, there are scenarios, where abstracting the data requires points that are not necessarily part of the raw data. For example, in many single-cell trajectory inference methods, it is common to define cluster centers (which are not part of the actual data) to describe the trajectory. Also in skeletonization the SPs become relevant as argued below in response "CST and Skeleton". We will emphasize these aspects in the final manuscript to provide a more comprehensive understanding of the applicability of the SPs.

Reviewer comment (Theory):
Why do the authors exclude $\alpha>1$ from their investigations? Did they study the complexity of the optimization problem (which is proved to be NP-hard only for $0<\alpha\leq 1$) in this case?

Author response:
- **Case $\alpha>1$**: Apart from the limit case detailed in Appendix D.1, we refrained from further exploration due to practical observations indicating an excessive star-shaped tree, even in low dimensions. For instance, the toy dataset in Figure 2 became a star tree for $\alpha=1.15$. Regarding NP-hardness, it remains unproven in this regime as the strictly concavity argument applicable for $0<\alpha\leq1$ loses its validity. We hypothesize that in practice the problem might be generally easier to solve for higher $\alpha$ values, especially considering that, for the BCST problem, the SPs tend to collapse into a single point.

Reviewer comment (Experiments):
Why do the authors not consider the central spanning tree (but only the branched version) in the application to 3D plant skeletonization?

Author response:
- **CST and Skeleton**: We exclusively utilized the BCST for skeletonization, primarily because it is often desired that the skeleton goes through the "middle" of the volume. The SPs in the BCST offer the flexibility to be situated in the volume's center, ensuring that the BCST skeleton traverses this central region. In contrast, the CST lacks this flexibility and would, at best, zigzag over surface points. Hence, the CST was not considered for this particular application.

Reviewer comment (Theory):
On page 2: Is it obvious that the model turns into the minimum routing cost tree for $\alpha=1$?

Author response:
- **Equivalence MRCT and CST with $\alpha=1$**: As argued in the paper, the term $m_{ij}(1-m_{ij})$ is proportional to the betweenness centrality of the edge $(i, j)$. This centrality quantifies the number of shortest paths that traverse the given edge. Thus, the multiplication of the length of each edge by its frequency in a shortest path is a rearrangement of the sum over all shortest path costs, i.e. the MRCT cost. Formally,
 
$\displaystyle\sum_{i,j\in V\times V}d_T(i,j)=\sum_{i,j\in V\times V}\sum_{(u,v)\in P_{ij}} ||x_u-x_v||=\sum_{(u,v)\in T} \sum_{i,j\in V\times V}I_{(u,v)\in P_{ij}}||x_u-x_v||\propto \sum_{(u,v)\in T}m_{uv}(1-m_{uv})||x_u-x_v||$

  where $P_{ij}$ is the path from $i$ to $j$ in tree $T$ and $I_{(u,v)\in P_{ij}}$ is an indicator set.

Reviewer comment (Theory):
On page 6: Why is the optimization problem (2) not everywhere differentiable?

Author response:
- **Differentiability of (2)**: Equation (2) contains the term $||x_i-x_j||$. The Euclidean norm is not differentiable at 0, thus when two Steiner points $x_i$ and $x_j$ share coordinates, the function is not differentiable.

Reviewer comment (Theory):
Theoretical interest is noted in the study of branching angles and degree of Steiner points, but the link with the rest of the paper is unclear. Could this be of any use in addressing the paper's main questions?

Author response:
- **Branching angles study**: The study of the angles has practical applications on our algorithms. Having an analytical expression of the branching angles may serve as a criterion of the convergence of the iterative reweighted least square approach used to compute the SPs positions. Moreover, the infeasability of degree four SPs can ease the detection of non-optimal solutions and identify areas where the tree can be further improved, potentially leading to better results.

Reviewer comment (Evaluation):
In Section 2, a single cell gene expression measurement example is provided. However, it seems that CST also cannot detect the trajectory bifurcation. Moreover, as mST cannot detect the trajectory bifurcation even on the original data, the example seems hard to support the superiority of proposed model regarding the robustness.

Author response:
- **(Q2) CST on single cell data**: While CST may not fully capture the bifurcation, it does exhibit a discernible split, indicated by two wide edges. The fact that the mST is not able to detect the bifurcation, does not diminish the value of the robustness argument as the mST undergoes drastic changes post-perturbation, in contrast to the CST. This highlights the CST's superiority in terms of robustness, even if not optimal. Nevertheless, we acknowledge that BCST might be more suitable for such applications due to better trajectory representation and preservation. We remark, that this is not the only example we show; the Appendix illustrates another instance leading to the same conclusions.

Reviewer comment (Theory):
Is there any theoretical guarantee regarding the returned results of the heuristic algorithms proposed in section 4?

Author response:
- **(Q4) Guarantees of the heuristic**: Currently, our heuristic lacks theoretical guarantees regarding the cost of the solutions it provides. However, through empirical evaluation, we have demonstrated that our heuristic consistently produces competitive solutions. It is worth noting that many algorithms employed for this class of problems rely on genetic algorithms or local search strategies, neither of which inherently provide any formal guarantee. While some algorithms may offer worst-case scenario bounds, they often struggle to scale effectively for large instance problems [1,2].

[1] Ravelo et al., "A PTAS for the metric case of the optimum weighted source–destination communication spanning tree problem", (2019).

[2] Wu et al., "Approximation algorithms for some optimum communication spanning tree problems", (2000).