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arxiv: 2504.03865 · v3 · submitted 2025-04-04 · 💻 cs.CG

Towards an Optimal Bound for the Interleaving Distance on Mapper Graphs

Erin Wolf Chambers , Ishika Ghosh , Elizabeth Munch , Sarah Percival , Bei Wang This is my paper

Pith reviewed 2026-05-22 21:05 UTC · model grok-4.3

classification 💻 cs.CG
keywords mapper graphsinterleaving distanceinteger linear programmingtopological data analysisReeb graphsassignment problemdata classificationsimilarity measures
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The pith

Integer linear programs provide the first framework for bounding the interleaving distance on mapper graphs

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces the first framework to bound the interleaving distance between mapper graphs by formulating integer linear programs. One ILP checks whether an n-interleaving exists for a chosen n, while the other finds the assignment between the graphs that minimizes the loss for that n. Mapper graphs discretize the shape of point-cloud data with a real-valued function, and their interleaving distance measures how much the graphs must stretch to match. The method turns a previously hard-to-compute quantity into a practical optimization task that supports classification of datasets by mapper-graph similarity.

Core claim

The authors show that two integer linear programs can bound the interleaving distance on mapper graphs. For any fixed n, the first program decides if an n-interleaving exists between two given mapper graphs, and the second program identifies the assignment of maps that achieves the smallest loss value. Because the loss from prior work upper-bounds the true distance, the minimal loss supplies a computable upper bound. The approach is evaluated on small graphs with known distances and on benchmark and simulated data to illustrate its use in classification tasks.

What carries the argument

Integer linear programs that encode the existence of an n-interleaving or minimize the loss over possible assignments between two mapper graphs

If this is right

  • If an ILP for a fixed n returns a feasible solution, the interleaving distance between the two mapper graphs is at most n.
  • The smallest loss found for a given n supplies a concrete numerical upper bound on the distance.
  • The resulting bounds can be used directly for classification or clustering tasks that compare mapper graphs from different datasets.
  • The same ILP formulations apply both to small examples where the distance is known exactly and to larger simulated and benchmark collections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the ILPs remain tractable on larger mapper graphs, the bounds could support similarity queries on streaming or incrementally built data summaries.
  • Because the exact interleaving distance on the underlying Reeb graphs is NP-hard, these ILP bounds offer a practical relaxation whose tightness can be measured on the discrete mapper case.
  • Optimizing the ILP over the choice of n itself might yield the exact distance in cases where the minimal loss reaches zero.

Load-bearing premise

The loss function is assumed to give a valid upper bound on the true interleaving distance, and the ILP constraints are assumed to correctly capture when an n-interleaving exists.

What would settle it

On any pair of mapper graphs whose exact interleaving distance is known by other means, the minimal loss returned by the second ILP is strictly smaller than that known distance.

Figures

Figures reproduced from arXiv: 2504.03865 by Bei Wang, Elizabeth Munch, Erin Wolf Chambers, Ishika Ghosh, Sarah Percival.

Figure 1
Figure 1. Figure 1: An example of an input space in 2D (a cup from the MPEG-7 data set), and the mapper graph computed using filtration function given by pixel location on the y-axis is shown at right. and practice [49, 52, 54, 61, 48]. The basic idea used in practice for defining the mapper graph is as follows. Given a dataset χ which has a distance measure d : χ × χ → R and a filter function f : χ → R, fix a cover U of the … view at source ↗
Figure 2
Figure 2. Figure 2: (Left) An example of a line and torus mapper graph, where the latter has a loop of height [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Left) Example images for the used categories of images from the MPEG-7 dataset. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (Left) Point clouds generated for the letter [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The notation used for the cover of R. At left, we emphasize the difference between the cover element Uσ, and the open set in the Alexandrov topology, Sσ, which is a discrete set of three objects. At right, we show the 1-thickenings Sσi and Sτi for the two types of basis open sets. form F : Open(U) → Set a mapper cosheaf. For notational simplicity, we write the induced map as F[⊆] : F(S) → F(T) when S ⊆ T i… view at source ↗
Figure 6
Figure 6. Figure 6: Example portion of a mapper cosheaf graph representation. For example, the vertices [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An example pair of input mapper cosheaf graphs, [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example boundary matrices of the graph of [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The inclusion matrices I V F (left) and I E F (right) giving the inclusion natural transformation F ⇒ Fn for the example shown in [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distance matrix example for F n from [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: All the matrices for a random input unnatural transformation [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Examples of the matrix multiplications used to determine the value of the loss function. [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Time taken by different solvers to optimize the loss for the torus and line mappers [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: (Left) Relationship between the computed upper bound and true interleaving distance [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (Left) An example of a pair of mapper graphs where the binary search on [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: (Left) Accuracy of KNN classifier for different [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Accuracy of KNN classifier for different k values during 5-fold cross-validation. The [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
read the original abstract

Mapper graphs are widely used tools in topological data analysis and visualization. They can be understood as discrete approximations of Reeb graphs, providing insight into the shape and connectivity of complex data. Given a high-dimensional point cloud together with a real-valued function defined on it, a mapper graph summarizes the induced topological structure: each node represents a local neighborhood, and edges connect nodes whose corresponding neighborhoods overlap. Our focus is the interleaving distance for mapper graphs, arising as a discretized analogue of the interleaving distance for Reeb graphs-a quantity known to be NP-hard to compute. This distance measures how similar two mapper graphs are by quantifying how much they must be ``stretched'' to be made comparable. Recent work introduced a loss function that gives an upper bound on this distance. The loss evaluates how far a given collection of maps, called an assignment, is from being a true interleaving. Importantly, it is computationally tractable, offering a practical way to bound the distance, however the quality of the bound is dependent on the choice of assignment. In this paper, we develop the first framework for bounding the interleaving distance on mapper graphs. We present the bound in two ways: first, by formulating an integer linear program (ILP) that determines whether an $n$-interleaving exists for a given $n$; and second, by constructing an ILP that identifies an assignment with minimal loss for that $n$. We also evaluate the method on small examples where the interleaving distance is known, and on benchmark and simulated datasets, demonstrating the utility of the approach for classification tasks based on mapper graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to develop the first framework for bounding the interleaving distance on mapper graphs. It does so by presenting two ILP formulations—one that decides the existence of an n-interleaving for a fixed n, and one that finds an assignment minimizing the loss from prior work for that n—thereby producing a computable upper bound. The approach is evaluated on small instances whose true distances are known by other means as well as on benchmark and simulated datasets, with demonstrated utility for classification tasks based on mapper graphs.

Significance. If the ILP encodings are faithful and the loss function supplies a valid upper bound, the work supplies the first practical algorithmic method for a quantity whose exact computation is NP-hard. The explicit ILP constructions together with the evaluation on instances with independently known distances constitute a concrete, falsifiable contribution that could be adopted for mapper-graph comparisons in topological data analysis.

minor comments (3)
  1. The manuscript should include a short table or paragraph in the evaluation section listing the sizes of the small instances (number of nodes/edges) and the wall-clock times required to solve the two ILPs; without these data it is difficult to judge scalability even for the “small” regime.
  2. In the ILP formulations, the meaning of each binary variable and each family of constraints should be stated explicitly in a single dedicated paragraph or table immediately after the mathematical program is displayed; current notation leaves some auxiliary variables implicit.
  3. A brief comparison, even on the small examples, between the bound obtained by the new minimal-loss ILP and the bound obtained by the heuristic assignment method of the cited prior work would strengthen the claim that the ILP improves upon existing practice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our contribution and for recommending minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution is the explicit construction of two new ILP formulations: one to decide existence of an n-interleaving and one to minimize the loss of an assignment for fixed n. These are presented as algorithmic encodings of combinatorial conditions that were previously intractable. The loss function itself is imported from prior work as an external upper bound; the paper does not redefine the distance in terms of the loss or fit parameters to the target quantity and then relabel the fit as a prediction. No self-citation chain is invoked to justify uniqueness or to smuggle an ansatz, and no renaming of known empirical patterns occurs. The derivation chain is therefore self-contained: the ILPs are new objects whose correctness is asserted by direct encoding rather than by reduction to the paper's own fitted outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, axioms, or invented entities; none can be identified.

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