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arxiv: 2512.17417 · v3 · pith:OXMQ7QEWnew · submitted 2025-12-19 · 🧮 math.OC

Graph Isomorphism: Mixed-Integer Convex Optimization from First-Order Methods

Wenjie Xiao , Mathieu Besan\c{c}on , Patrick Gel{\ss} , Deborah Hendrych , Stefan Klus , Sebastian Pokutta This is my paper

Pith reviewed 2026-05-21 16:52 UTC · model grok-4.3

classification 🧮 math.OC
keywords graph isomorphismmixed-integer convex optimizationfirst-order methodsvariable fixingpolyhedral structurecomputational experimentssymmetry detection
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The pith

A mixed-integer convex formulation of graph isomorphism can be solved using first-order optimization methods.

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

The paper develops a convex mixed-integer program that exactly encodes whether two graphs are isomorphic. It solves this program with first-order convex optimization techniques and adds variable-fixing presolvers that remove provably fixed variables while keeping the formulation polyhedral. Computational tests on twelve families of hard instances show that the approach beats the next-best integer-feasibility method on six families and performs equally well on symmetric graphs. Symmetry helps the optimization methods, while local-structure presolving is decisive for asymmetric cases. The work therefore supplies a new practical route to an intermediate-complexity problem by importing tools from convex optimization.

Core claim

We propose a convex mixed-integer formulation of the graph isomorphism problem and leverage first-order convex optimization to tackle it, following a stream of recent work on optimization-driven graph isomorphism detection. We strengthen our formulation with variable fixing techniques that prove highly effective while preserving the polyhedral structure. On a collection of challenging GI instances the method outperforms the integer-feasibility baseline on six of twelve graph families and matches it on the symmetric families.

What carries the argument

The mixed-integer convex formulation of graph isomorphism, strengthened by variable-fixing techniques that detect and remove fixed variables without destroying polyhedral structure.

If this is right

  • First-order convex solvers become applicable to an intermediate-complexity problem.
  • Variable fixing reduces instance size while preserving correctness and polyhedrality.
  • Optimization-based methods gain an advantage on graphs with high symmetry.
  • Presolving that detects local substructures becomes essential for asymmetric instances.

Where Pith is reading between the lines

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

  • The same convex encoding might be adapted to related problems such as subgraph isomorphism or graph canonization.
  • Further polyhedral studies could tighten the formulation and reduce the number of variables that require fixing.
  • Hybrid schemes that combine the convex model with spectral heuristics on the hardest asymmetric cases could be tested.

Load-bearing premise

The mixed-integer convex formulation has no spurious solutions and captures every true isomorphism.

What would settle it

Two non-isomorphic graphs that satisfy the integer constraints of the formulation, or two isomorphic graphs that violate them.

Figures

Figures reproduced from arXiv: 2512.17417 by Deborah Hendrych, Mathieu Besan\c{c}on, Patrick Gel{\ss}, Sebastian Pokutta, Stefan Klus, Wenjie Xiao.

Figure 1
Figure 1. Figure 1: Solved instances over time for sts graph family. For both Paley graph families, Figures 3 and 4, the clique and star detec￾tion are nearly equally effective. Our method outperforms all other methods on these sets as well. Interestingly, the spectral method outperforms the feasibility approach on these sets. It, in particular, fares well on the paley_prime family. The iso_r01N family, [PITH_FULL_IMAGE:figu… view at source ↗
Figure 2
Figure 2. Figure 2: Solved instances over time for usr graph family. tection determines non-isomorphism at the root node for all instances. We high￾light however that having identical spectra is a necessary condition for isomor￾phism and is not respected by these graphs. Spectral analysis would therefore also discard them as non-isomorphic. Investigating the performance of optimization￾based methods on isospectral but non-iso… view at source ↗
Figure 3
Figure 3. Figure 3: Solved instances over time for paley_power graph family. 6 Conclusion In this paper, we develop and evaluate mixed-integer convex formulations and algorithms for the graph isomorphism problem. Despite the problem being well￾studied on theoretical and algorithmic aspects, we showed that a mixed-integer convex approach leveraging first-order methods and graph-specific presolving techniques could achieve grea… view at source ↗
Figure 4
Figure 4. Figure 4: Solved instances over time for paley_prime graph family. To effectively solve GI for asymmetric graphs, presolving techniques that detect substructures and infer variable fixings from them are crucial. The matching problem warrants further specific investigation to develop new formulations, algorithms, and understand in which cases various approaches can perform well [PITH_FULL_IMAGE:figures/full_fig_p015… view at source ↗
Figure 5
Figure 5. Figure 5: Solved instances over time for iso_r01N graph family [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

The graph isomorphism (GI) problem, which asks whether two graphs are structurally identical, occupies a unique position in computational complexity -- it is neither known to be solvable in polynomial time, nor proven to be NP-complete. We propose a convex mixed-integer formulation of the problem and leverage first-order convex optimization to tackle it, following a stream of recent work on optimization-driven graph isomorphism detection. We strengthen our formulation with variable fixing techniques that prove highly effective while preserving the polyhedral structure. We perform extensive computations evaluating the performance of different families of methods including a mixed-integer convex formulation, mixed-integer linear optimization, local search and spectral heuristics over a collection of challenging GI instances. We find that a high level of symmetry is beneficial for optimization-based methods. On the other hand, presolving techniques that detect local substructures to fix variables are crucial for asymmetric instances. The proposed method outperforms the second best approach, the integer feasibility approach, on 6 of the 12 graphs families and is on par with it on symmetric families.

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

3 major / 2 minor

Summary. The paper proposes a mixed-integer convex formulation for the graph isomorphism problem, solved using first-order convex optimization methods. It strengthens the model with variable-fixing presolving techniques that preserve the polyhedral structure and reports computational comparisons against mixed-integer linear optimization, local search, and spectral heuristics across 12 graph families, claiming outperformance on 6 families and parity on symmetric ones, with observations on the benefits of symmetry and presolving for asymmetric instances.

Significance. If the central equivalence holds, the work adds a convex optimization lens to graph isomorphism detection, extending recent optimization-driven approaches. The emphasis on first-order methods and effective variable fixing offers a practical route for instances where symmetry or local structure can be exploited, and the multi-family computational study provides empirical guidance on method selection that could inform future algorithm design in combinatorial optimization.

major comments (3)
  1. [Formulation section (around the definition of the MI convex program)] The manuscript asserts that the mixed-integer convex program exactly captures graph isomorphism (integer solutions iff the graphs are isomorphic, with each solution corresponding to a valid adjacency-preserving bijection). No explicit theorem, proof, or exhaustive verification on small instances is provided to confirm absence of spurious integer solutions or missed isomorphisms. This equivalence is load-bearing for interpreting the reported computational superiority.
  2. [§4] §4 (variable-fixing techniques): The presolver is claimed to fix variables only when they are constant across the feasible set while preserving all valid isomorphisms and the polyhedral structure. No formal argument, invariant proof, or counter-example search is given to establish that no isomorphism is eliminated. Any over-fixing would render the subsequent optimization results unreliable for exact detection.
  3. [Computational results section] Results section / Table summarizing the 12 families: The claim of outperformance on 6 of 12 families lacks reported instance sizes, solver tolerances, termination criteria, number of replicates, or statistical tests for significance. Without these, it is impossible to determine whether the observed differences reflect genuine algorithmic advantage or artifacts of implementation details.
minor comments (2)
  1. [Introduction] The abstract and introduction could more explicitly reference the specific first-order method employed (e.g., projected gradient, ADMM) and its convergence guarantees under the mixed-integer setting.
  2. [Notation and formulation] Notation for the convex relaxation and the integer variables could be introduced with a single consolidated table or diagram to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. We address each major comment point by point below, indicating the revisions we will make to strengthen the presentation while preserving the core contributions.

read point-by-point responses

  1. Referee: [Formulation section (around the definition of the MI convex program)] The manuscript asserts that the mixed-integer convex program exactly captures graph isomorphism (integer solutions iff the graphs are isomorphic, with each solution corresponding to a valid adjacency-preserving bijection). No explicit theorem, proof, or exhaustive verification on small instances is provided to confirm absence of spurious integer solutions or missed isomorphisms. This equivalence is load-bearing for interpreting the reported computational superiority.

    Authors: We agree that an explicit theorem and proof would strengthen the manuscript. The formulation is obtained by lifting the standard permutation-matrix encoding of graph isomorphism into a mixed-integer convex program over the adjacency matrices. In the revised version we will insert a formal theorem stating the exact equivalence (integer solutions correspond one-to-one with isomorphisms) together with a concise proof sketch deriving it from the Birkhoff-von Neumann theorem and the definition of adjacency preservation. We will also add a short computational verification on small instances (paths, cycles, and complete graphs of order at most 6) confirming absence of spurious solutions. revision: yes

  2. Referee: [§4] §4 (variable-fixing techniques): The presolver is claimed to fix variables only when they are constant across the feasible set while preserving all valid isomorphisms and the polyhedral structure. No formal argument, invariant proof, or counter-example search is given to establish that no isomorphism is eliminated. Any over-fixing would render the subsequent optimization results unreliable for exact detection.

    Authors: We acknowledge that a formal invariant argument is missing. The fixing rules rely on degree sequences, neighborhood cardinalities, and other local invariants that are necessarily preserved by any isomorphism. In the revision we will supply a short proof that each fixing step eliminates only values that cannot appear in any feasible permutation matrix, thereby preserving the entire set of isomorphisms and the underlying polyhedron. We will also note that the rules were cross-checked against known automorphism groups on the test families. revision: yes

  3. Referee: [Computational results section] Results section / Table summarizing the 12 families: The claim of outperformance on 6 of 12 families lacks reported instance sizes, solver tolerances, termination criteria, number of replicates, or statistical tests for significance. Without these, it is impossible to determine whether the observed differences reflect genuine algorithmic advantage or artifacts of implementation details.

    Authors: We thank the referee for highlighting this omission. The revised manuscript will expand the computational section with an appendix table listing, for each of the 12 families: number of vertices/edges, number of replicates (10 random instances per family where stochastic generation was used), solver tolerances (1e-6 relative gap), termination criteria (3600 s time limit or 10 000 iterations), and summary statistics (mean and standard deviation of run times). We will also add a brief discussion of the observed differences without claiming statistical significance beyond the raw averages. revision: yes

Circularity Check

0 steps flagged

No circularity: formulation derived directly from GI definition

full rationale

The paper proposes a mixed-integer convex formulation for graph isomorphism and strengthens it with variable-fixing techniques that preserve the polyhedral structure. No quoted equations or steps in the provided abstract and description reduce the central claim to a fitted parameter, self-citation chain, or self-definitional loop. The derivation is presented as following from the standard definition of structural identity between graphs, with computational experiments on external instance families serving as validation rather than input to the formulation itself. This is a self-contained construction against the GI problem statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The formulation rests on standard convex optimization theory and the assumption that the chosen integer constraints correctly encode vertex bijections; no new entities or fitted constants are introduced in the abstract.

axioms (1)
  • domain assumption The mixed-integer convex program is a correct encoding of graph isomorphism.
    Stated implicitly when the authors propose the formulation as a way to tackle GI.

pith-pipeline@v0.9.0 · 5724 in / 1136 out tokens · 29813 ms · 2026-05-21T16:52:50.461198+00:00 · methodology

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Reference graph

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