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arXiv:2611.05048 · v2

Graph Similarity Meets Optimal Transport

Jens BreitungTimo GervensMartin Grohe
cs.DM
首次提交 2026-11-03 23:16
版本更新 2026-11-05 01:28
雷达观测 2026-11-05 12:12
arXiv ↗PDF ↗

摘要

A natural way to compare two graphs is to align their vertex sets by a bijection and to measure the difference of the aligned adjacency matrices in a matrix norm. Depending on the norm, this yields, for example, the edit distance, the cut distance, the spectral distance, and the local edit distance. To make these distances applicable to graphs of different order, one can consider blowups of the graphs or, equivalently, relax bijections to fractional alignments, that is, optimal transport plans between the uniform distributions on the vertex sets, in the spirit of Gromov-Wasserstein distances. We investigate how much the optimal transport versions can differ from their bijective counterparts, that is, we quantify their integrality gaps. For the edit distance, we improve the best known upper bound on the integrality gap from 3 to 2 and the lower bound from 6/5 to 4/3. For the cut distance, we approach Lovász's conjecture that $δ_\square \le 2δ_\square^{\rm OT}$ from both sides. We improve the previously best known upper bound $δ_\square \le 32 \cdot (δ_\square^{\rm OT})^{1/67}$ (Borgs, Chayes, Lovász, Sós and Vesztergombi, Adv. Math. 2008) to $δ_\square \le C_\varepsilon \cdot (δ_\square^{\rm OT})^{1-\varepsilon}$ for every $\varepsilon > 0$, and we construct graphs $G,H$ with $δ_\square(G,H) = 2δ_\square^{\rm OT}(G,H)$, establishing a factor-2 lower bound that matches Lovász's conjecture. For the local edit and spectral distances, we obtain upper bounds of order $\sqrt{t\ln(1/t)}$ and $\sqrt{t}$, respectively, where $t$ is the corresponding optimal transport distance, and we show that, unlike for the edit and cut distances, no linear upper bounds are possible. Finally, we show that computing the optimal transport distances is NP-hard, and we give hardness of approximation results for the cut, spectral, and local edit distances.

作者备注：v2: fixed typos and broken link to Lean repository

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摘要变化

A natural way to compare two graphs is to align their vertex sets by a bijection and to measure the difference of the aligned adjacency matrices in a matrix norm. Depending on the norm, this yields, for example, the edit distance, the cut distance, the spectral distance, and the local edit distance. To make these distances applicable to graphs of different order, one can consider blowups of the graphs or, equivalently, relax bijections to fractional alignments, that is, optimal transport plans between the uniform distributions on the vertex sets, in the spirit of Gromov-Wasserstein distances. We investigate how much the optimal transport versions can differ from their bijective counterparts, that is, we quantify their integrality gaps. For the edit distance, we improve the best known upper bound on the integrality gap from 3 to 2 and the lower bound from 6/5 to 4/3. For the cut distance, we approach Lovász's conjecture that $δ_\square \le 2δ_\square^{\rm OT}$ from both sides. We improve the previously best known upper bound $δ_\square \le 32 \cdot (δ_\square^{\rm OT})^{1/67}$ (Borgs, Chayes, Lovász, Sós and Vesztergombi, Adv. Math. 2008) to $δ_\square \le C_\varepsilon \cdot (δ_\square^{\rm OT})^{1-\varepsilon}$ for every $\varepsilon > 0$, and we construct graphs $G,H$ with $δ_\square(G,H) = 2δ_\square^{\rm OT}(G,H)$, establishing a factor-2 lower bound that matches Lovász's conjecture. For the local edit and spectral distances, we obtain upper bounds of order $\sqrt{t\ln(1/t)}$ and $\sqrt{t}$, respectively, where $t$ is the corresponding optimal transport distance, and we show that, unlike for the edit and cut distances, no linear upper bounds are possible. Finally, we show that computing the optimal transport distances is NP-hard, and we give hardness of approximation results for the cut, spectral, and local edit distances.

作者备注变化

v2: fixed typos and broken link to Lean repository

观测到的版本

相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。

v2 · 2026-11-05 01:28 · 最新观测版本

Graph Similarity Meets Optimal Transport

A natural way to compare two graphs is to align their vertex sets by a bijection and to measure the difference of the aligned adjacency matrices in a matrix norm. Depending on the norm, this yields, for example, the edit distance, the cut distance, the spectral distance, and the local edit distance. To make these distances applicable to graphs of different order, one can consider blowups of the graphs or, equivalently, relax bijections to fractional alignments, that is, optimal transport plans between the uniform distributions on the vertex sets, in the spirit of Gromov-Wasserstein distances. We investigate how much the optimal transport versions can differ from their bijective counterparts, that is, we quantify their integrality gaps. For the edit distance, we improve the best known upper bound on the integrality gap from 3 to 2 and the lower bound from 6/5 to 4/3. For the cut distance, we approach Lovász's conjecture that $δ_\square \le 2δ_\square^{\rm OT}$ from both sides. We improve the previously best known upper bound $δ_\square \le 32 \cdot (δ_\square^{\rm OT})^{1/67}$ (Borgs, Chayes, Lovász, Sós and Vesztergombi, Adv. Math. 2008) to $δ_\square \le C_\varepsilon \cdot (δ_\square^{\rm OT})^{1-\varepsilon}$ for every $\varepsilon > 0$, and we construct graphs $G,H$ with $δ_\square(G,H) = 2δ_\square^{\rm OT}(G,H)$, establishing a factor-2 lower bound that matches Lovász's conjecture. For the local edit and spectral distances, we obtain upper bounds of order $\sqrt{t\ln(1/t)}$ and $\sqrt{t}$, respectively, where $t$ is the corresponding optimal transport distance, and we show that, unlike for the edit and cut distances, no linear upper bounds are possible. Finally, we show that computing the optimal transport distances is NP-hard, and we give hardness of approximation results for the cut, spectral, and local edit distances.

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v1 · 2026-11-03 23:16

Graph Similarity Meets Optimal Transport

A natural way to compare two graphs is to align their vertex sets by a bijection and to measure the difference of the aligned adjacency matrices in a matrix norm. Depending on the norm, this yields, for example, the edit distance, the cut distance, the spectral distance, and the local edit distance. To make these distances applicable to graphs of different order, one can consider blowups of the graphs or, equivalently, relax bijections to fractional alignments, that is, optimal transport plans between the uniform distributions on the vertex sets, in the spirit of Gromov-Wasserstein distances. We investigate how much the optimal transport versions can differ from their bijective counterparts, that is, we quantify their integrality gaps. For the edit distance, we improve the best known upper bound on the integrality gap from 3 to 2 and the lower bound from 6/5 to 4/3. For the cut distance, we approach Lovász's conjecture that $δ_\square \le 2δ_\square^{OT}$ from both sides. We improve the previously best known upper bound $δ_\square \le 32 \cdot (δ_\square^{OT})^{1/67}$ (Borgs, Chayes, Lovász, Sós and Vesztergombi, Adv. Math. 2008) to $δ_\square \le C_\varepsilon \cdot (δ_\square^{OT})^{1-\varepsilon}$ for every $\varepsilon > 0$, and we construct graphs $G,H$ with $δ_\square(G,H) = 2δ_\square^{OT}(G,H)$, establishing a factor-2 lower bound that matches Lovász's conjecture. For the local edit and spectral distances, we obtain upper bounds of order $\sqrt{t\ln(1/t)}$ and $\sqrt{t}$, respectively, where $t$ is the corresponding optimal transport distance, and we show that, unlike for the edit and cut distances, no linear upper bounds are possible. Finally, we show that computing the optimal transport distances is NP-hard, and we give hardness of approximation results for the cut, spectral, and local edit distances.

查看此版本解读 →查看此版本原文 ↗