图相似性与最优传输
Graph Similarity Meets Optimal Transport
Jens Breitung, Timo Gervens, Martin Grohe
AI 速览：该论文研究用最优传输松弛双射对齐来定义图距离时，最优传输版本与双射版本之间的整性间隙，并给出若干距离的上下界与计算复杂性结果。
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.