Interplay Between Graph Isomorphism and Earth Mover’s Distance in the Query and Communication Worlds
The graph isomorphism distance between two graphs G_u and G_k is the fraction of entries in the adjacency matrix that has to be changed to make G_u isomorphic to G_k. We study the problem of estimating, up to a constant additive factor, the graph isomorphism distance between two graphs in the query model. In other words, if G_k is a known graph and G_u is an unknown graph whose adjacency matrix has to be accessed by querying the entries, what is the query complexity for testing whether the graph isomorphism distance between G_u and G_k is less than γ₁ or more than γ₂, where γ₁ and γ₂ are two constants with 0 ≤ γ₁ < γ₂ ≤ 1. It is also called the tolerant property testing of graph isomorphism in the dense graph model. The non-tolerant version (where γ₁ is 0) has been studied by Fischer and Matsliah (SICOMP'08).
In this paper, we prove a (interesting) connection between tolerant graph isomorphism testing and tolerant testing of the well studied Earth Mover’s Distance (EMD). We prove that deciding tolerant graph isomorphism is equivalent to deciding tolerant EMD testing between multi-sets in the query setting. Moreover, the reductions between tolerant graph isomorphism and tolerant EMD testing (in query setting) can also be extended directly to work in the two party Alice-Bob communication model (where Alice and Bob have one graph each and they want to solve tolerant graph isomorphism problem by communicating bits), and possibly in other sublinear models as well.
Testing tolerant EMD between two probability distributions is equivalent to testing EMD between two multi-sets, where the multiplicity of each element is taken appropriately, and we sample elements from the unknown multi-set with replacement. In this paper, our (main) contribution is to introduce the problem of {(tolerant) EMD testing between multi-sets (over Hamming cube) when we get samples from the unknown multi-set without replacement} and to show that this variant of tolerant testing of EMD is as hard as tolerant testing of graph isomorphism between two graphs. {Thus, while testing of equivalence between distributions is at the heart of the non-tolerant testing of graph isomorphism, we are showing that the estimation of the EMD over a Hamming cube (when we are allowed to sample without replacement) is at the heart of tolerant graph isomorphism.} We believe that the introduction of the problem of testing EMD between multi-sets (when we get samples without replacement) opens an entirely new direction in the world of testing properties of distributions.
Graph Isomorphism
Earth Mover Distance
Query Complexity
Theory of computation~Streaming, sublinear and near linear time algorithms
34:1-34:23
RANDOM
https://eccc.weizmann.ac.il/report/2020/135/
The authors would like to thank an anonymous reviewer for pointing out a mistake in an earlier version of this paper, as well as the reviewers of RANDOM for various suggestions that improved the presentation of the paper.
Sourav
Chakraborty
Sourav Chakraborty
Indian Statistical Institute, Kolkata, India
https://www.isical.ac.in/~sourav/
Arijit
Ghosh
Arijit Ghosh
Indian Statistical Institute, Kolkata, India
https://sites.google.com/site/homepagearijitghosh/
Gopinath
Mishra
Gopinath Mishra
Indian Statistical Institute, Kolkata, India
https://sites.google.com/view/gopinathmishra/
Sayantan
Sen
Sayantan Sen
Indian Statistical Institute, Kolkata, India
https://sites.google.com/view/sayantans
10.4230/LIPIcs.APPROX/RANDOM.2021.34
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Sourav Chakraborty, Arijit Ghosh, Gopinath Mishra, and Sayantan Sen
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