[Paper Review] Graph isomorphism and volumes of convex bodies

Shmuel Friedland|ArXiv.org|Nov 9, 2009

Point processes and geometric inequalities15 references3 citations

TL;DR

This paper establishes a novel equivalence between the graph isomorphism problem and the equality of volumes of specific convex polytopes intersected with balls of increasing radius centered at the origin. By leveraging known fully randomized polynomial-time approximation schemes (fpras) for volume computation, the authors derive a semi-fpras of complexity O*(n¹⁴) for testing isomorphism between two graphs, offering a potential pathway to resolving the long-standing computational complexity of the graph isomorphism problem.

ABSTRACT

We show that a nontrivial graph isomorphism problem of two undirected graphs, and more generally, the permutation similarity of two given $n imes n$ matrices, is equivalent to equalities of volumes of the induced three convex bounded polytopes intersected with a given sequence of balls, centered at the origin with radii $t_i\in (0,\sqrt{n-1})$, where $\{t_i\}$ is an increasing sequence converging to $\sqrt{n-1}$. These polytopes are characterized by $n^2$ inequalities in at most $n^2$ variables. The existence of fpras for computing volumes of convex bodies gives rise to a semi-frpas of order $O^*(n^{14})$ at most to find if given two undirected graphs are isomorphic.

Motivation & Objective

• To establish a new characterization of graph isomorphism in terms of geometric volume equality of convex polytopes.
• To reduce the graph isomorphism problem to a sequence of volume comparisons of bounded convex sets.
• To leverage existing fpras for convex volume computation to design a semi-fpras for graph isomorphism testing.
• To provide a geometric framework for permutational similarity of matrices using coherent algebras and doubly stochastic constraints.
• To offer a potential polynomial-time approach to the graph isomorphism problem, which remains in the complexity class NP but is not known to be NP-complete or in P.

Proposed method

• Define three convex polytopes P(A,A), P(A,B), and P(B,B) as solutions to linear matrix equations SX - XT = 0 with non-negativity constraints X ≥ -Jₙ.
• Use the Frobenius norm radius of these polytopes to characterize isomorphism: G and H are isomorphic iff ||P(A,B)||_F = √(n-1).
• Construct orthogonal projections T₁, T₂, T₃ from the solution spaces P₀(A,A), P₀(A,B), P₀(B,B) to ℝ^Δ, preserving inner products and transforming the polytopes into new bounded polytopes X₁, X₂, X₃ in Δ-dimensional space.
• Represent the projected polytopes Xᵢ as intersections of half-spaces defined by inequalities u_{(i,j),l}ᵀx ≥ -1/n, where the vectors u are computable in polynomial time.
• Apply a randomized volume estimation algorithm over a sequence of nested balls B(0, tⱼ) with radii tⱼ = 2^{j/N}/n, covering the interval (1/n, √(n-1)).
• Test equality of volume ratios vol(Xⱼ,ᵢ)/vol(Xⱼ₋₁,ᵢ) across i=1,2,3 for each j; equality within relative error ε implies isomorphism with high probability.

Experimental results

Research questions

• RQ1 Can the graph isomorphism problem be reduced to a sequence of volume comparisons of convex polytopes?
• RQ2 Is there a geometric characterization of permutational similarity of two matrices using convex polytopes and their intersections with balls?
• RQ3 Can a fully randomized polynomial-time approximation scheme (fpras) for convex volume computation be used to design a semi-fpras for graph isomorphism?
• RQ4 What is the computational complexity of testing isomorphism via volume equality of projected polytopes?
• RQ5 Does the equality of volumes of polytopes P(A,A), P(A,B), P(B,B) intersected with balls of increasing radius imply graph isomorphism?

Key findings

• The graph isomorphism problem is equivalent to the equality of volumes of the intersections of the polytopes P(A,A), P(A,B), and P(B,B) with balls of radius t ∈ (0, √(n−1)), for all t in an increasing sequence converging to √(n−1).
• The dimension of the solution spaces P₀(A,A), P₀(A,B), and P₀(B,B) must be equal for isomorphism to hold, and this is a necessary condition.
• The existence of a fpras for computing the volume of convex bodies enables a semi-fpras of complexity O*(n¹⁴) for testing graph isomorphism.
• The volume ratios vol(Xⱼ,₁)/vol(Xⱼ₋₁,₁), vol(Xⱼ,₂)/vol(Xⱼ₋₁,₂), and vol(Xⱼ,₃)/vol(Xⱼ₋₁,₃) must be equal within relative error ε for all j to confirm isomorphism with high probability.
• The algorithm runs in O*(n¹⁴) time, where the O* notation absorbs polynomial factors in log n and Δ, with Δ ≤ (n−1)² being the dimension of the solution space.
• The method relies on projecting the solution spaces into Δ-dimensional space via orthonormal transformations, enabling efficient volume estimation using known randomized algorithms.