A Geometric Approach to Graph Isomorphism

• Published in: Algorithms and Computation pp. 674 - 685
• Main Authors: Aurora, Pawan, Mehta, Shashank K.
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing 2014
• Series: Lecture Notes in Computer Science
• Subjects: Graph isomorphism problem; Linear programming; Polyhedral combinatorics
• ISBN: 3319130749; 9783319130743
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-13075-0_53

We present an integer linear program (IP), for the Graph Isomorphism (GI) problem, which has non-empty feasible solution if and only if the input pair of graphs are isomorphic. We study the polytope of the convex hull of the solution points of IP, denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}^{[2]}$$\end{document}. Exponentially many facets of this polytope are known. We show that in case of non-isomorphic pairs of graphs if a feasible solution exists for the linear program relaxation (LP) of the IP, then it violates a unique facet of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}^{[2]}$$\end{document}. We present an algorithm for GI based on the solution of LP and prove that it detects non-isomorphism in polynomial time if the solution of the LP violates any of the known facets.