QAPV: a polynomial invariant for graph isomorphism testing

Abstract

ABSTRACT.To each instance of the Quadratic Assignment Problem (QAP) a relaxed instance can be associated. Both variances of their solution values can be calculated in polynomial time. The graph isomorphism problem (GIP) can be modeled as a QAP, associating its pair of data matrices with a pair of graphs of the same order and size. We look for invariant edge weight functions for the graphs composing the instances in order to try to find quantitative differences between variances that could be associated with the absence of isomorphism. This technique is sensitive enough to show the effect of a single edge exchange between two regular graphs of up to 3,000 vertices and 300,000 edges with degrees up to 200. Planar graph pairs from a dense family up to 300,000 vertices were also discriminated. We conjecture the existence of functions able to discriminate non-isomorphic pairs for every instance of the problem.

Development and in silico evaluation of large-scale metabolite identification methods using functional group detection for metabolomics

“…The pseudo-linear behavior of our new algorithm as shown in Figure 9 is stable for values of m up to 150 and remains sufficiently fast for large values of m during functional group searching, making CASS tractable for systematic functional group searches in KEGG and HMDB. Furthermore, the demonstrated polynomial behavior of our algorithm (Figure 9E ) is the best expected performance, given the debate on whether the common subgraph isomorphism problem has polynomial or NP-complete behavior (de Melo et al, 2013 ). Also, Figure 10 further highlights the relative differences between the Ullmann algorithm and CASS on a log scale.…”