Itemset Isomorphism: GI-Complete

This paper addresses the problem of finding a class of representative itemsets up to subitemset isomorphism. An efficient algorithm is of practical importance in the domain of optimal sorting networks. Although only exponential algorithms for solving the problem exist in the literature, the complexity classification of the problem has never been addressed. In this paper, we present a complexity classification of the itemset isomorphism and subitemset isomorphism problems. We prove that the problem of checking whether two itemsets are isomorphic to each other is GI-Hard; the Graph Isomorphism (GI) problem is known to be in NP and LWPP, but widely believed to be neither P nor NP-Complete. As an immediate consequence, we prove that finding a class representative itemsets up to subitemset isomorphism is GI-Hard — at least as hard as the graph isomorphism problem.