A Comparative Study of Graph Isomorphism Applications

Somkunwar, Rachna; Vaze, Vinod

doi:10.5120/ijca2017913414

Cited by 3 publications

(1 citation statement)

References 15 publications

(12 reference statements)

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“…1], Graph (Non-)Isomorphism played a special role in the creation of the class AM [Bab85, GMR85,BM88], and it still stands today as one of the few natural candidates for a problem that is "NP-intermediate," that is, in NP, but neither in P nor NP-complete [Lad75] (see [Exc] for additional candidates). Beyond its practical applications (e. g., [SV17,Irn05] and references therein) and its naturality, part of its fascination comes from its universal property: GI is universal for isomorphism problems for "explicitly given" structures [ZKT85,Sec. 15], that is, first-order structures on a set V where, e. g., a k-ary relation on V is given by listing out a subset R ⊆ V k .…”

Section: Introductionmentioning

confidence: 99%

Isomorphism problems for tensors, groups, and cubic forms: completeness and reductions

Grochow,

Qiao

2019

Preprint

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In this paper we consider the problems of testing isomorphism of tensors, p-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. These problems can all be cast as orbit problems on multi-way arrays under different group actions. Our first two main results are:1. All the aforementioned isomorphism problems are equivalent under polynomial-time reductions, in conjunction with the recent results of Futorny-Grochow-Sergeichuk (Lin. Alg. Appl., 2019).2. Isomorphism of d-tensors reduces to isomorphism of 3-tensors, for any d ≥ 3.All but one of the reductions for the preceding contributions work over arbitrary fields. Together they suggest that the aforementioned isomorphism problems form a rich and robust equivalence class, which we call Tensor Isomorphism-complete, or TI-complete for short. Furthermore, this provides a unified viewpoint on these hard isomorphism testing problems arising from a variety of areas.We then leverage the techniques used in the above results to prove two first-of-their-kind results for Group Isomorphism (GpI):3. We give a reduction from testing isomorphism of p-groups of exponent p and small class (c < p) to isomorphism of p-groups of exponent p and class 2. The latter are widely believed to be the hardest cases of GpI, but as far as we know, this is the first reduction from any more general class of groups to this class.4. We give a search-to-decision reduction for isomorphism of p-groups of exponent p and class 2 in time |G| O(log log |G|) . While search-to-decision reductions for Graph Isomorphism (GI) have been known for more than 40 years, as far as we know this is the first non-trivial search-to-decision reduction in the context of GpI.Our main technique for (1), (3), and ( 4) is a linear-algebraic analogue of the classical graph coloring gadget, which was used to obtain the search-to-decision reduction for GI. This gadget construction may be of independent interest and utility. The technique for (2) gives a method for encoding an arbitrary tensor into an algebra.

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