On the complexity of isomorphism problems for tensors, groups, and polynomials IV: linear-length reductions and their applications

Many isomorphism problems for tensors, groups, algebras, and polynomials were recently shown to be equivalent to one another under polynomial-time reductions, prompting the introduction of the complexity class TI (Grochow Qiao, ITCS '21; SIAM J. Comp., '23). Using the tensorial viewpoint, Grochow Qiao (CCC '21) then gave moderately exponential-time search- and counting-to-decision reductions for a class of p-groups. A significant issue was that the reductions usually incurred a quadratic increase in the length of the tensors involved. When the tensors represent p-groups, this corresponds to an increase in the order of the group of the form |G|^Θ(log |G|), negating any asymptotic gains in the Cayley table model. In this paper, we present a new kind of tensor gadget that allows us to replace those quadratic-length reductions with linear-length ones, yielding the following consequences: 1. If Graph Isomorphism is in P, then testing equivalence of cubic forms in n variables over F_q, and testing isomorphism of n-dimensional algebras over F_q, can both be solved in time q^O(n), improving from the brute-force upper bound q^O(n^2) for both of these. 2. Combined with the |G|^O((log |G|)^5/6)-time isomorphism-test for p-groups of class 2 and exponent p (Sun, STOC '23), our reductions extend this runtime to p-groups of class c and exponent p where c更多