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 $|G|^{\Theta(\log |G|)}$, 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. Combined with the recent breakthrough $|G|^{O((\log |G|)^{5/6})}$-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 $c<p$. 2. Our reductions show that Sun's algorithm solves several TI-complete problems over $F_p$, such as isomorphism problems for cubic forms, algebras, and tensors, in time $p^{O(n^{1.8} \log p)}$. 3. Polynomial-time search- and counting-to-decision reduction for testing isomorphism of $p$-groups of class $2$ and exponent $p$ in the Cayley table model. This answers questions of Arvind and T\'oran (Bull. EATCS, 2005) for this group class, thought to be one of the hardest cases of Group Isomorphism. 4. If Graph Isomorphism is in P, then testing equivalence of cubic forms and testing isomorphism of algebra over a finite field $F_q$ can both be solved in time $q^{O(n)}$, improving from the brute-force upper bound $q^{O(n^2)}$.

翻译：近年来，张量、群、代数和多项式的许多同构问题被证明在多项式时间归约下彼此等价，由此引入了复杂性类TI（Grochow & Qiao, ITCS '21; SIAM J. Comp., '23）。基于张量视角，Grochow & Qiao（CCC '21）进一步给出了一类$p$-群的适度指数时间搜索到判定归约及计数到判定归约。一个显著问题是这些归约通常会导致涉及张量长度的二次增长。当张量表示$p$-群时，这对应群阶的$|G|^{\Theta(\log |G|)}$形式增长，从而抵消了凯莱表模型中的任何渐近增益。本文提出一种新型张量工具，使我们能够用线性长度归约替代上述二次长度归约，并得到以下结果：1. 结合近期针对类2指数$p$的$p$-群取得的突破性$|G|^{O((\log |G|)^{5/6})}$时间同构检验算法（Sun, STOC '23），我们的归约将该运行时间推广至类$c$且$c<p$的指数$p$的$p$-群。2. 我们的归约表明，Sun的算法可在$p^{O(n^{1.8} \log p)}$时间内求解$F_p$上多个TI完全问题，包括三次型、代数和张量的同构问题。3. 在凯莱表模型中，我们给出了针对类2指数$p$的$p$-群同构检验的多项式时间搜索到判定归约及计数到判定归约。这回答了Arvind与Tóran（Bull. EATCS, 2005）对该群类提出的开放问题——该群类被认为是群同构问题中最困难的情形之一。4. 若图同构属于P，则有限域$F_q$上三次型等价性检验和代数同构检验均可$q^{O(n)}$时间内求解，改进了暴力搜索上界$q^{O(n^2)}$。