Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] claims a strong submultiplicative upper bound on the rank of a three-tensor obtained as an iterated Kronecker product of a constant-size base tensor. The conjecture, if true, most notably would put square matrix multiplication in quadratic time. We note here that some more-or-less unexpected algorithmic results in the area of exponential-time algorithms would also follow. Specifically, we study the so-called set cover conjecture, which states that for any $\epsilon>0$ there exists a positive integer constant $k$ such that no algorithm solves the $k$-Set Cover problem in worst-case time $\mathcal{O}((2-\epsilon)^n|\mathcal F|\operatorname{poly}(n))$. The $k$-Set Cover problem asks, given as input an $n$-element universe $U$, a family $\mathcal F$ of size-at-most-$k$ subsets of $U$, and a positive integer $t$, whether there is a subfamily of at most $t$ sets in $\mathcal F$ whose union is $U$. The conjecture was formulated by Cygan et al. in the monograph Parameterized Algorithms [Springer, 2015] but was implicit as a hypothesis already in Cygan et al. [CCC 2012, ACM Trans. Algorithms 2016], there conjectured to follow from the Strong Exponential Time Hypothesis. We prove that if the asymptotic rank conjecture is true, then the set cover conjecture is false. Using a reduction by Krauthgamer and Trabelsi [STACS 2019], in this scenario we would also get a $\mathcal{O}((2-\delta)^n)$-time randomized algorithm for some constant $\delta>0$ for another well-studied problem for which no such algorithm is known, namely that of deciding whether a given $n$-vertex directed graph has a Hamiltonian cycle.

翻译：Strassen的渐近秩猜想[Progr. Math. 120 (1994)]声称，通过常数规模基张量的迭代Kronecker积得到的三阶张量，其秩满足强次乘性上界。该猜想若成立，最引人注目的结果是将方阵乘法的时间复杂度降至二次时间。我们在此指出，该猜想还将导致指数时间算法领域的一些或多或少出人意料的算法结果。具体而言，我们研究所谓的集合覆盖猜想，该猜想断言：对任意$\epsilon>0$，存在正整数常数$k$，使得没有算法能在最坏情况时间复杂度$\mathcal{O}((2-\epsilon)^n|\mathcal F|\operatorname{poly}(n))$内解决$k$集合覆盖问题。$k$集合覆盖问题要求：给定一个$n$元素全集$U$、一个由$U$的大小不超过$k$的子集构成的族$\mathcal F$，以及一个正整数$t$，判断$\mathcal F$中是否存在至多$t$个集合构成的子族，其并集为$U$。该猜想由Cygan等人在专著《参数化算法》[Springer, 2015]中正式提出，但作为假设早已隐含在Cygan等人[CCC 2012, ACM Trans. Algorithms 2016]的工作中，当时推测该猜想可由强指数时间假说推出。我们证明：若渐近秩猜想成立，则集合覆盖猜想不成立。利用Krauthgamer和Trabelsi [STACS 2019]的归约技术，在该情形下，我们还能得到一个时间复杂度为$\mathcal{O}((2-\delta)^n)$的随机化算法（其中$\delta>0$为某常数），用于解决另一个尚无此类算法的经典问题——判定给定$n$顶点有向图是否包含哈密顿回路。