证伪SETH和正交向量猜想的更多结果
More Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture
摘要：
强指数时间假设和OV猜想是两个流行的困难假设，用于证明大量的下界，特别是在多项式时间算法领域。中维的OV猜想指出，不存在$ε>0$，对于该$ε>0$，$O(N^{2-ε})\mathrm{poly}(D)$时间算法可以判定在包含$D$维二进制向量的给定大小的集合$N$中是否存在一对正交向量。我们加强了这些硬度假设的证据。特别是，我们表明，如果OV-猜想失败，那么两个问题，我们远远没有得到甚至微小的改进，在穷举搜索将有惊人的快速算法。如果OV猜想不成立，则存在一个固定的$ε>0$，使得：（1）对于所有$d$和所有足够大的$k$，存在一个随机化算法，该算法需要$O(n^{(1-ε)k})$时间来求解具有$n$顶点的$d$ -超图上的零权$k$ -团和最小权$k$-团问题.因此，OV-猜想被加权团猜想所隐含。(2)对于所有$c$，稀疏TC 1电路对$n$输入的可满足性（即，具有$cn$线、深度$c\log n$以及否定、AND、OR和阈值门的电路）可以在时间${O((2-ε)^n)}$中计算。
英文摘要：
The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomial-time algorithms. The OV-conjecture in moderate dimension states there is no $ε>0$ for which an $O(N^{2-ε})\mathrm{poly}(D)$ time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size $N$ that contains $D$-dimensional binary vectors. We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed $ε>0$ such that: (1) For all $d$ and all large enough $k$, there is a randomized algorithm that takes $O(n^{(1-ε)k})$ time to solve the Zero-Weight-$k$-Clique and Min-Weight-$k$-Clique problems on $d$-hypergraphs with $n$ vertices. As a consequence, the OV-conjecture is implied by the Weighted Clique conjecture. (2) For all $c$, the satisfiability of sparse TC1 circuits on $n$ inputs (that is, circuits with $cn$ wires, depth $c\log n$, and negation, AND, OR, and threshold gates) can be computed in time ${O((2-ε)^n)}$.
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