We provide improved space-time tradeoffs for permutation problems over additively idempotent semi-rings. In particular, there is an algorithm for the Traveling Salesperson Problem that solves $N$-vertex instances using space $S$ and time $T$ where $S\cdot T \leq 3.7493^{N}$. This improves a previous work by Koivisto and Parviainen [SODA'10] where $S\cdot T \leq 3.9271^N$, and overcomes a barrier they identified, as their bound was shown to be optimal within their framework. To get our results, we introduce a new parameter of a set system that we call the chain efficiency. This relates the number of maximal chains contained in the set system with the cardinality of the system. We show that set systems of high efficiency imply efficient space-time tradeoffs for permutation problems, and give constructions of set systems with high chain efficiency, disproving a conjecture by Johnson, Leader and Russel [Comb. Probab. Comput.'15].

翻译：我们针对加法幂等半环上的置换问题，提出了改进的时空权衡方法。特别地，对于旅行商问题，存在一种算法，可在空间$S$和时间$T$内求解$N$顶点实例，满足$S\cdot T \leq 3.7493^{N}$。这改进了Koivisto和Parviainen [SODA'10] 此前的工作（其中$S\cdot T \leq 3.9271^N$），并突破了他们识别的障碍——因为其界限在其框架中被证明是最优的。为获得结果，我们引入了一个集合系统的新参数，称为链效率。该参数将集合系统中包含的最大链数量与系统基数联系起来。我们证明了高效率的集合系统意味着置换问题的高效时空权衡，并构造了具有高链效率的集合系统，推翻了Johnson、Leader和Russel [Comb. Probab. Comput.'15] 的一个猜想。