AI 中文总结
研究关于[n]上集合系统规模与全链数量权衡问题，通过转化为信息与熵权衡，给出γ≈3.1819的最优界，弥合差距，得到时空乘积为O(3.1819^n)的TSP算法，还改进了布尔格中纤维最小规模的界。
AI 中文摘要
我们几乎解决了一个关于[n]上集合系统的自然极值问题：集合的数量（规模）与全链数量之间的权衡。这个问题最初由约翰逊、利德和拉塞尔提出[组合概率计算，2015]，作为组合学中斯佩纳型结果的对应问题。最近，阿梅利、内德洛夫和王引入的一个框架，以及达兰特和科兹马独立提出的框架[FOCS 2026]，将这个问题与贝尔曼 - 赫尔德 - 卡尔普风格的动态规划算法在排列问题（如旅行商问题（TSP））中的时空复杂度联系起来。具体而言，他们表明，只要存在规模为S且链密度为D的集合系统，时空乘积γ^(n + o(n))对于TSP是可行的，其中γ = S^2 / D。在本文中，我们给出了这个量的一个本质上最优的界γ≈3.1819，弥合了之前最好的下界γ≥3.015和上界γ≤3.572之间的差距。这意味着对于输入规模n，有一个时空乘积为O(3.1819^n)的TSP算法，以及在这个广泛框架中进一步改进的一个限制。更一般地，对于任何可行值S，我们都可以得到接近最优的值D，有效地解决了每个规模下全链数量的问题。我们结果的关键步骤是将极值组合学问题转化为涉及两个随机变量的信息与熵的权衡。这种重新表述精确地捕捉了组合问题的最优权衡，导致了一个可以推导原始对偶证书的框架，证明了γ的严格上下界。我们还给出了我们技术的进一步应用，改进了达福斯、桑兹和温克勒关于布尔格中纤维最小规模的一个界。
英文摘要
We nearly settle a natural extremal question about set systems over $[n]$: the tradeoff between the {size} (number of sets) and the number of {full chains}. This question was initially raised by Johnson, Leader, and Russell [Combin.~Probab.~Comp., 2015] as a counterpart to Sperner-type results in combinatorics. Recently, a framework introduced by Ameli, Nederlof, and Wang, and independently by Dallant and Kozma [FOCS 2026] linked this question to the space- and time-complexity of Bellman-Held-Karp-style dynamic programming algorithms for permutation problems such as the traveling salesman (TSP). Precisely, they showed that a space-time product $γ^{n+o(n)}$ is feasible for the TSP, whenever a set system of (normalized) size $S$ and chain density $D$ exists, with $ γ= S^2/D$. In this paper we show an essentially {optimal} bound of $γ\approx 3.1819$ for this quantity, closing the gap between the previous best lower and upper bounds of $γ\geq 3.015$ and $ γ\leq 3.572$ respectively. This implies a TSP algorithm with space-time product $O(3.1819^n)$ for input size $n$, as well as a limit to further improvements in this broad framework. More generally, we can obtain close to optimal values $D$ for any feasible value $S$, effectively settling the question of the number of full chains at every size. The crucial step towards our results is casting the extremal combinatorics question as an {information~vs.~entropy} tradeoff involving two random variables. This reformulation {exactly} captures the optimal tradeoff for the combinatorial problem, leading to a framework in which primal-dual certificates can be derived, proving rigorous upper and lower bounds on $γ$. We also give a further application of our techniques, improving a bound of Duffus, Sands, and Winkler on the minimum size of fibres in the Boolean lattice.
