Optimal chain density, entropy, and space-time tradeoffs for the TSP

2026-07-13

AI Summary

This study investigates the trade-off between the number of full chains and the size of a set system, applying this insight to optimize the time–space complexity of the Traveling Salesman Problem (TSP). By formulating an extremal combinatorial problem precisely as an information–entropy trade-off for the first time, the authors develop a novel framework that integrates entropy methods, duality certificates, and extremal combinatorial analysis to derive rigorous bounds. This framework nearly optimally characterizes the relationship between chain density and system size, yielding tight upper and lower bounds on the number of full chains. As a consequence, the time–space product complexity of TSP algorithms based on the Bellman–Held–Karp paradigm is reduced to $O(3.1819^n)$, and the known bound on the minimum fiber size in Boolean lattices is improved.

Abstract

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.

Problem

• set systems
• full chains
• space-time tradeoffs
• TSP
• extremal combinatorics

Innovation

• entropy-information tradeoff
• chain density
• space-time tradeoff
• TSP algorithm
• extremal combinatorics