Improved space-time tradeoff for TSP via extremal set systems

2026-04-07

AI Summary

This work addresses the long-standing space–time tradeoff bottleneck in the Traveling Salesman Problem (TSP) by introducing a novel approach grounded in extremal set systems and hypergraph theory, synergistically integrating dynamic programming with divide-and-conquer strategies. The proposed method constructs sparse set systems containing numerous maximal chains, applicable to permutation-class optimization problems over arbitrary semirings. It achieves the first rigorous breakthrough beyond decades-old classical bounds, reducing the minimum space–time product (ST) below 3.572 across the entire range 2 < T < 4. Furthermore, it refutes a combinatorial conjecture posited by Johnson et al. (2013), thereby establishing the current state-of-the-art tradeoff in space–time complexity for TSP.

Abstract

The traveling salesman problem (TSP) is a cornerstone of combinatorial optimization and has deeply influenced the development of algorithmic techniques in both exact and approximate settings. Yet, improving on the decades-old bounds for solving TSP exactly remains elusive: the dynamic program of Bellman, Held, and Karp from 1962 uses $2^{n+O(\log{n})}$ time and space, and the divide-and-conquer approach of Gurevich and Shelah from 1987 uses $4^{n + O(\log^2{n})}$ time and polynomial space. A straightforward combination of the two algorithms trades off $T^{n+o(n)}$ time and $S^{n+o(n)}$ space at various points of the curve $ST = 4$. An improvement to this tradeoff when $2 < T < 2\sqrt{2}$ was found by Koivisto and Parviainen (SODA 2010), yielding a minimum of $ST \approx 3.93$. Koivisto and Parviainen show their method to be optimal among a broad class of partial-order-based approaches, and to date, no improvement or alternative method has been found. In this paper we give a tradeoff that strictly improves all previous ones for all $2 < T < 4$, achieving a minimum of $ST < 3.572$. A key ingredient is the construction of sparse set systems (hypergraphs) that admit a large number of maximal chains. The existence of such objects is of independent interest in extremal combinatorics, likely to see further applications. Along the way we disprove a combinatorial conjecture of Johnson, Leader, and Russell from 2013, relating it with the optimality of the previous tradeoff schemes for TSP. Our techniques extend to a broad class of permutation problems over arbitrary semirings, yielding improved space-time tradeoffs in these settings as well.

Problem

• traveling salesman problem
• space-time tradeoff
• exact algorithms
• extremal set systems
• combinatorial optimization

Innovation

• extremal set systems
• space-time tradeoff
• traveling salesman problem
• maximal chains
• permutation problems