Improved Space-Time Tradeoffs for Permutation Problems via Extremal Combinatorics

2026-04-07

AI Summary

This work investigates space–time trade-offs for permutation problems—such as the Traveling Salesman Problem—over additively idempotent semirings. By introducing a novel parameter called “chain efficiency” for set systems and constructing set systems with high chain efficiency, the study overcomes theoretical limitations of existing algorithmic frameworks and refutes a conjecture by Johnson et al. regarding the structure of set systems. Combining tools from extremal combinatorics and complexity analysis, the authors achieve, for the first time, an algorithm for the N-vertex Traveling Salesman Problem whose space S and time T satisfy S·T ≤ 3.7493^N, significantly improving upon the previous best bound of 3.9271^N.

Abstract

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].

Problem

• space-time tradeoffs
• permutation problems
• Traveling Salesperson Problem
• additively idempotent semi-rings

Innovation

• space-time tradeoff
• chain efficiency
• permutation problems
• extremal combinatorics
• traveling salesperson problem