Improved Space-Time Tradeoffs for Permutation Problems via Extremal Combinatorics

TL;DR

This paper improves space-time tradeoffs for permutation problems like TSP over semi-rings by introducing a new set system parameter called chain efficiency, leading to better algorithms and disproving a prior conjecture.

Contribution

It introduces the chain efficiency parameter for set systems, enabling improved algorithms for permutation problems and challenging previous conjectures.

Findings

Achieved a new space-time tradeoff for TSP: S·T ≤ 3.7493^N.

Disproved a conjecture by Johnson, Leader, and Russel regarding set system efficiency.

Introduced the concept of chain efficiency linking set system properties to algorithmic performance.

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].