Graph Threading with Turn Costs

TL;DR

This paper investigates the problem of threading a string through tubes to form a graph while minimizing total turn costs, revealing NP-hardness and providing efficient algorithms for special cases.

Contribution

It introduces the turn-cost threading problem, proves its NP-hardness, and offers polynomial-time algorithms for specific graph classes and constraints.

Findings

Minimum-turn threading is NP-hard for degree 4 graphs.

Special cases of the problem can be solved efficiently.

Algorithms are provided for edge-restricted and grid graph variants.

Abstract

How should we thread a single string through a set of tubes so that pulling the string taut self-assembles the tubes into a desired graph? While prior work [ITCS 2024] solves this problem with the goal of minimizing the length of string, we study here the objective of minimizing the total turn cost. The frictional force required to pull the string through the tubes grows exponentially with the total absolute turn angles (by the Capstan equation), so this metric often dominates the friction in real-world applications such as deployable structures. We show that minimum-turn threading is NP-hard, even for graphs of maximum degree 4, and even when restricted to some special cases of threading. On the other hand, we show that these special cases can in fact be solved efficiently for graphs of maximum degree 4, thereby fully characterizing their dependence on maximum degree. We further provide polynomial-time exact and approximation algorithms for variants of turn-cost threading: restricting to threading each edge exactly twice, and on rectangular grid graphs.