Graph Threading with Turn Costs

TL;DR

This work reframes graph threading as a turn-cost optimization problem, motivated by frictional forces that scale with turn angles. It establishes NP-hardness for general graphs and for max degree $4$, sharply contrasting with the previous polynomial-time results for minimum-length threading. The authors give polynomial-time algorithms for maximum degree $3$ cases (Perfect and Double Threading) and explore rich special cases, including Exactly-Double Threading (equivalent to TSP) and grid-graph threading with tight approximation guarantees. They also present a practical grid-threading strategy and a $2r$-approximation for bounded turn costs, offering a spectrum of tractable and intractable regimes with clear performance tradeoffs. These results illuminate how turn costs shape algorithmic feasibility and provide concrete tools for designing deployable structures with friction-aware threading.

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.

Figures (9)

• Figure 1: Two distinct threadings of a tetrahedron, where each junction graph is either a tree (a) or a cycle (b) (demaine2024graph) Their threading graphs are shown in the corners.
• Figure 2: A tree-like graph with degree-3 junctions as marked by the circles. A green line connecting an edge pair indicates a turn cost of $0$, whereas a red line indicates a turn cost of $1$. All other turns have zero cost. The minimum-turn threading of this graph deviates far from a perfect threading --- it traverses the "root" triangle as many times as there are "leaf" triangles.
• Figure 3: Constructing an instance $H$ of Threading from an instance $G$ of Hamiltonian Cycle. The turn costs at the central vertex in $H$ are depicted inside the right-most circle, where a green line connecting a pair of edges indicates a turn cost of $0$ and a red line indicates a turn cost of $1$. Turn costs for unlabeled turns are assumed to be $1$. The edges of the junction graph induced by the threading in orange are in bold.
• Figure 4: (a) The junction representing variable $x$, with green indicating a cost of $1$ and red indicating a cost of $2$. (b) Two possible minimum-cost threadings of the junction and their corresponding junction graphs; double-threading the left and right edges of the junction corresponds to setting $x$ and $\lnot x$ to true, respectively.
• Figure 5: (a) The gadget for clause $\mathbbm{1}(x, y, z)$ and (b) an example threading that corresponds to assigning $x$ to true.

Theorems & Definitions (22)

• Definition 1: Threading demaine2024graph
• Definition 2: Turn Cost
• Definition 5: Minimum-Turn Perfect Threading
• Theorem 8
• Theorem 9
• Lemma 10
• Theorem 11
• Corollary 12
• Definition 13: 1-in-3 SAT
• Claim 14