Covering Tours and Cycle Covers with Turn Costs: Hardness and Approximation

TL;DR

This paper studies the computational complexity and approximation algorithms for finding minimal-turn tours and cycle covers in grid graphs, revealing NP-hardness results and introducing the first constant-factor approximation algorithms for various problem variants.

Contribution

It proves NP-hardness for minimum-turn cycle covers in grid graphs, solving a long-standing open problem, and develops the first constant-factor approximation algorithms for subset and penalty variants.

Findings

NP-hardness of minimum-turn cycle cover in 2D grid graphs

NP-hardness of subset cycle cover in thin grid graphs

First constant-factor approximation algorithms for subset and penalty variants

Abstract

We investigate a variety of problems of finding tours and cycle covers with minimum turn cost. Questions of this type have been studied in the past, with complexity and approximation results as well as open problems dating back to work by Arkin et al. in 2001. A wide spectrum of practical applications have renewed the interest in these questions, and spawned variants: for full coverage, every point has to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring an individual penalty. We make a number of contributions. We first show that finding a minimum-turn (full) cycle cover is NP-hard even in 2-dimensional grid graphs, solving the long-standing open Problem 53 in The Open Problems Project edited by Demaine, Mitchell and O'Rourke. We also prove NP-hardness of finding a subset cycle cover of minimum turn cost in thin grid graphs, for which Arkin et al. gave a polynomial-time algorithm for full coverage; this shows that their boundary techniques cannot be applied to compute exact solutions for subset and penalty variants. On the positive side, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants, making use of LP/IP techniques. For full coverage in more general grid graphs (e.g., hexagonal grids), our approximation factors are better than the combinatorial ones of Arkin et al. Our approach can also be extended to other geometric variants, such as scenarios with obstacles and linear combinations of turn and distance costs.