What Goes Around Comes Around: Covering Tours and Cycle Covers with Turn Costs

We investigate several geometric problems of finding tours and cycle covers with minimum turn cost, which have been studied in the past, with complexity, approximation results, and open problems dating back to work by Arkin et al. in 2001. Many new practical applications have spawned variants: For full coverage, all points have to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring a penalty. We 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. We also provide a number of positive results. In particular, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants for grid-based instances, based on LP/IP techniques. These geometric versions allow many possible edge directions (and thus, turn angles, such as in hexagonal grids or higher-dimensional variants); our approximation factors improve the combinatorial ones of Arkin et al.