Abstract
We investigate a variety of geometric problems of finding tours and cycle covers with minimum turn cost, which have been studied in the past, with complexity and 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, 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 a 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 for very general classes of instances, making use of LP/IP techniques. For these problems with many possible edge directions (and thus, turn angles, such as in hexagonal grids or higher-dimensional variants), our approximation factors also improve 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.
A full version of this extended abstract can be found at [17].
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Fekete, S.P., Krupke, D. (2019). Covering Tours and Cycle Covers with Turn Costs: Hardness and Approximation. In: Heggernes, P. (eds) Algorithms and Complexity. CIAC 2019. Lecture Notes in Computer Science(), vol 11485. Springer, Cham. https://doi.org/10.1007/978-3-030-17402-6_19
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