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

08/13/2018
by   Sandor P. Fekete, et al.
0

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.

READ FULL TEXT

page 5

page 6

page 23

research
02/19/2019

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (...
research
05/08/2018

The Computational Complexity of Finding Hamiltonian Cycles in Grid Graphs of Semiregular Tessellations

Finding Hamitonian Cycles in square grid graphs is a well studied and im...
research
07/03/2023

The Lawn Mowing Problem: From Algebra to Algorithms

For a given polygonal region P, the Lawn Mowing Problem (LMP) asks for a...
research
10/18/2018

FPT algorithms to recognize well covered graphs

Given a graph G, let vc(G) and vc^+(G) be the sizes of a minimum and a m...
research
08/15/2020

Finding a Shortest Even Hole in Polynomial Time

An even (respectively, odd) hole in a graph is an induced cycle with eve...
research
01/07/2020

Hardness results for three kinds of colored connections of graphs

The concept of rainbow connection number of a graph was introduced by Ch...
research
09/09/2021

Worbel: Aggregating Point Labels into Word Clouds

Point feature labeling is a classical problem in cartography and GIS tha...

Please sign up or login with your details

Forgot password? Click here to reset