On Covering Points with Minimum Turns

Minghui Jiang
2015 International journal of computational geometry and applications  
The problem of covering a set of points by a minimum number of lines is one of the oldest problems in computational geometry. • Megiddo and Tamir (1982) proved that the line cover problem is NP-hard even in R 2 . Axis-parallel Lines For the rectilinear version of the problem in which the lines must be axis-parallel, • Hassin and Megiddo (1991) observed that the problem in R 2 reduces to vertex cover in bipartite graphs and hence is solvable in polynomial time, and proved that the problem in R 3
more » ... in NP-hard by a reduction from 3-SAT, • Gaur and Bhattacharya (2007) presented a (d − 1)approximation algorithm for the problem in R d for all d ≥ 3. Hyperplanes For the more general problem of covering n points in R d by k (not necessarily axis-parallel) hyperplanes, • Langerman and Morin (2005) presented FPT algorithms with both d and k as parameters. • Grantson and Levcopoulos (2006) and Wang, Li, and Chen (2010) improved the running times. Polygonal Chain with Minimum Turns Instead of using lines, we can cover the points using a polygonal chain of line segments, with the goal of minimizing the number of links or turns in the chain. Given a set of n points in R d , a chain of line segments that covers all n points is called a covering tour if the chain is closed, and is called a spanning path if the chain is open. A covering tour (or a spanning path) is rectilinear if all segments in the tour (or the path) are axis-parallel.
doi:10.1142/s0218195915500016 fatcat:doe4blchxjagzc65r3ragl6zxu