S. Whitesides | Searcharxiv

arXiv2009

Abstract Milling with Turn Costs

The Abstract Milling problem is a natural and quite general graph-theoretic model for geometric milling problems. Given a graph, one asks for a walk that covers all its vertices with a minimum number of turns, as specified in the graph model by a 0/1 turncost function fx at each vertex x giving, for each ordered pair of edges (e,f) incident at x, the turn cost at x of a walk that enters the vertex on edge e and departs on edge f. We describe an initial study of the parameterized complexity of the problem. Our main positive result shows that Abstract Milling, parameterized by: number of turns, treewidth and maximum degree, is fixed-parameter tractable, We also show that Abstract Milling parameterized by (only) the number of turns and the pathwidth, is hard for W[1] -- one of the few parameterized intractability results for bounded pathwidth.

cs.CC↗

arXiv1999

Locked and Unlocked Polygonal Chains in 3D

In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D. All our algorithms require only O(n) basic ``moves'' and run in linear time.

cs.CG↗

arXiv1998

Locked and Unlocked Polygonal Chains in 3D

In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D with a polynomial number of moves.

cs.CG↗