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.
@article{arxiv.0912.1050,
title = {Abstract Milling with Turn Costs},
author = {M. Fellows and P. Giannopoulos and C. Knauer and C. Paul and F. Rosamond and S. Whitesides and N. Yu},
journal= {arXiv preprint arXiv:0912.1050},
year = {2009}
}