Abstract Milling with Turn Costs

Abstract Milling with Turn Costs ⋆ M. Fellows 1 ⋆⋆ , P. Giannopoulos 2⋆⋆⋆ , C. Knauer 2 , Christophe Paul 3 † ,F. Rosamond 1 ‡ , S. Whitesides 4 § , and N. Yu 4 1 PCRU, Office of DVC(Research), University of Newcastle, Australia 2 Institut fu¨r Informatik, Freie Universit¨at, Berlin, Germany 3 CNRS - LIRMM, Montpellier, France 4 McGill University, School of Computer Science, Canada.Abstract. The Abstract Milling problem is a natural and quitegeneral graph-theoretic model for geometric milling problems. Given agraph, one asks for a walk that covers all its vertices with a minimumnumber of turns , as specified in the graph model by a 0/1 turncost func-tion f x 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 edgee and departs on edge f. We describe an initial study of the parame-terized complexity of the problem. Our main positive result shows thatAbstract Milling, parameterized by: number of turns, treewidth andmaximum degree, is fixed-parameter tractable, We also show that Ab-stract Milling parameterized by (only) the number of turns and thepathwidth, is hard for W[1] — one of the few parameterized intractabilityresults for bounded pathwidth.