Abstract Milling with Turn Costs

TL;DR

This paper studies the Abstract Milling problem, a graph-theoretic model for geometric milling, focusing on its parameterized complexity and showing fixed-parameter tractability under certain parameters.

Contribution

It provides the first fixed-parameter tractability results for Abstract Milling parameterized by turns, treewidth, and degree, and establishes W[1]-hardness for parameters involving pathwidth.

Findings

FPT when parameterized by turns, treewidth, and degree.

W[1]-hardness when parameterized by turns and pathwidth.

Initial complexity analysis of the Abstract Milling problem.

Abstract

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.