Abstract
We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range delta >0. In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most delta from one of these facilities. We investigate this covering problem in terms of the rational parameter delta . We prove that the problem is polynomially solvable whenever delta is a unit fraction, and that the problem is NP-hard for all non unit fractions delta . We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for delta <3/2, and it is W[2]-hard for delta ge 3/2.
Highlights
We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges
We investigate the algorithmic behavior of a continuous covering problem on graphs
We provide a complete picture of the complexity of computing the δcovering number for connected graphs G = (V, E) and positive rational numbers δ
Summary
We investigate the algorithmic behavior of a continuous covering problem on graphs. Consider an undirected connected graph G = (V , E), whose edges are rectifiable and have unit length. The objective in [9] is not to cover, but to pack, and is dual to our objective: Place as many facilities as possible on the graph, subject to the condition that any two facilities have at least distance δ from each other This packing problem is polynomially solvable, if δ is a rational number with numerator 1 or 2, and it is NP-hard for all other rational values of δ. If δ is large (say δ ≥ 4), the main goal should be to cover all the vertices of the input graph with the facilities, whereas the edges only play a minor role and will be covered without much additional effort These cases have the flavor of the DOMINATING SET problem, which belongs to the intractable problems in the parametrized world.
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