On independent set on B1-EPG graphs [article]

Marin Bougeret, Stephane Bessy, Daniel Gonçalves, Cristophe Paul
2015 arXiv   pre-print
In this paper we consider the Maximum Independent Set problem (MIS) on B_1-EPG graphs. EPG (for Edge intersection graphs of Paths on a Grid) was introduced in edgeintersinglebend as the class of graphs whose vertices can be represented as simple paths on a rectangular grid so that two vertices are adjacent if and only if the corresponding paths share at least one edge of the underlying grid. The restricted class B_k-EPG denotes EPG-graphs where every path has at most k bends. The study of MIS
more » ... B_1-EPG graphs has been initiated in wadsMIS where authors prove that MIS is NP-complete on B_1-EPG graphs, and provide a polynomial 4-approximation. In this article we study the approximability and the fixed parameter tractability of MIS on B_1-EPG. We show that there is no PTAS for MIS on B_1-EPG unless P=NP, even if there is only one shape of path, and even if each path has its vertical part or its horizontal part of length at most 3. This is optimal, as we show that if all paths have their horizontal part bounded by a constant, then MIS admits a PTAS. Finally, we show that MIS is FPT in the standard parameterization on B_1-EPG restricted to only three shapes of path, and W_1-hard on B_2-EPG. The status for general B_1-EPG (with the four shapes) is left open.
arXiv:1510.00598v1 fatcat:uu6ymc23wnep7chkymxdytuww4