Approximation Algorithms for B1-EPG Graphs
The edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. We consider the case of single-bend paths, namely, the class known as B1-EPG graphs. The motivation for studying these graphs comes from the context of circuit layout problems. It is known that recognizing B1-EPG graphs is NP-complete, nevertheless, optimization problems when given a set of paths in the grid are of considerable practical interest.
In this paper, we show that the coloring problem and the maximum independent set problem are both NP-complete for B1-EPG graphs, even when the EPG representation is given. We then provide efficient 4-approximation algorithms for both of these problems, assuming the EPG representation is given. We conclude by noting that the maximum clique problem can be optimally solved in polynomial time for B1-EPG graphs, even when the EPG representation is not given.
Unable to display preview. Download preview PDF.
- 4.Cameron, K., Chaplick, S., Hoang, C.T.: Recognizing Edge Intersection Graphs of \(\llcorner\)-Shaped Grid Paths. In: LAGOS 2013 (to appear, 2013)Google Scholar
- 5.Cohen, E., Golumbic, M.C., Ries, B.: Characterizations of cographs as intersection graphs of paths on a grid (submitted)Google Scholar
- 8.Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980); Annals of Discrete Mathematics, 2nd edn., vol. 57. Elsevier, Amsterdam (2004)Google Scholar
- 11.Heldt, D., Knauer, K., Ueckerdt, T.: Edge-intersection graphs of grid paths: the bend-number, Arxiv preprint arXiv:1009.2861, arxiv.org (September 2010)Google Scholar