Approximation Algorithms for B 1-EPG Graphs

Abstract

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 B 1-EPG graphs. The motivation for studying these graphs comes from the context of circuit layout problems. It is known that recognizing B 1-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 B 1-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 B 1-EPG graphs, even when the EPG representation is not given.