Approximation Algorithms for Packing Directed Acyclic Graphs into Two-Size Blocks

In this paper we consider the following variant of clustering or laying out problems of graphs: Given a directed acyclic graph (DAG for short) and an integer B, the objective is to find a mapping of its nodes into blocks of size at most B that minimizes the maximum number of external arcs during traversals of the acyclic structure by following paths from the roots to the leaves. An external arc is defined as an arc connecting two distinct blocks. This paper focuses on the case B = 2. Even if B = 2 and the height of the DAG is three, it is known that the problem is NP-hard, and furthermore, there is no 3/2 - epsilon factor approximation algorithm for B = 2 and a small positive epsilon unless P = NP. On the other hand, the best approximation ratio previously shown is 3. In this paper we improve the approximation ratio into strictly smaller than 2. Also, we investigate the relationship between the height of input DAGs and the inapproximability, since the above inapproximability bound 3/2 - epsilon is shown only for DAGs of height 3.