How to pack directed acyclic graphs into small blocks

This paper studies the following variant of clustering or laying out problems for directed acyclic graphs (DAG for short), called the minimum block transfer problem. The objective of this problem is to find a partition of a node set which satisfies the following two conditions: (i) Each element (called a block) of the partition has a size that is at most B and (ii) the maximum number of external arcs among directed paths from the roots to the leaves is minimized. Here, an external arc is defined as an arc connecting two distinct blocks. This paper mainly studies the case B=2. First, we show that the problem is NP-hard even if the height of DAGs is three and its maximum indegree and outdegree are two and three, respectively. Then, we design (i) linear-time optimal algorithms for DAGs of height at most two, (ii) a very simple 2-approximation algorithm, and moreover, (iii) a (2−2∕h)-approximation algorithm for the case that the height h of the input DAG is even and the other one for odd h, whose approximation ratio is 2−2∕(h+1). As for the inapproximability of the problem, for any ε>0 and unless P = NP, we show that the problem does not admit any polynomial time (3∕2−ε)-approximation ((4∕3−ε)-approximation, resp.) algorithm if the height of the input DAGs is restricted to at most five (at least six, resp.). Also, in the case B≥3, we show the NP-hardness and prove a (3∕2−ε)-inapproximability of this case.