TY - GEN

T1 - How to pack directed acyclic graphs into small blocks

AU - Asahiro, Yuichi

AU - Furukawa, Tetsuya

AU - Ikegami, Keiichi

AU - Miyano, Eiji

PY - 2006/1/1

Y1 - 2006/1/1

N2 - The paper studies the following variant of clustering or laying out problems of graphs: Given a directed acyclic graph (DAG for short), 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. The problem can be shown to be NP-hard generally, and to remain intractable even if B = 2 and the height of DAGs is three. In this paper we provide a 3/2 factor linear time approximation algorithm for B = 2, and prove that the 3/2 ratio is optimal in terms of approximation guarantee. In the case of B ≥ 3, we also show that there is no 3/2 - ε factor approximation algorithm assuming P ≠ NP, where ε is arbitrarily small positive. Furthermore, we give a 2 factor approximation algorithm for B = 3 if the input is restricted to a set of layered graphs.

AB - The paper studies the following variant of clustering or laying out problems of graphs: Given a directed acyclic graph (DAG for short), 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. The problem can be shown to be NP-hard generally, and to remain intractable even if B = 2 and the height of DAGs is three. In this paper we provide a 3/2 factor linear time approximation algorithm for B = 2, and prove that the 3/2 ratio is optimal in terms of approximation guarantee. In the case of B ≥ 3, we also show that there is no 3/2 - ε factor approximation algorithm assuming P ≠ NP, where ε is arbitrarily small positive. Furthermore, we give a 2 factor approximation algorithm for B = 3 if the input is restricted to a set of layered graphs.

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UR - http://www.scopus.com/inward/citedby.url?scp=33746104069&partnerID=8YFLogxK

U2 - 10.1007/11758471_27

DO - 10.1007/11758471_27

M3 - Conference contribution

AN - SCOPUS:33746104069

SN - 354034375X

SN - 9783540343752

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 272

EP - 283

BT - Algorithms and Complexity - 6th Italian Conference, CIAC 2006, Proceedings

PB - Springer Verlag

T2 - 6th Italian Conference on Algorithms and Complexity, CIAC 2006

Y2 - 29 May 2006 through 31 May 2006

ER -