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**On a graph partition problem with application to VLSI layout.**
*(English)*
Zbl 0764.68132

Summary: We discuss a graph partition problem with application to VLSI layout. It is not difficult to show that the general partition problem is NP- complete, so we restrict our attention to some special classes of graphs. A graph is called a circle graph if the nodes of the graph represent the chords of a circle and there is an edge between the two nodes if the corresponding chords intersects. K. J. Supowit [IEEE Trans. Computer Aided Design 6, 93-94 (1987)] had posed the partition problem of circle graphs as an open problem. However, it is also not too difficult to show that the partition problem on circle graphs remains NP-complete. We give an integer linear programming formulation for the general graph partition problem. The problem can then be solved using one of the several techniques for solving integer linear programming problems. We show that the partition problem can be solved in polynomial time if the graph is a tree. A graph is called perfect if the graph and each of its induced subgraphs have chromatic numbers equal to the maximum cardinality of its cliques. When the partition problem is restricted to perfect graphs we observe an interesting phenomenon. In the integer linear programming formulation of the partition problem we always obtain an integral solution when the graph is perfect. This phenomenon is unusual because the constraint matrix does not belong to the class of totally unimodular, balanced, or perfect matrices [M. W. Padberg, Math. Program. 6, 180-196 (1974; Zbl 0284.90061)], for which such a phenomenon is guaranteed. In our extensive experimentation we always found an optimal integral solution when the graph is perfect. We conjecture that an optimal integral solution for the perfect graph always exists. If the conjecture is true then it is likely to lead to the development of a polynomial time algorithm for some subclasses of perfect graphs, such as the interval graphs. We show that for some other subclasses of perfect graphs, such as the permutation graphs, the number of maximal cliques may be exponentially large and as such, even if the conjecture is true, may not lead to a polynomial time algorithm for the partition problem. The partition problem is the generalization of the chromatic sum problem introduced by E. Kubicka and A. Schwenk in [Introduction to chromatic sums, Proc. ACM Computer Science Conf. (1989)]. The problem was referred to as weighted coloring problem by K. J. Supowit. However, L. Lovasz in [M. Groetschel, L. Lovasz and A. Schrijver, Geometric algorithms and combinatorial optimization (1988; Zbl 0634.05001)] uses weighted coloring problem to describe a different problem. To avoid confusion we refer to it as optimum cost chromatic partition (OCCP) problem. The problem is the following: Given a graph \(G=(V,E)\) and a cost vector \(C=\langle C_ 1,C_ 2,\dots,C_ n\rangle (| V|=n)\), partition the vertex set \(V\) into independent sets \(V_ 1,V_ 2,\dots,V_ k\), \(1\leq k\leq n\), such that \(\sum^ k_{i=1}C_ i| V_ i|\) is minimum. If we set cost \(C_ i=i\), then this problem reduces to the chromatic sum problem of E. Kubicka and A. Schwenk [Introduction to chromatic sums, Proc. ACM Computer Science Conf. (1989)].

### MSC:

68R10 | Graph theory (including graph drawing) in computer science |

05C15 | Coloring of graphs and hypergraphs |

94C15 | Applications of graph theory to circuits and networks |

90C10 | Integer programming |

94C99 | Circuits, networks |

68R05 | Combinatorics in computer science |

### Keywords:

graph partition; VLSI layout; circle graph; NP-complete; linear programming; perfect graphs; polynomial time algorithm; interval graphs; permutation graphs; chromatic sum
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\textit{A. Sen} et al., Inf. Process. Lett. 43, No. 2, 87--94 (1992; Zbl 0764.68132)

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