Packing \(r\)-cliques in weighted chordal graphs.

*(English)*Zbl 1091.90073Summary: In P. Hell et al. [Discrete Appl. Math. 141, No. 1–3, 185–194 (2004; Zbl 1043.05097)], we have previously observed that, in a chordal graph \(G\), the maximum number of independent \(r\)-cliques (i.e., of vertex disjoint subgraphs of \(G\), each isomorphic to \(K_r\), with no edges joining any two of the subgraphs) equals the minimum number of cliques of \(G\) that meet all the \(r\)-cliques of \(G\). When \(r = 1\), this says that chordal graphs have independence number equal to the clique covering number. When \(r = 2\), this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph \(G\), with a nonnegative weight for each \(r\)-clique in \(G\), we seek a set of independent \(r\)-cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from \(G\), and is an adaptation of an algorithm of M. Farber [Oper. Res. Let. 1, 134–138 (1982; Zbl 0495.05053)]. The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the \(r\)-cliques have the same weight, we simplify the algorithms reported in Hell et al. (loc. cit.), although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.

##### MSC:

90C35 | Programming involving graphs or networks |

05C85 | Graph algorithms (graph-theoretic aspects) |

##### Keywords:

chordal graphs; min-max theorems; greedy algorithms; linear programming duality; complementary slackness
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DOI

##### References:

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