# Clique partitioning of interval graphs with submodular costs on the cliques

Dion Gijswijt; Vincent Jost; Maurice Queyranne

RAIRO - Operations Research (2007)

- Volume: 41, Issue: 3, page 275-287
- ISSN: 0399-0559

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topGijswijt, Dion, Jost, Vincent, and Queyranne, Maurice. "Clique partitioning of interval graphs with submodular costs on the cliques." RAIRO - Operations Research 41.3 (2007): 275-287. <http://eudml.org/doc/250125>.

@article{Gijswijt2007,

abstract = {
Given a graph G = (V,E) and a “cost function” $f: 2^V\rightarrow\mathbb\{R\}$ (provided by an oracle), the problem [PCliqW] consists in finding a partition into cliques of V(G) of minimum cost. Here, the cost of a partition is the sum of the costs of the cliques in the partition.
We provide a polynomial time dynamic program for the case where G is an interval graph and f belongs to a subclass of submodular set functions, which we call “value-polymatroidal”.
This provides a common solution for various generalizations of the coloring
problem in co-interval graphs such as max-coloring,
“Greene-Kleitman's dual”, probabilist coloring and chromatic entropy. In the last two cases, this is the first polytime algorithm for co-interval graphs. In contrast, NP-hardness of related problems is discussed. We also describe an ILP formulation for [PCliqW] which gives a common polyhedral framework to express min-max relations such as $\{\overline\{\chi\}\}=\alpha$
for perfect graphs and the polymatroid intersection theorem. This approach allows to provide a min-max formula for [PCliqW] if G is the line-graph of a bipartite graph and f is submodular.
However, this approach fails to provide a min-max relation for [PCliqW] if G is an interval graphs and f is value-polymatroidal.
},

author = {Gijswijt, Dion, Jost, Vincent, Queyranne, Maurice},

journal = {RAIRO - Operations Research},

keywords = {Partition into cliques; Interval graphs; Circular arc graphs; Max-coloring; Probabilist coloring; Chromatic entropy; Partial q-coloring; Batch-scheduling; Submodular functions; Bipartite matchings; Split graphs; partition into cliques; interval graphs; circular arc graphs; probabilist coloring; chromatic entropy; partial -coloring; batch-scheduling; submodular functions; bipartite matchings},

language = {eng},

month = {8},

number = {3},

pages = {275-287},

publisher = {EDP Sciences},

title = {Clique partitioning of interval graphs with submodular costs on the cliques},

url = {http://eudml.org/doc/250125},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Gijswijt, Dion

AU - Jost, Vincent

AU - Queyranne, Maurice

TI - Clique partitioning of interval graphs with submodular costs on the cliques

JO - RAIRO - Operations Research

DA - 2007/8//

PB - EDP Sciences

VL - 41

IS - 3

SP - 275

EP - 287

AB -
Given a graph G = (V,E) and a “cost function” $f: 2^V\rightarrow\mathbb{R}$ (provided by an oracle), the problem [PCliqW] consists in finding a partition into cliques of V(G) of minimum cost. Here, the cost of a partition is the sum of the costs of the cliques in the partition.
We provide a polynomial time dynamic program for the case where G is an interval graph and f belongs to a subclass of submodular set functions, which we call “value-polymatroidal”.
This provides a common solution for various generalizations of the coloring
problem in co-interval graphs such as max-coloring,
“Greene-Kleitman's dual”, probabilist coloring and chromatic entropy. In the last two cases, this is the first polytime algorithm for co-interval graphs. In contrast, NP-hardness of related problems is discussed. We also describe an ILP formulation for [PCliqW] which gives a common polyhedral framework to express min-max relations such as ${\overline{\chi}}=\alpha$
for perfect graphs and the polymatroid intersection theorem. This approach allows to provide a min-max formula for [PCliqW] if G is the line-graph of a bipartite graph and f is submodular.
However, this approach fails to provide a min-max relation for [PCliqW] if G is an interval graphs and f is value-polymatroidal.

LA - eng

KW - Partition into cliques; Interval graphs; Circular arc graphs; Max-coloring; Probabilist coloring; Chromatic entropy; Partial q-coloring; Batch-scheduling; Submodular functions; Bipartite matchings; Split graphs; partition into cliques; interval graphs; circular arc graphs; probabilist coloring; chromatic entropy; partial -coloring; batch-scheduling; submodular functions; bipartite matchings

UR - http://eudml.org/doc/250125

ER -

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