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- INSTANCE:
Graph
.
- SOLUTION:
A coloring of
*G*, i.e., a partition of *V* into disjoint sets
such that each
is an independent set for *G*.
- MEASURE:
The sum of the coloring, i.e.
.

*Good News:*
As easy as MAXIMUM INDEPENDENT SET[57], within a constant factor.
*Bad News:*
As hard as MINIMUM GRAPH COLORING[57,156].
*Comment:*
The good news hold for any hereditary class of graphs.
In particular, perfect graphs are approximable within 4 [57].
For bipartite graphs, the problem is APX-complete and approximable
within 10/9 [59]. Approximable within 2 on line graphs
[57] and on interval graphs [425].
Generalization of MINIMUM COLOR SUM where the input is extended by
positive coloring costs
,
and *i* is
changed to
in the objective function, is not approximable within
even for split, chordal, permutation and
comparability graphs. For bipartite and interval graphs, the problem
is approximable within
but not within
[270].
In a multicoloring generalization, we are given a graph *G* and an
integral color requirement *x(v)* for each vertex *v*, and are to assign
each vertex to *x(v)* sets
such that each
is an
independent set. The objective is to minimize
,
where *f(v)* is the largest *k* such that *v* is assigned to .
In the non-preemptive variant, a vertex *v* must be assigned to contiguous
sets
.
This variant is
approximable within
on perfect graphs, 2.796 on bipartite
graphs, and
on bounded degree graphs [58],
within *O(1)* on interval graphs and 12 on line graphs [227],
and admits an
FPTAS on partial *k*-trees and a PTAS on planar graphs [226].
The preemptive variant is approximable within
on graphs
where MAXIMUM INDEPENDENT SET is approximable within ,
within 2
on line graphs, 1.5 on bipartite graphs, and
on
bounded-degree graphs [58], and admits a PTAS on trees
[228] and on partial *k*-trees and planar graphs [226].

** Next:** MAXIMUM ACHROMATIC NUMBER
** Up:** Covering and Partitioning
** Previous:** MINIMUM GRAPH COLORING
** Index**
*Viggo Kann*

*2000-03-20*