Approximation algorithms for the chromatic sum
The chromatic sum of a graph G is the smallest total among all proper colorings of G using natural numbers. It was shown that computing the chromatic sum is NP-hard. In this article we prove that a simple greedy algorithm applied to sparse graphs gives a "good" approximation of the chromatic sum. For all graphs the existence of a polynomial time algorithm that approximates the chromatic sum with a linear function error implies P = NP.
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