# Minimizing Average Completion of Dedicated Tasks and Interval Graphs

## Abstract

Scheduling dependent jobs on multiple machines is modeled as a graph *(multi)coloring* problem. The focus of this work is on the *sum of completion times* measure. This is known as the *sum (multi)coloring* of the conflict graph. We also initiate the study of the *waiting time* and the *robust throughput* of colorings. For uniform-length tasks we give an algorithm which simultaneously approximates these two measures, as well as sum coloring and the chromatic number, within constant factor, for any graph in which the *k*-colorable subgraph problem is polynomially solvable. In particular, this improves the best approximation ratio known for sum coloring interval graphs from 2 to 1.665.

We then consider the problem of scheduling non-preemptively tasks (of non-uniform lengths) that require exclusive use of dedicated processors. The objective is to minimize the sum of completion times. We obtain the first constant factor approximations for this problem, when each task uses a constant number of processors.

## Keywords

Completion Time Chromatic Number Interval Graph Chordal Graph Average Completion## Preview

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