The energy costs for cooling a data center constitute a significant portion of the overall running costs. Thermal imbalance and hot spots that arise due to imbalanced workloads lead to significant wasted cooling effort -in order to ensure that no equipment is operating above a certain temperature, the data center may be cooled more than necessary. Therefore it is desirable to schedule the workload in a data center in a thermally aware manner, assigning jobs to machines not just based on local

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... ad of the machines, but based on the overall thermal profile of the data center. This is challenging because of the spatial cross-interference between machines, where a job assigned to a machine may impact not only that machine's temperature, but also nearby machines. Here, we continue formal analysis of the thermal scheduling problem that we initiated recently [25] . In that work, the notion of effective load of a machine which is a function of the local load on the machine as well as the load on nearby machines, was introduced, and optimal scheduling policies for a simple model (where cross-effects are restricted within a rack) were presented, under the assumption that jobs can be split among different machines. Here we consider the more realistic problem of integral assignment of jobs, and allow for cross-interference among different machines in adjacent racks in the data center. The integral assignment problem with cross-interference is NP-hard, even for a simple two machine model. We consider three different heat flow models, and give constant factor approximation algorithms for maximizing the number (or total profit) of jobs assigned in each model, without violating thermal constraints. We also consider the problem of minimizing the maximum temperature on any machine when all jobs need to be assigned, and give constant factor algorithms for this problem.