Energy efficient scheduling via partial shutdown

Motivated by issues of saving energy in data centers we define a collection of new problems referred to as "machine activation" problems. The central framework we introduce considers a collection of <i>m</i> machines (unrelated or related) with each machine <i>i</i> having an <i>activation cost</i> of <i>a</i>_<i>i</i>. There is also a collection of <i>n</i> jobs that need to be performed, and <i>p</i>_{<i>i, j</i>} is the processing time of job <i>j</i> on machine <i>i</i>. Standard scheduling models assume that the set of machines is fixed and all machines are available. However, in our setting, we assume that there is an activation cost budget of <i>A</i> - we would like to <i>select</i> a subset <i>S</i> of the machines to activate with total cost <i>a</i>(<i>S</i>) ≤ <i>A</i> and <i>find</i> a schedule for the <i>n</i> jobs on the machines in <i>S</i> minimizing the makespan (or any other metric). We consider both the unrelated machines setting, as well as the setting of scheduling uniformly related parallel machines, where machine <i>i</i> has activation cost <i>a</i>_<i>i</i> and speed <i>s</i>_<i>i</i>, and the processing time of job <i>j</i> on machine <i>i</i> is <i>p</i>_{<i>i, j</i>} = <i>p</i>_<i>j</i><i>s</i>_<i>i</i>, where <i>p</i>_<i>j</i> is the processing requirement of job <i>j</i>. For the general unrelated machine activation problem, our main results are that if there is a schedule with makespan <i>T</i> and activation cost <i>A</i> then we can obtain a schedule with makespan (2 + ε)T and activation cost 2(1 + 1/ε)(ln <i>n</i>/<i>OPT</i> + 1)<i>A</i>, for any ε > 0. We also consider assignment costs for jobs as in the generalized assignment problem, and using our framework, provide algorithms that minimize the machine activation and the assignment cost simultaneously. In addition, we present a greedy algorithm which only works for the basic version and yields a makespan of 2<i>T</i> and an activation cost <i>A</i>(1 + ln <i>n</i>). For the uniformly related parallel machine sceduling problem, we develop a polynomial time approximation scheme that outputs a schedule with the property that the activation cost of the subset of machines is at most <i>A</i> and the makespan is at most (1 + ε)<i>T</i> for any ε > 0. For the special case of <i>m</i> identical speed machines, the machine activation problem is trivial, since the cheapest subset of <i>k</i> machines is always the best choice if the optimal solution activates <i>k</i> machines. In addition, we consider the case whem some jobs can be dropped (and are treated as outliers).