Compact LP Relaxations for Allocation Problems *

Klaus Jansen, Lars Rohwedder
BY 1st Symposium on Simplicity in Algorithms   unpublished
We consider the restricted versions of Scheduling on Unrelated Machines and the Santa Claus problem. In these problems we are given a set of jobs and a set of machines. Every job j has a size p j and a set of allowed machines Γ(j), i.e., it can only be assigned to those machines. In the first problem, the objective is to minimize the maximum load among all machines; in the latter problem it is to maximize the minimum load. For these problems, the strongest LP relaxation known is the
more » ... n LP. The configuration LP has an exponential number of variables and it cannot be solved exactly unless P = NP. Our main result is a new LP relaxation for these problems. This LP has only O(n 3) variables and constraints. It is a further relaxation of the configuration LP, but it obeys the best bounds known for its integrality gap (11/6 and 4). For the configuration LP these bounds were obtained using two local search algorithm. These algorithms, however, differ significantly in presentation. In this paper, we give a meta algorithm based on the local search ideas. With an instantiation for each objective function, we prove the bounds for the new compact LP relaxation (in particular, for the configuration LP). This way, we bring out many analogies between the two proofs, which were not apparent before. 1 Introduction We consider the problem of allocating jobs J to machines M. A popular variation is the restricted case, where j ∈ J has a size p j and can only be assigned to Γ(j) ⊆ M. Two natural objective functions are to minimize the maximum load or to maximize the minimum load among all machines, where the load of a machine is defined as the sum of the sizes over the jobs assigned to it. The first objective will be referred to as Makespan and the latter as Max-min. These problems are special cases of Scheduling on Unrelated Machines and the Santa Claus problem. Recent breakthroughs in both problems can be attributed to the study of the exponential size configuration LP which started with [4]. It was shown for the Max-Min problem that the LP has an integrality gap of at most 4 [3], which was the first constant factor guarantee there. Later, Svensson transferred these ideas to the Makespan problem and proved an upper bound of 33/17 for the integrality gap in this case [11], thereby giving the first improvement over * Research was supported by German Research Foundation (DFG) project JA 612/15-2