### Scheduling with conflicts: online and offline algorithms

Guy Even, Magnús M. Halldórsson, Lotem Kaplan, Dana Ron
2008 Journal of Scheduling
We consider the following problem of scheduling with conflicts (SWC): Find a minimum makespan schedule on identical machines where conflicting jobs cannot be scheduled concurrently. We study the problem when conflicts between jobs are modeled by general graphs. Our first main positive result is an exact algorithm for two machines and job sizes in {1, 2}. For jobs sizes in {1, 2, 3}, we can obtain a 4 3 -approximation, which improves on the 3 2 -approximation that was previously known for this
more » ... se. Our main negative result is that for jobs sizes in {1, 2, 3, 4}, the problem is APX-hard. Our second contribution is the initiation of the study of an online model for SWC, where we present the first results in this model. Specifically, we prove a lower bound of 2 − 1 m on the competitive ratio of any deterministic online algorithm for m machines and unit jobs, and an upper bound of 2 when the algorithm is not restricted computationally. For three machines we can show that an efficient greedy algorithm achieves this bound. For two machines we present a more complex algorithm that achieves a competitive ratio of 2 − 1 7 when the number of jobs is known in advance to the algorithm. We consider the problem of scheduling with conflicts (SWC), defined as follows. There are m identical machines. The input consists of a set J of n jobs with processing times {p j } j∈J , and a conflict graph G = (J, E) over the jobs. Each edge in E models a pair of conflicting jobs that cannot be scheduled concurrently (on different machines). A schedule is an assignment of time intervals on the m machines to the jobs that satisfies the following conditions: (i) each job j ∈ J is assigned an interval of length p j on one machine; (ii) intervals on the same machine do not overlap; and (iii) intervals assigned to conflicting jobs do not overlap. The makespan of a schedule is the largest endpoint of an interval assigned to a job. We are interested in schedules that minimize the makespan. Scheduling with conflicts generally arises as resource-constrained scheduling [GG75] . In this setting there is a set of resources, each with a certain supply. Each job has a specified demand for each resource. A conflict arises between a subset of jobs if their cumulative demand of a resource exceeds its supply. In general this setting can be modeled by a conflict hypergraph. In special cases, a conflict graph suffices, e.g., if the resources are nonsharable. In [BC96] an application of this type is presented for balancing the load in a parallel computation. In [HKP + 03] other applications are mentioned in traffic intersection control, frequency assignment in cellular networks, and session management in local area networks. An application derived from a problem of assigning operations to processors, where the operations are given in a flow graph, is described in [BJ95] . In the online version of SWC, each job j has a release time r j that specifies when the job arrives. A job j arrives together with the conflict edges to all jobs whose release time is at most r j . Hence, the conflict graph is revealed as the jobs arrive. A job j can be scheduled starting any time t ≥ r j . Scheduling decisions at time t can only depend on jobs that have arrived before or at time t. This model is often called scheduling over time. We compare the makespan of the schedule computed by an online algorithm to the makespan of an optimal offline algorithm whose input consists of the whole conflict graph and the release times of all the jobs. We are interested in bounding the competitive ratio, i.e., the worst case ratio between the online makespan and the optimal offline makespan. Problems involving conflict scheduling can be classified according to the following parameters: (i) The objective: minimum makespan, minimum sum of completion times, maximum response time (latency). (ii) Job model: unit jobs, arbitrary processing times with/without preemption, batch scheduling. (iii) Number of machines m: fixed or unlimited. In this paper we focus on the case characterized by the minimum makespan objective, nonpreemptive processing, and a fixed number of machines. We consider the cases of unit jobs, short jobs, and arbitrary processing times. One may further characterize problems by: (iv) Online/Offline model: In the online model, jobs arrive at different release times. (v) Conflict graph: general or belonging to specific classes (e.g., trees, interval graphs, etc.) We consider both the online and offline models, and treat general graphs.