The Geometry of Scheduling

We consider the following general scheduling problem: The input consists of n jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. Our main result is a randomized polynomial-time algorithm with an approximation ratio O(log log nP), where P is the maximum job size. We also give an O(1) approximation in the special case when all jobs have identical release times. The main idea is to reduce this scheduling problem to a particular geometric set-cover problem which is then solved using the local ratio technique and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. Our geometric interpretation of scheduling may be of independent interest.Comment: Conference version in FOCS 201

Autonomous Algorithms for Centralized and Distributed Interference Coordination: A Virtual Layer Based Approach

Interference mitigation techniques are essential for improving the performance of interference limited wireless networks. In this paper, we introduce novel interference mitigation schemes for wireless cellular networks with space division multiple access (SDMA). The schemes are based on a virtual layer that captures and simplifies the complicated interference situation in the network and that is used for power control. We show how optimization in this virtual layer generates gradually adapting power control settings that lead to autonomous interference minimization. Thereby, the granularity of control ranges from controlling frequency sub-band power via controlling the power on a per-beam basis, to a granularity of only enforcing average power constraints per beam. In conjunction with suitable short-term scheduling, our algorithms gradually steer the network towards a higher utility. We use extensive system-level simulations to compare three distributed algorithms and evaluate their applicability for different user mobility assumptions. In particular, it turns out that larger gains can be achieved by imposing average power constraints and allowing opportunistic scheduling instantaneously, rather than controlling the power in a strict way. Furthermore, we introduce a centralized algorithm, which directly solves the underlying optimization and shows fast convergence, as a performance benchmark for the distributed solutions. Moreover, we investigate the deviation from global optimality by comparing to a branch-and-bound-based solution.Comment: revised versio