Nearly optimal perfectly-periodic schedules

We consider the problem of scheduling a set of jobs on a single shared resource using time-multiplexing. A perfectly-periodic schedule is one where resource time is divided into equal size "time-slots" quanta, and each job gets a time slot precisely every fixed interval of time (the period of the job). Periodic schedules are advantageous in distributed settings with synchronized clocks, since they require very little communication to establish, and thereafter no additional communication overhead is needed. In this work we study the case where each job i has a given demand probability w_i, and the goal is to design a perfectly-periodic schedule that minimizes the average time a random client waits until its job is executed. The problem is known to be NP-hard. The best known polynomial algorithm to date guarantees average waiting time of at most 3/2 opt + O(log M), where opt is the optimal waiting time. In this paper, we develop a tree-based methodology for periodic scheduling, and using new general results, we derive algorithms with better bounds. A key quantity in our methodology is a₁ ^def= √max{w_i}/ ∑ √w_i. We compare the cost of a solution provided by our algorithms to the cost of a solution to a relaxed (non-integral) version of the problem. Our asymptotic tree-based algorithm guarantees cost of at most 1 + a₁^O(1) times the cost of the relaxed problem; on the other hand, we prove that the cost of any integral solution is bounded from below by the cost of the relaxed solution times 1 + a₁^Ω(1). We also provide three other tree-based algorithms with cost bounded by the cost of the relaxed solution times 3/2, 4/3+O(a₁), and 9/8+O(a₁). Each one of our four algorithms is the best known for some range of values of a₁.