Contact Center Scheduling with Strict Resource Requirements

Consider the following problem which often arises in contact center scheduling scenarios. We are given a set of employees where each employee can be deployed for shifts consisting of L consecutive time units. Further, each employee specifies a set of possible start times, and can be deployed for a bounded number of shifts only. At each point of time t, we are also given a lower bound r(t) on the number of employees that should be present at this time. The goal is to find a schedule for the employees such that the number of time slots whose requirements are met is maximized. Such problems naturally arise in many other situations, e.g., sensor networks and cloud computing. The strict nature of the resource requirement makes this problem very hard to approximate. In this paper, we give a bicriteria approximation algorithm for this problem. Given a parameter epsilon > 0, we give an O(1/epsilon(3). log 1/epsilon)-approximation algorithm for this problem, where we count those time slots for which we satisfy at least (1-epsilon)-fraction of the requirement. Our techniques involve a configuration LP relaxation for this problem, and we use non-trivial structural properties of an optimal solution to solve this LP relaxation. We even consider the more general problem where shift lengths of different employees can vary significantly. In this case, we show that even finding a good bicriteria approximation is hard (under standard complexity theoretic assumptions).