Capacitated Arc Stabbing

In the Capacitated Arc Stabbing problem (CAS) we are given a set of arcs and a set of points on a circle. We say that a point p covers, or stabs, an arc A if p is contained in A. Each point has a weight and a capacity that determines the number of arcs it may cover. The goal is to find a minimum weight set of points that stabs all the arcs. CAS models a periodic multi-item lot sizing problem in which we are given a set of production requests each with its own periodic release time and deadline. Production takes place in batches of bounded capacity: each time unit t   is associated with a capacity c(t)c(t) and weight w(t)w(t), where c(t)c(t) bounds the number of requests that can be manufactured at time t  , and w(t)w(t) is a fixed cost for manufacturing any positive number of requests up to c(t)c(t) at time t. The goal is to find a minimum weight periodic schedule. We present a polynomial time algorithm for CAS that is based on a non-trivial reduction to Capacitated Interval Stabbing. Our approach applies to both hard and soft capacities. We also consider two variants of CAS in which some arcs may remain uncovered: in the partial variant there is a covering requirement g, and the goal is to find a minimum weight set of points that covers at least g arcs; and in the prize collecting variant each arc has a penalty that must be paid if this arc is not covered.