The Online Specialization Problem

We study the online specialization problem, where items arrive in an online fashion for processing by one of n different methods. Each method has two costs: a processing cost (paid once for each item processed), and a set-up cost (paid only once, on the method’s first use). There are n possible types of items; an item’s type determines the set of methods available to process it. Each method has a different degree of specialization. Highly specialized methods can process few item types while generic methods may process all item types. This is a generalization of ski-rental and closely related to the capital investment problem of Azar et al. (1999). We primarily study the case where method i + 1 is always more specialized than method i and the set-up cost for a more specialized method is always higher than that of a less specialized method. We describe an algorithm with competitive ratio O(logn), and also show an Ω(logn) lower bound on the competitive ratio for this problem; this shows our ratio is tight up to constant factors.