Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems

摘要

We consider two new variants of online integer programs that are dual to each other. In the packing problem we are given a set of items and a collection of knapsack constraints over these items that are revealed over time in an online fashion. Upon arrival of a constraint we may need to remove several items (irrevocably) so as to maintain feasibility of the solution. Hence, the set of packed items becomes smaller over time. The goal is to maximize the number, or value, of packed items. The problem originates from a buffer-overflow model in communication networks, where items represent information units broken to multiple packets. The other problem considered is online covering: There is a universe we need to cover. Sets arrive online, and we must decide whether we take each set to the cover or give it up, so the number of sets in the solution grows over time. The cost of a solution is the total cost of sets taken, plus a penalty for each uncovered element. This problem is motivated by team formation, where the universe consists of skills, and sets represent candidates we may hire. The packing problem was introduced in [8] for the special case where the matrix is binary; in this paper we extend the solution to general matrices with non-negative integer entries. The covering problem is introduced in this paper; we present matching upper and lower bounds on its competitive ratio.