Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems
- 195 Downloads
We consider two new variants of online integer programs that are duals. 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 into multiple packets. The other problem considered is online covering: there is a universe to be covered. Sets arrive online, and we must decide for each set whether we add it to the cover or give it up. The cost of a solution is the total cost of sets taken, plus a penalty for each uncovered element. The number of sets in the solution grows over time, but its cost goes down. 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 Emek et al. (SIAM J Comput 41(4):728–746, 2012) 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.
KeywordsCompetitive analysis Randomized algorithm Packing integer programs Online set packing Team formation Prize-collecting multi-covering
We thank Moti Medina for going to Berkeley to represent us.
- 2.Awerbuch, B., Azar, Y., Plotkin, S.A.: Throughput-competitive on-line routing. In: 34th IEEE Annual Symposium on Foundations of Computer Science, pp. 32–40 (1993)Google Scholar
- 3.Bateni, M., Hajiaghayi, M., Zadimoghaddam, M.: Submodular secretary problem and extensions. In: 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, Volume 6302 of LNCS, pp. 39–52 (2010)Google Scholar
- 7.Cygan, M.: Improved approximation for 3-dimensional matching via bounded pathwidth local search. In: 54th IEEE Annual Symposium on Foundations of Computer Science, pp. 509–518 (2013)Google Scholar
- 9.Feldman, M., Naor, J.S., Schwartz, R.: Improved competitive ratios for submodular secretary problems. In: 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, Volume 6845 of LNCS, pp. 218–229 (2011)Google Scholar
- 22.Srinivasan, A.: Improved approximations of packing and covering problems. In: 27th Annual ACM Symposium on the Theory of Computing, pp. 268–276 (1995)Google Scholar