https://hal.inria.fr/hal-01423640Fraigniaud, PierrePierreFraigniaudGANG - Networks, Graphs and Algorithms - IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique - Inria de Paris - Inria - Institut National de Recherche en Informatique et en AutomatiqueIRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche ScientifiqueHalldorsson, Magnus M.Magnus M.HalldorssonSchool of Computer Science [Reykjavik] - Reykjavík UniversityPatt-Shamir, BoazBoazPatt-ShamirDepartment of Electrical Engineering - TAU - Tel Aviv UniversityRawitz, DrorDrorRawitzTAU-CS - School of Computer Science - TAU - Tel Aviv UniversityRosén, AdiAdiRosénIRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche ScientifiqueShrinking Maxima, Decreasing Costs: New Online Packing and Covering ProblemsHAL CCSD2016[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS][INFO.INFO-DC] Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC]Fraigniaud, Pierre2016-12-30 17:57:462023-02-08 17:11:002016-12-30 17:57:46enJournal articles1Weconsidertwonewvariantsofonlineintegerprogramsthatareduals.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.