TY - JOUR

T1 - Shrinking Maxima, Decreasing Costs

T2 - New Online Packing and Covering Problems

AU - Fraigniaud, Pierre

AU - Halldórsson, Magnús M.

AU - Patt-Shamir, Boaz

AU - Rawitz, Dror

AU - Rosén, Adi

N1 - Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2016/4/1

Y1 - 2016/4/1

N2 - 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.

AB - 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.

KW - Competitive analysis

KW - Online set packing

KW - Packing integer programs

KW - Prize-collecting multi-covering

KW - Randomized algorithm

KW - Team formation

UR - http://www.scopus.com/inward/record.url?scp=84926341750&partnerID=8YFLogxK

U2 - 10.1007/s00453-015-9995-8

DO - 10.1007/s00453-015-9995-8

M3 - מאמר

AN - SCOPUS:84926341750

VL - 74

SP - 1205

EP - 1223

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 4

ER -