TY - GEN
T1 - Primal-Dual Schemes for Online Matching in Bounded Degree Graphs
AU - Cohen, Ilan Reuven
AU - Peng, Binghui
N1 - Publisher Copyright: © Ilan Reuven Cohen and Binghui Peng.
PY - 2023/9
Y1 - 2023/9
N2 - We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [8], the allocation problem [5], and the AdWords problem [13] in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc [14]. Such graphs model natural properties on the degrees of advertisers and queries in the allocation and AdWords problems. While previous work only considers the scenario where k ≥ d, we consider the interesting intermediate regime of k ≤ d and prove a tight competitive ratio as a function of k, d (under the small-bid assumption) of τ(k, d) = 1 − (1 − k/d) · (1 − 1/d)d−k for the b-matching and allocation problems. We exploit primal-dual schemes [6, 3] to design and analyze the corresponding tight upper and lower bounds. Finally, we show a separation between the allocation and AdWords problems. We demonstrate that τ(k, d) competitiveness is impossible for the AdWords problem even in (k, d)-bounded degree graphs.
AB - We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [8], the allocation problem [5], and the AdWords problem [13] in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc [14]. Such graphs model natural properties on the degrees of advertisers and queries in the allocation and AdWords problems. While previous work only considers the scenario where k ≥ d, we consider the interesting intermediate regime of k ≤ d and prove a tight competitive ratio as a function of k, d (under the small-bid assumption) of τ(k, d) = 1 − (1 − k/d) · (1 − 1/d)d−k for the b-matching and allocation problems. We exploit primal-dual schemes [6, 3] to design and analyze the corresponding tight upper and lower bounds. Finally, we show a separation between the allocation and AdWords problems. We demonstrate that τ(k, d) competitiveness is impossible for the AdWords problem even in (k, d)-bounded degree graphs.
KW - Online Matching
KW - Primal-dual analysis
KW - bounded-degree graph
KW - the AdWords problem
UR - http://www.scopus.com/inward/record.url?scp=85173488750&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ESA.2023.35
DO - 10.4230/LIPIcs.ESA.2023.35
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 31st Annual European Symposium on Algorithms, ESA 2023
A2 - Li Gortz, Inge
A2 - Farach-Colton, Martin
A2 - Puglisi, Simon J.
A2 - Herman, Grzegorz
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 31st Annual European Symposium on Algorithms, ESA 2023
Y2 - 4 September 2023 through 6 September 2023
ER -