Abstract
We study a bilevel optimization problem which is a zerosum Stackelberg game. In this problem, there are two players, a leader and a follower, who pick items from a common set. Both the leader and the follower have their own (multidimensional) budgets, respectively. Each item is associated with a profit, which is the same to the leader and the follower, and will consume the leader’s (follower’s) budget if it is selected by the leader (follower). The leader and the follower will select items in a sequential way: First, the leader selects items within the leader’s budget. Then the follower selects items from the remaining items within the follower’s budget. The goal of the leader is to minimize the maximum profit that the follower can obtain. Let s_A and s_B be the dimension of the leader’s and follower’s budget, respectively. A special case of our problem is the bilevel knapsack problem studied by Caprara et al. [SIAM Journal on Optimization, 2014], where s_A = s_B = 1. We consider the general problem and obtain an (s_B+ε)approximation algorithm when s_A and s_B are both constant. In particular, if s_B = 1, our algorithm implies a PTAS for the bilevel knapsack problem, which is the first 𝒪(1)approximation algorithm. We also complement our result by showing that there does not exist any (4/3ε)approximation algorithm even if s_A = 1 and s_B = 2. We also consider a variant of our problem with resource augmentation when s_A and s_B are both part of the input. We obtain an 𝒪(1)approximation algorithm with 𝒪(1)resource augmentation, that is, we give an algorithm that returns a solution which exceeds the given leader’s budget by 𝒪(1) times, and the objective value achieved by the solution is 𝒪(1) times the optimal objective value that respects the leader’s budget.
BibTeX  Entry
@InProceedings{chen_et_al:LIPIcs.ICALP.2022.39,
author = {Chen, Lin and Wu, Xiaoyu and Zhang, Guochuan},
title = {{Approximation Algorithms for Interdiction Problem with Packing Constraints}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {39:139:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772358},
ISSN = {18688969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16380},
URN = {urn:nbn:de:0030drops163806},
doi = {10.4230/LIPIcs.ICALP.2022.39},
annote = {Keywords: Bilevel Integer Programming, Interdiction Constraints, Knapsack}
}
Keywords: 

Bilevel Integer Programming, Interdiction Constraints, Knapsack 
Collection: 

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) 
Issue Date: 

2022 
Date of publication: 

28.06.2022 