Algorithms for budgeted auctions and multi-agent covering problems

ACKNOWLEDGEMENTS I would like to thank my advisor, Vijay Vazirani, for all his guidance, and from whom I have learnt the most. Especially during the last two years, he has played the role of a great mentor and a friend, always giving the right advice at the right moment. His knowledge and wisdom about research and life is among the best, and I feel lucky to have acquired at least a few bits of that, both consciously and subconsciously. Thanks to Milena Mihail for her support especially when I needed it the most, the initial phases when I would think of quitting. Thanks to Aranyak Mehta for being a great collaborator, friend, and mentor during my internship at Google Inc. My initial work with him, in many ways, defined my research interests. Thanks to my co-authors and friends and Pushkar Tripathi for all the great moments we shared working together and writing papers. Thanks to and many other friends with whom I have shared some of the best moments in Atlanta. Thanks to Adam O'Neill for all the philosophical breaks. Thanks also to the rest of the theory and ACO group members for creating a great research environment. Finally, thanks to my parents and family members for always being there, even though I could not never explain to them what exactly I was working on all these years. SUMMARY In this thesis, we do an algorithmic study of optimization problems in budgeted auctions, and some well known covering problems in the multi-agent setting. We give new results for the design of approximation algorithms, online algorithms and hardness of approximation for these problems. Along the way we give new insights for many other related problems. Budgeted Auctions. We study the following allocation problem which arises in budgeted auctions: Given a set of m indivisible items and n agents; agent i is willing to pay b ij for item j and has an overall budget of B i (i.e. the maximum total amount agent i is willing to pay). The goal is to allocate items to the agents so as to maximize the total revenue obtained. We give two approximation algorithms with 3/4-approximation factor, improving upon the previous best known factor of 0.632. We use linear programming based techniques, and our approximation ratio is optimal in the sense that it matches the integrality gap of the linear program used by us and …