Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max |S|≤k f(S), the greedy algorithm yields a (1 − 1/e)-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires |X | to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

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In the problem of finding an efficient allocation when agents' utilities are privately known, we examine the effect of restricting attention to mechanisms using demand queries, which ask agents to report an optimal allocation given a price list. We construct a combinatorial allocation problem with m items and two agents whose valuations lie in a certain class, such that (i) e ciency can be obtained with a mechanism using O(m) bits, but (ii) any demand-query mechanism guaranteeing a higher efficiency than giving all items to one agent uses a number of queries that is exponential in m. The same is proven for any demand-query mechanism achieving an improvement in expected efficiency, for a constructed joint probability distribution over agents' valuations from the class. These results cast doubt on the usefulness of such common combinatorial allocation mechanisms as iterative auctions and other preference elicitation mechanisms using demand queries, as well as value queries and order queries (which are easily replicated with demand queries in our setting).

We design an expected polynomial-time, truthful-in-expectation, (1 − 1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass mostconcreteexamplesofsubmodularfunctionsstudiedinthiscontext,includingcoveragefunctions, matroid weighted-rank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P ̸ = NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly

We consider the problem of max-min fair allocation of indivisible goods. Our focus will be on the restricted version of the problem in which there are m items, each of which associated with a non-negative value. There are also n players and each player is only interested in some of the items. The goal is to distribute the items between the players such that the least happy person is as happy as possible, i.e. one wants to maximize the minimum of the sum of the values of the items given to any player. This problem is also known as the Santa Claus problem [3]. Feige [9] proves that the integrality gap of a certain configuration LP, described by Bansal and Sviridenko [3], is bounded from below by some (unspecified) constant. This gives an efficient way to estimate the optimum value of the problem within a constant factor. However, the proof in [9] is nonconstructive: it uses the Lovasz local lemma and does not provide a polynomial time algorithm for finding an allocation. In this paper, we take a different approach to this problem, based upon local search techniques for finding perfect matchings in certain classes of hypergraphs. As a result, we prove that the integrality gap of the configuration LP is bounded by 1 5. Our proof is nonconstructive in the following sense: it does provide a local search algorithm which finds the corresponding allocation, but this algorithm is not known to converge to a local optimum in a polynomial number of steps. 1

Very recently, Hartline and Lucier [14] studied singleparameter mechanism design problems in the Bayesian setting. They proposed a black-box reduction that converted Bayesian approximation algorithms into Bayesian-Incentive-Compatible (BIC) mechanisms while preserving social welfare. It remains a major open question if one can find similar reduction in the more important multi-parameter setting. In this paper, we give positive answer to this question when the prior distribution has finite and small support. We propose a black-box reduction for designing BIC multi-parameter mechanisms. The reduction converts any algorithm into an ɛ-BIC mechanism with only marginal loss in social welfare. As a result, for combinatorial auctions with sub-additive agents we get an ɛ-BIC mechanism that achieves constant approximation. 1

I dedicate this thesis to my parents, Anjani and Sitanshu Mehta. iii ACKNOWLEDGEMENTS I would like to thank Dick Lipton and Vijay Vazirani, for their continuous support and guidance throughout these five years. Thanks also to the rest of the theory and ACO group at Georgia Tech, faculty and students, who have created such a motivating and enriching research environment. Thanks, of course, to my collaborators and co-authors- the results presented here are products of our joint work. Finally, thanks to Swati, the fixed point of the map of my life, without whose support and patience I would never have finished this thesis.

Abstract. We give an improved lower bound for the approximation ratio of truthful mechanisms for the unrelated machines scheduling problem. The mechanism design version of the problem which was proposed and studied in a seminal paper of Nisan and Ronen is at the core of the emerging area of Algorithmic Game Theory. The new lower bound 1 + φ ≈ 2.618 is a step towards the final resolution of this important problem. 1

Abstract. We provide a 3/2-approximation algorithm for an offline budgeted allocations problem, an improvement over the e/(e − 1) approximation of Andelman and Mansour [1] and the e/(e − 1) − ɛ approximation (for ɛ ≈ 0.0001) of Feige and Vondrak [5] for the more general Maximum Submodular Welfare (SMW) problem. For a special case of our problem, we improve this ratio to √ 2. Finally, we prove that it is APX-hard. The problem we study has applications to sponsored search auctions. 1

Congestion games are non-cooperative games where the utility of a player from using a certain resource depends on the total number of players that are using the same resource. While most work so far took a distributed game-theoretic approach to this problem, this paper studies centralized solutions for congestion games. The first part of the paper analyzes the problem from a computational perspective. We analyze the computational complexity of the welfare-maximization problem, for which we provide both approximation algorithms and lower bounds. We study this optimization problem under different kinds of congestion effects (externalities) among the players: positive, negative, and unrestricted. Our main algorithmic result is a constant approximation algorithm for congestion games with unrestricted externalities. In the second part of the paper, we also take the strategic behavior of the players into account, and present centralized truthful mechanisms for congestion-game environments. Our main result in this part is an incentive-compatible mechanism for m-resource n-player congestion games that achieves an O ( √ m log n) approximation to the optimal welfare. We also describe an important and useful connection between congestion games and combinatorial auctions. This connection allows us to use insights and methods from the combinatorial-auction literature for solving congestion-game problems.