Abstract. Approximating the optimal social welfare while preserving truthfulness is a well studied problem in algorithmic mechanism design. Assuming that the social welfare of a given mechanism design problem can be optimized by an integer program whose integrality gap is at most α, Lavi and Swamy [1] propose a general approach to designing a randomized α-approximation mechanism which is truthful in expectation. Their method is based on decomposing an optimal solution for the relaxed linear program into a convex combination of integer solutions. Unfortunately, Lavi and Swamy's decomposition technique relies heavily on the ellipsoid method, which is notorious for its poor practical performance. To overcome this problem, we present an alternative decomposition technique which yields an α(1 + ǫ) approximation and only requires a quadratic number of calls to an integrality gap verifier.

In markets such as digital advertising auctions, bidders want to maximize value rather than payoff. This is different to the utility functions typically assumed in auction theory and leads to different strategies and outcomes. We refer to bidders who maximize value as value bidders. While simple single-object auction formats are truthful, standard multi-object auction formats allow for manipulation. It is straightforward to show that there cannot be a truthful and revenue-maximizing deterministic auction mechanism with value bidders and general valuations. Approximation has been used as a means to achieve truthfulness, and we study which approximation ratios we can get from truthful approximation mechanisms. We show that the approximation ratio that can be achieved with a deterministic and truthful approximation mechanism with n bidders and m items cannot be higher than 1/n for general valuations. For randomized approximation mechanisms there is a framework with a ratio of O( √ m ǫ 3 ) with probability at least 1 − ǫ, for 0 < ǫ < 1. We provide better ratios for environments with restricted valuations.

We study the problem of maximizing constrained non-monotone submodular functions and provide approximation algorithms that improve existing algorithms in terms of either the approximation factor or simplicity. Our algorithms combine existing local search and greedy based algorithms. Different constraints that we study are exact cardinality and multiple knapsack constraints. For the multiple-knapsack constraints we achieve a (0.25 − 2ǫ)-factor algorithm.We also show, as our main contribution, how to use the continuous greedy process for non-monotone functions and, as a result, obtain a 0.13-factor approximation algorithm for maximization over any solvable down-monotone polytope. The continuous greedy process has been previously used for maximizing smooth monotone submodular function over a downmonotone polytope [CCPV08]. This implies a 0.13-approximation for several discrete problems, such as maximizing a non-negative submodular function subject to a matroid constraint and/or multiple knapsack constraints.

In this paper, we study a mechanism design problem for a strategic variant of the generalized assignment problem (GAP) in a both payment-free and priorfree environment. In GAP, a set of items has to be optimally assigned to a set of bins without exceeding the capacity of any singular bin. In the strategic variant of the problem we study, bins are held by strategic agents, and each agent may hide its compatibility with some items in order to obtain items of higher values. The compatibility between an agent and an item encodes the willingness of the agent to receive the item. Our goal is to maximize total value (sum of agents' values, or social welfare) while certifying no agent can benefit from hiding its compatibility with items. The model has applications in auctions with budgeted bidders.

We propose a truthful-in-expectation, (1 − 1 e )-approximation mechanism for the generalized assignment auction. In such an auction, each bidder has a knapsack valuation function and bidders' values for items are private. We present a novel convex optimization program for the auction which results in a maximal-in-distributional-range (MIDR) allocation rule. The presented program contains at least a (1 − 1 e ) ratio of the optimal social welfare. We show how to implement the convex program in polynomial time using a fractional local search algorithm which approximates the optimal solution within an arbitrarily small error. This leads to an approximately MIDR allocation rule which in turn can be transformed to an approximately truthful-in-expectation mechanism. Our contribution has algorithmic importance, as well; it simplifies the existing optimization algorithms for the GAP while the approximation ratio is comparable to the best given approximation.