arXiv subjects

Salman Fadaei

Publications and source records attributed to Salman Fadaei.

7 recordsLinked to original sources

arXiv2017

Generalized Assignment Problem: Truthful Mechanism Design without Money

In this paper, we study a problem of truthful mechanism design for a strategic variant of the generalized assignment problem (GAP) in a both payment-free and prior-free 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. For two variants of the problem, namely \emph{multiple knapsack problem} in which each item has the same size and value over bins, and \emph{density-invariant GAP} in which each item has the same value density over the bins, we propose truthful $4$-approximation algorithms. For the general problem, we propose an $O(\ln{(U/L)})$-approximation mechanism where $U$ and $L$ are the upper and lower bounds for value densities of the compatible item-bin pairs.

cs.GT↗

arXiv2016

A Note on Relaxation and Rounding in Algorithmic Mechanism Design

In this note, we revisit the \emph{relaxation and rounding} technique employed several times in algorithmic mechanism design. We try to introduce a general framework which covers the most significant algorithms in mechanism design that use the relaxation and rounding technique. We believe that this framework is not only a generalization of the existing algorithms but also can be leveraged for further results in algorithmic mechanism design. Before presenting the framework, we briefly define algorithmic mechanism design, its connections to game theory and computer science, and the challenges in the field.

cs.GT↗

arXiv2016

A Truthful Mechanism for the Generalized Assignment Problem

We propose a truthful-in-expectation, $(1-1/e)$-approximation mechanism for a strategic variant of the generalized assignment problem (GAP). 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, values for assigning items to bins are the private information of bidders and the mechanism should provide bidders with incentives to truthfully report their values. The approximation ratio of the mechanism is a significant improvement over the approximation ratio of the existing truthful mechanism for GAP. The proposed mechanism comprises a novel convex optimization program as the allocation rule as well as an appropriate payment rule. To implement the convex program in polynomial time, we propose a fractional local search algorithm which approximates the optimal solution within an arbitrarily small error leading to an approximately truthful-in-expectation mechanism. The presented algorithm improves upon the existing optimization algorithms for GAP in terms of simplicity and runtime while the approximation ratio closely matches the best approximation ratio given for GAP when all inputs are publicly known.

cs.GT↗

arXiv2016

Mechanism Design via Dantzig-Wolfe Decomposition

In random allocation rules, typically first an optimal fractional point is calculated via solving a linear program. The calculated point represents a fractional assignment of objects or more generally packages of objects to agents. In order to implement an expected assignment, the mechanism designer must decompose the fractional point into integer solutions, each satisfying underlying constraints. The resulting convex combination can then be viewed as a probability distribution over feasible assignments out of which a random assignment can be sampled. This approach has been successfully employed in combinatorial optimization as well as mechanism design with or without money. In this paper, we show that both finding the optimal fractional point as well as its decomposition into integer solutions can be done at once. We propose an appropriate linear program which provides the desired solution. We show that the linear program can be solved via Dantzig-Wolfe decomposition. Dantzig-Wolfe decomposition is a direct implementation of the revised simplex method which is well known to be highly efficient in practice. We also show how to use the Benders decomposition as an alternative method to solve the problem. The proposed method can also find a decomposition into integer solutions when the fractional point is readily present perhaps as an outcome of other algorithms rather than linear programming. The resulting convex decomposition in this case is tight in terms of the number of integer points according to the Carath{é}odory's theorem.

cs.GT↗

arXiv2016

Truthfulness with Value-Maximizing Bidders

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(sqrt(m)). We provide better ratios for environments with restricted valuations.

cs.GT↗

arXiv2016

Maximizing Non-monotone Submodular Set Functions Subject to Different Constraints: Combined Algorithms

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 $(0.25-2ε)$-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 down-monotone polytope \cite{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.

cs.DS↗

arXiv2014

Fast Convex Decomposition for Truthful Social Welfare Approximation

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~\cite{Lavi11} 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 $α(1 + ε)$ approximation and only requires a quadratic number of calls to an integrality gap verifier.

cs.GT↗