Budget feasible mechanisms on matroids

Motivated by many practical applications, in this paper we study budget feasible mechanisms where the goal is to procure independent sets from matroids. More specifically, we are given a matroid =(,) where each ground (indivisible) element is a selfish agent. The cost of each element (i.e., for selling the item or performing a service) is only known to the element itself. There is a buyer with a budget having additive valuations over the set of elements E. The goal is to design an incentive compatible (truthful) budget feasible mechanism which procures an independent set of the matroid under the given budget that yields the largest value possible to the buyer. Our result is a deterministic, polynomial-time, individually rational, truthful and budget feasible mechanism with 4-approximation to the optimal independent set. Then, we extend our mechanism to the setting of matroid intersections in which the goal is to procure common independent sets from multiple matroids. We show that, given a polynomial time deterministic blackbox that returns -approximation solutions to the matroid intersection problem, there exists a deterministic, polynomial time, individually rational, truthful and budget feasible mechanism with (3+1) -approximation to the optimal common independent set

Budget Feasible Mechanisms on Matroids

Motivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid M= (E, I). Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair M= (E, I) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an α-approximation to the problem of finding a maximum-value independent set in M, there exists an individually rational, truthful and budget feasible mechanism which is (3 α+ 1) -approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids

Robust Algorithms for the Secretary Problem

In classical secretary problems, a sequence of n elements arrive in a uniformly random order, and we want to choose a single item, or a set of size K. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin’s popular 1/e-secretary algorithm is sensitive to even a single adversarial arrival: if the adversary gives one large bid at the beginning of the stream, the algorithm does not select any element at all. We investigate a robust version of the secretary problem. In the Byzantine Secretary model, we have two kinds of elements: green (good) and red (rogue). The values of all elements are chosen by the adversary. The green elements arrive at times uniformly randomly drawn from [0,1]. The red elements, however, arrive at adversarially chosen times. Naturally, the algorithm does not see these colors: how well can it solve secretary problems? We show that selecting the highest value red set, or the single largest green element is not possible with even a small fraction of red items. However, on the positive side, we show that these are the only bad cases, by giving algorithms which get value comparable to the value of the optimal green set minus the largest green item. (This benchmark reminds us of regret minimization and digital auctions, where we subtract an additive term depending on the "scale" of the problem.) Specifically, we give an algorithm to pick K elements, which gets within (1-ε) factor of the above benchmark, as long as K ≥ poly(ε^{-1} log n). We extend this to the knapsack secretary problem, for large knapsack size K. For the single-item case, an analogous benchmark is the value of the second-largest green item. For value-maximization, we give a poly log^* n-competitive algorithm, using a multi-layered bucketing scheme that adaptively refines our estimates of second-max over time. For probability-maximization, we show the existence of a good randomized algorithm, using the minimax principle. We hope that this work will spur further research on robust algorithms for the secretary problem, and for other problems in sequential decision-making, where the existing algorithms are not robust and often tend to overfit to the model.ISSN:1868-896

Balancing Relevance and Diversity in Online Bipartite Matching via Submodularity

In bipartite matching problems, vertices on one side of a bipartite graph are paired with those on the other. In its online variant, one side of the graph is available offline, while the vertices on the other side arrive online. When a vertex arrives, an irrevocable and immediate decision should be made by the algorithm; either match it to an available vertex or drop it. Examples of such problems include matching workers to firms, advertisers to keywords, organs to patients, and so on. Much of the literature focuses on maximizing the total relevance---modeled via total weight---of the matching. However, in many real-world problems, it is also important to consider contributions of diversity: hiring a diverse pool of candidates, displaying a relevant but diverse set of ads, and so on. In this paper, we propose the Online Submodular Bipartite Matching (\osbm) problem, where the goal is to maximize a submodular function $f$ over the set of matched edges. This objective is general enough to capture the notion of both diversity (\emph{e.g.,} a weighted coverage function) and relevance (\emph{e.g.,} the traditional linear function)---as well as many other natural objective functions occurring in practice (\emph{e.g.,} limited total budget in advertising settings). We propose novel algorithms that have provable guarantees and are essentially optimal when restricted to various special cases. We also run experiments on real-world and synthetic datasets to validate our algorithms.Comment: To appear in AAAI 201

Single-Sample Prophet Inequalities via Greedy-Ordered Selection

We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.</p

Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives: Offline and Online

The framework of budget-feasible mechanism design studies procurement auctions where the auctioneer (buyer) aims to maximize his valuation function subject to a hard budget constraint. We study the problem of designing truthful mechanisms that have good approximation guarantees and never pay the participating agents (sellers) more than the budget. We focus on the case of general (non-monotone) submodular valuation functions and derive the first truthful, budget-feasible and $O(1)$-approximate mechanisms that run in polynomial time in the value query model, for both offline and online auctions. Prior to our work, the only $O(1)$-approximation mechanism known for non-monotone submodular objectives required an exponential number of value queries. At the heart of our approach lies a novel greedy algorithm for non-monotone submodular maximization under a knapsack constraint. Our algorithm builds two candidate solutions simultaneously (to achieve a good approximation), yet ensures that agents cannot jump from one solution to the other (to implicitly enforce truthfulness). Ours is the first mechanism for the problem where---crucially---the agents are not ordered with respect to their marginal value per cost. This allows us to appropriately adapt these ideas to the online setting as well. To further illustrate the applicability of our approach, we also consider the case where additional feasibility constraints are present. We obtain $O(p)$-approximation mechanisms for both monotone and non-monotone submodular objectives, when the feasible solutions are independent sets of a $p$-system. With the exception of additive valuation functions, no mechanisms were known for this setting prior to our work. Finally, we provide lower bounds suggesting that, when one cares about non-trivial approximation guarantees in polynomial time, our results are asymptotically best possible.Comment: Accepted to EC 201

Mechanism Design without Money via Stable Matching

Mechanism design without money has a rich history in social choice literature. Due to the strong impossibility theorem by Gibbard and Satterthwaite, exploring domains in which there exist dominant strategy mechanisms is one of the central questions in the field. We propose a general framework, called the generalized packing problem (\gpp), to study the mechanism design questions without payment. The \gpp\ possesses a rich structure and comprises a number of well-studied models as special cases, including, e.g., matroid, matching, knapsack, independent set, and the generalized assignment problem. We adopt the agenda of approximate mechanism design where the objective is to design a truthful (or strategyproof) mechanism without money that can be implemented in polynomial time and yields a good approximation to the socially optimal solution. We study several special cases of \gpp, and give constant approximation mechanisms for matroid, matching, knapsack, and the generalized assignment problem. Our result for generalized assignment problem solves an open problem proposed in \cite{DG10}. Our main technical contribution is in exploitation of the approaches from stable matching, which is a fundamental solution concept in the context of matching marketplaces, in application to mechanism design. Stable matching, while conceptually simple, provides a set of powerful tools to manage and analyze self-interested behaviors of participating agents. Our mechanism uses a stable matching algorithm as a critical component and adopts other approaches like random sampling and online mechanisms. Our work also enriches the stable matching theory with a new knapsack constrained matching model