Gagan Goel

Approximability of Combinatorial Problems with Multi-agent Submodular Cost Functions

Abstract— Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems

Budget constrained auctions with heterogeneous items

In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders have arbitrary demand and budget constraints (and additive valuations). Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with a discrete type space for each bidder, where her valuations for different items can be correlated. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles.

Approximability of combinatorial problems with multi-agent submodular cost functions

Abstract— Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems

Mechanism design for fair division: allocating divisible items without payments

We revisit the classic problem of fair division from a mechanism design perspective and provide an elegant truthful mechanism that yields surprisingly good approximation guarantees for the widely used solution of Proportional Fairness. This solution, which is closely related to Nash bargaining and the competitive equilibrium, is known to be not implementable in a truthful fashion, which has been its main drawback. To alleviate this issue, we propose a new mechanism, which we call the Partial Allocation mechanism, that discards a carefully chosen fraction of the allocated resources in order to incentivize the agents to be truthful in reporting their valuations. This mechanism introduces a way to implement interesting truthful outcomes in settings where monetary payments are not an option. For a multi-dimensional domain with an arbitrary number of agents and items, and for the very large class of homogeneous valuation functions, we prove that our mechanism provides every agent with at least a 1/e ≈ 0.368 fraction of her Proportionally Fair valuation. To the best of our knowledge, this is the first result that gives a constant factor approximation to every agent for the Proportionally Fair solution. To complement this result, we show that no truthful mechanism can guarantee more than 0.5 approximation, even for the restricted class of additive linear valuations. In addition to this, we uncover a connection between the Partial Allocation mechanism and VCG-based mechanism design. We also ask whether better approximation ratios are possible in more restricted settings. In particular, motivated by the massive privatization auction in the Czech republic in the early 90s we provide another mechanism for additive linear valuations that works really well when all the items are highly demanded.

On the Approximability of Budgeted Allocations and Improved Lower Bounds for Submodular Welfare Maximization and GAP

In this paper we consider the following maximum budgeted allocation (MBA) problem: Given a set of m indivisible items and n agents; each agent i willing to pay bij on item j and with a maximum budget of Bi, the goal is to allocate items to agents to maximize revenue. The problem naturally arises as auctioneer revenue maximization in budget-constrained auctions and as winner determination problem in combinatorial auctions when utilities of agents are budgeted-additive.We give a 3/4-approximation algorithm for MBA improving upon the previous best of sime0.632[2, 10]. Our techniques are based on a natural LP relaxation of MBA and our factor is optimal in the sense that it matches the integrality gap of the LP.We prove it is NP-hard to approximate MBA to any factor better than 15/16, previously only NP-hardness was known [21, 17]. Our result also implies NP- hardness of approximating maximum submodular welfare with demand oracle to a factor better than 15/16, improving upon the best known hardness of 275/276[10].Our hardness techniques can be modified to prove that it is NP-hard to approximate the Generalized Assignment Problem (GAP) to any factor better than 10/11. This improves upon the 422/423 hardness of [7, 9].We use iterative rounding on a natural LP relaxation of MBA to obtain the 3/4-approximation. We also give a (3/4 - epsiv) -factor algorithm based on the primal-dual schema which runs in O(nm) time, for any constant epsiv > 0.

Mechanism Design for Crowdsourcing: An Optimal 1-1/e Competitive Budget-Feasible Mechanism for Large Markets

In this paper we consider a mechanism design problem in the context of large-scale crowdsourcing markets such as Amazons Mechanical Turk mturk, ClickWorker clickworker, CrowdFlower crowdflower. In these markets, there is a requester who wants to hire workers to accomplish some tasks. Each worker is assumed to give some utility to the requester on getting hired. Moreover each worker has a minimum cost that he wants to get paid for getting hired. This minimum cost is assumed to be private information of the workers. The question then is -- if the requester has a limited budget, how to design a direct revelation mechanism that picks the right set of workers to hire in order to maximize the requesters utility? We note that although the previous work (Singer (2010) chen et al. (2011)) has studied this problem, a crucial difference in which we deviate from earlier work is the notion of large-scale markets that we introduce in our model. Without the large market assumption, it is known that no mechanism can achieve a competitive ratio better than 0.414 and 0.5 for deterministic and randomized mechanisms respectively (while the best known deterministic and randomized mechanisms achieve an approximation ratio of 0.292 and 0.33 respectively). In this paper, we design a budget-feasible mechanism for large markets that achieves a competitive ratio of 1 - 1/e ≃ 0.63. Our mechanism can be seen as a generalization of an alternate way to look at the proportional share mechanism, which is used in all the previous works so far on this problem. Interestingly, we can also show that our mechanism is optimal by showing that no truthful mechanism can achieve a factor better than 1 - 1/e, thus, fully resolving this setting. Finally we consider the more general case of submodular utility functions and give new and improved mechanisms for the case when the market is large.

Towards Topology Aware Networks

We focus on efficient protocols that enhance a network with topology awareness. We discuss centralized algorithms with provable performance, and introduce decentralized asynchronous heuristics that use only local information and local computations. These algorithms are based on distributed solutions of convex programs expressing optimization of various spectral properties of the matrix associated with the graph of the network topology. For example, these algorithms assign special weights to links crossing or directed towards small cuts by minimizing the second eigenvalue. Our main technical ingredient is to perform the decentralized asynchronous computations in a manner that preserves critical invariants of the exact second eigenvalue of the adjacency matrix associated with the network topology.

Allocating tasks to workers with matching constraints: truthful mechanisms for crowdsourcing markets

Designing optimal pricing policies and mechanisms for allocating tasks to workers is central to the online crowdsourcing markets. In this paper, we consider the following realistic setting of online crowdsourcing markets -- there is a requester with a limited budget and a heterogeneous set of tasks each requiring certain skills; there is a pool of workers and each worker has certain expertise and interests which define the set of tasks she can and is willing to do. Under the matching constraints given by this bipartite graph between workers and tasks, we design our incentive-compatible mechanism truthuniform which allocates the tasks to the workers, while ensuring budget feasibility and achieves near-optimal utility for the requester. Apart from strong theoretical guarantees, we carry out experiments on a realistic case study of Wikipedia translation project on Mechanical Turk. We note that this is the first paper to address this setting from a mechanism design perspective.

A Perfect Price Discrimination Market Model with Production, and a Rational Convex Program for It

Recent results showing PPAD-completeness of the problem of computing an equilibrium for Fishers market model under additively separable, piecewise-linear, concave (PLC) utilities have dealt a serious blow to the program of obtaining efficient algorithms for computing equilibria in “traditional” market models, and has prompted a search for alternative models that are realistic as well as amenable to efficient computation. In this paper, we show that introducing perfect price discrimination into the Fisher model with PLC utilities renders its equilibrium polynomial time computable. Moreover, its set of equilibria are captured by a convex program that generalizes the classical Eisenberg-Gale program, and always admits a rational solution. We also give a combinatorial, polynomial time algorithm for computing an equilibrium. Next, we introduce production into our model, and again give a rational convex program that captures its equilibria. We use this program to obtain surprisingly simple proofs of both welfare theorems for this model. Finally, we also give an application of our price discrimination market model to online display advertising marketplaces.

Efficiency of (revenue-)optimal mechanisms

We compare the expected efficiency of revenue maximizing (or optimal) mechanisms with that of efficiency maximizing ones. We show that the efficiency of the revenue maximizing mechanism for selling a single item with (k + logeovere-1 k + 1) bidders is at least as much as the efficiency of the efficiency-maximizing mechanism with k bidders, when bidder valuations are drawn i.i.d. from a Monotone Hazard Rate distribution. Surprisingly, we also show that this bound is tight within a small additive constant of 4.7. In other words, Θ(log k) extra bidders suffice for the revenue-maximizing mechanism to match the efficiency of the efficiency-maximizing mechanism, while o(log k) do not. This is in contrast to the result of Bulow and Klemperer comparing the revenue of the two mechanisms, where only one extra bidder suffices. More precisely, they show that the revenue of the efficiency-maximizing mechanism with k + 1 bidders is no less than the revenue of the revenue-maximizing mechanism with k bidders. We extend our result for the case of selling t identical items and show that Θ(log k) + t Θ(log log k) extra bidders suffice for the revenue-maximizing mechanism to match the efficiency of the efficiency-maximizing mechanism. In order to prove our results, we do a classification of Monotone Hazard Rate (MHR) distributions and identify a family of MHR distributions, such that for each class in our classification, there is a member of this family that is pointwise lower than every distribution in that class. This lets us prove interesting structural theorems about distributions with Monotone Hazard Rate.