Afshin Nikzad

Approximation Algorithms for Computing Maximin Share Allocations

We study the problem of computing maximin share allocations, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of an agent is the best she can guarantee to herself, if she is allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then is to find a partition, where each agent is guaranteed her maximin share. Such allocations do not always exist, hence we resort to approximation algorithms. Our main result is a 2/3-approximation that runs in polynomial time for any number of agents and goods. This improves upon the algorithm of Procaccia and Wang (2014), which is also a 2/3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in Procaccia and Wang (2014), exploiting the construction of carefully selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty in establishing lower bounds, we undertake a probabilistic analysis. We prove that in randomly generated instances, maximin share allocations exist with high probability. This can be seen as a justification of previously reported experimental evidence. Finally, we provide further positive results for two special cases arising from previous works. The first is the intriguing case of three agents, where we provide an improved 7/8-approximation. The second case is when all item values belong to {0, 1, 2}, where we obtain an exact algorithm.

Optimal Iterative Pricing over Social Networks (Extended Abstract)

We study the optimal pricing for revenue maximization over social networks in the presence of positive network externalities. In our model, the value of a digital good for a buyer is a function of the set of buyers who have already bought the item. In this setting, a decision to buy an item depends on its price and also on the set of other buyers that have already owned that item. The revenue maximization problem in the context of social networks has been studied by Hartline, Mirrokni, and Sundararajan [4], following the previous line of research on optimal viral marketing over social networks [5,6,7].

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.

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.

Optimal online pricing with network externalities

We study the optimal pricing strategy for profit maximization in presence of network externalities where a decision to buy a product depends on the price offered to the buyer and also on the set of her friends who have already bought that product. We model the network influences by a weighted graph where the utility of each buyer is the sum of her initial value on the product, and the linearly additive influence from her friends. We assume that the buyers arrive online and the seller should offer a price to each buyer when she enters the market. We also take into account the manufacturing cost. In this paper, we first assume that the monopolist defines a unique price for the product and commits to it for all buyers. In this case, we present an FPTAS algorithm that approximates the optimal price with a high probability. We also prove that finding the optimum price is NP-hard. Second, we consider a market with positive network externalities and assume that the monopolist could offer a private price to each customer. We prove that this problem is also hard to approximate for linear influences. On the positive side, we present a polynomial time algorithm for the problem when influences are symmetric. At last, we show that the seller has more ability to extract influences with price discrimination.

Deliver or Hold: Approximation Algorithms for the Periodic Inventory Routing Problem

The inventory routing problem involves trading off inventory holding costs at client locations with vehicle routing costs to deliver frequently from a single central depot to meet deterministic client demands over a finite planing horizon. In this paper, we consider periodic solutions that visit clients in one of several specified frequencies, and focus on the case when the frequencies of visiting nodes are nested. We give the first constant-factor approximation algorithms for designing optimum nested periodic schedules for the problem with no limit on vehicle capacities by simple reductions to prize-collecting network design problems. For instance, we present a 2.55-approximation algorithm for the minimum-cost nested periodic schedule where the vehicle routes are modeled as minimum Steiner trees. We also show a general reduction from the capacitated problem where all vehicles have the same capacity to the uncapacitated version with a slight loss in performance. This reduction gives a 4.55-approximation for the capacitated problem. In addition, we prove several structural results relating the values of optimal policies of various types.

Sending Secrets Swiftly: Approximation Algorithms for Generalized Multicast Problems

We consider natural generalizations of the minimum broadcast time problem under the telephone model, where a rumor from a root node must be sent via phone calls to the whole graph in the minimum number of rounds; the telephone model implies that the set of edges involved in communicating in a round form a matching. The extensions we consider involve generalizing the number of calls that a vertex may participate in (the capacitated version), allowing conference calls (the hyperedge version) as well as a new multicommodity version we introduce where the rumors are no longer from a single node but from different sources and intended for specific destinations (the multicommodity version). Based on the ideas from [6,7], we present a very simple greedy algorithm for the basic multicast problem with logarithmic performance guarantee and adapt it to the extensions to design typically polylogarithmic approximation algorithms. For the multi-commodity version, we give the first approximation algorithm with performance ratio \(2^{ O\left(\log\log k \sqrt{\log k}\right)}\) for k source-sink pairs. We provide nearly matching lower bounds for the hypercasting problem. For the multicommodity multicasting problem, we present improved guarantees for other variants involving asymmetric capacities, small number of terminals and with larger additive guarantees.

Budget Feasible Procurement Auctions

We consider a simple and well-studied model for procurement problems and solve it to optimality. A buyer with a fixed budget wants to procure, from a set of available workers, a budget feasible subset that maximizes her utility: Any worker has a private reservation price and provides a publicly known utility to the buyer in case of being procured. The buyer’s utility function is additive over items. The goal is designing a direct revelation mechanism that solicits workers’ reservation prices and decides which workers to recruit and how much to pay them. Moreover, the mechanism has to maximize the buyer’s utility without violating her budget constraint. We study this problem in the prior-free setting; our main contribution is finding the optimal mechanism in this setting, under the “small bidders” assumption. This assumption, also known as the “small bid to budget ratio assumption,” states that the bid of each seller is small compared with the buyer’s budget. We also study a more general class of utility f...