Matching is a classic problem with a rich history and a significant impact, both on the theory of algorithms and in practice. Recently there has been a surge of interest in the online version of matching and its generalizations, due to the important new application domain of Internet advertising. The theory of online matching and allocation has played a critical role in designing algorithms for ad allocation. This monograph surveys the key problems, models and algorithms from online matchings, as well as their implication in the practice of ad allocation. The goal is to provide a classification of the problems in this area, an introduction into the techniques used, a glimpse into the practical impact, and to provide direction in terms of open questions. Matching continues to find core applications in diverse domains, and the advent of massive online and streaming data emphasizes the future applicability of the algorithms and techniques surveyed here.

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Online Matching and Ad Allocation

We give a simple proof that the ranking algorithm of Karp, Vazirani and Vazirani [KVV90] is 1-1/e competitive for the online bipartite matching problem. The proof is via a randomized primal-dual argument. Primal-dual algorithms have been successfully used for many online algorithm problems, but the dual constraints are always satisfied deterministically. This is the first instance of a non-trivial randomized primal-dual algorithm in which the dual constraints only hold in expectation. The approach also generalizes easily to the vertex-weighted version considered by Agarwal et al. [AGKM11]. Further we show that the proof is very similar to the deterministic primal-dual argument for the online budgeted allocation problem with small bids (also called the AdWords problem) of Mehta et al. [MSVV05].

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Randomized primal-dual analysis of RANKING for online bipartite matching

Crowdsourcing is a computing paradigm where humans are actively involved in a computing task, especially for tasks that are intrinsically easier for humans than for computers. Spatial crowdsourcing is an increasing popular category of crowdsourcing in the era of mobile Internet and sharing economy, where tasks are spatiotemporal and must be completed at a specific location and time. In fact, spatial crowdsourcing has stimulated a series of recent industrial successes including sharing economy for urban services (Uber and Gigwalk) and spatiotemporal data collection (OpenStreetMap and Waze). This survey dives deep into the challenges and techniques brought by the unique characteristics of spatial crowdsourcing. Particularly, we identify four core algorithmic issues in spatial crowdsourcing: (1) task assignment, (2) quality control, (3) incentive mechanism design, and (4) privacy protection. We conduct a comprehensive and systematic review of existing research on the aforementioned four issues. We also analyze representative spatial crowdsourcing applications and explain how they are enabled by these four technical issues. Finally, we discuss open questions that need to be addressed for future spatial crowdsourcing research and applications.

https://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=5938&context=sis_research

Spatial crowdsourcing: a survey

We study online variants of weighted bipartite matching on graphs and hypergraphs. In our model for online matching, the vertices on the right-hand side of a bipartite graph are given in advance and the vertices on the left-hand side arrive online in random order. Whenever a vertex arrives, its adjacent edges with the corresponding weights are revealed and the online algorithm has to decide which of these edges should be included in the matching. The studied matching problems have applications, e.g., in online ad auctions and combinatorial auctions where the right-hand side vertices correspond to items and the left-hand side vertices to bidders.

An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions

We introduce the online stochastic Convex Programming (CP) problem, a very general version of stochastic online problems which allows arbitrary concave objectives and convex feasibility constraints. Many well-studied problems like online stochastic packing and covering, online stochastic matching with concave returns, etc. form a special case of online stochastic CP. We present fast algorithms for these problems, which achieve near-optimal regret guarantees for both the i.i.d. and the random permutation models of stochastic inputs. When applied to the special case online packing, our ideas yield a simpler and faster primal-dual algorithm for this well studied problem, which achieves the optimal competitive ratio. Our techniques make explicit the connection of primal-dual paradigm and online learning to online stochastic CP.

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Fast algorithms for online stochastic convex programming

We consider a significant generalization of the Adwords problem by allowing arbitrary concave returns, and we characterize the optimal competitive ratio achievable. The problem considers a sequence of items arriving online that have to be allocated to agents, with different agents bidding different amounts. The objective function is the sum, over each agent i, of a monotonically non-decreasing concave function Mi : R+ -> R+ of the total amount allocated to i. All variants of online matching problems (including the Adwords problem) studied in the literature consider the special case of budgeted linear functions, that is, functions of the form Mi(ui) = min {ui,Bi} for some constant Bi. The distinguishing feature of this paper is in allowing arbitrary concave returns. The main result of this paper is that for each concave function M, there exists a constant F(M) ≤ 1 such that: there exists an algorithm with competitive ratio of miniF(Mi), independent of the sequence of items. No algorithm has a competitive ratio larger than F(M) over all instances with Mi= M for all i. Our algorithm is based on the primal-dual paradigm and makes use of convex programming duality. The upper bounds are obtained by formulating the task of finding the right counterexample as an optimization problem. This path takes us through the calculus of variations which deals with optimizing over continuous functions. The algorithm and the upper bound are related to each other via a set of differential equations, which points to a certain kind of duality between them.

Online matching with concave returns

We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and improve the known approximation ratio from Θ(m) to Θ(log m), where m is the number of machines. A local policy for each machine orders the set of jobs assigned to it only based on parameters of those jobs. A strongly local policy only uses the processing time of jobs on the same machine. We prove that the approximation ratio of any set of strongly local ordering policies in equilibria is at least Ω(m). In particular, it implies that the approximation ratio of a greedy shortest-first algorithm for machine scheduling is at least Ω(m). This closes the gap between the known lower and upper bounds for this problem and answers an open question raised by Ibarra and Kim (1977) [Ibarra OH, Kim CE (1977) Heuristic algori...

Optimal Coordination Mechanisms for Unrelated Machine Scheduling

Despite the pervasiveness of markets in our lives, little is known about the role of user interfaces (UIs) in promoting good decisions in market domains. How does the way we display market information to end users, and the set of choices we offer, influence users' decisions? In this paper, we introduce a new research agenda on "market user interface design." Our goal is to find the optimal market UI, taking into account that users incur cognitive costs and are boundedly rational. Via lab experiments we systematically explore the market UI design space, and we study the automatic optimization of market UIs given a behavioral (quantal response) model of user behavior. Surprisingly, we find that the behaviorally-optimized UI performs worse than the standard UI, suggesting that the quantal response model did not predict user behavior well. Subsequently, we identify important behavioral factors that are missing from the user model, including loss aversion and position effects, which motivates follow-up studies. Furthermore, we find significant differences between individual users in terms of rationality. This suggests future research on personalized UI designs, with interfaces that are tailored towards each individual user's needs, capabilities, and preferences.

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Market user interface design

Disclosed in some examples is a method of providing assistance to a consumer comprising authenticating with a shopping service over a network using a network adapter; downloading a shopping list from the shopping service over the network using a network adapter; displaying the shopping list to the user on a display screen, the shopping list comprising a name of an item, wherein each item in the shopping list may be displayed as a tile, wherein each tile is arranged in a series of rows and columns; receiving an input identifying one or more changes to the shopping list; processing the input using a computer processor; and sending the one or more changes to the shopping service over the network using the network adapter.

Kamal Jain

Papers

Randomized primal-dual analysis of RANKING for online bipartite matching

Online matching with concave returns

Optimal Coordination Mechanisms for Unrelated Machine Scheduling

Market user interface design

System and method for interactive and social shopping