Follow Your Star: New Frameworks for Online Stochastic Matching with Known and Unknown Patience

We study several generalizations of the Online Bipartite Matching problem. We consider settings with stochastic rewards, patience constraints, and weights (both vertex- and edge-weighted variants). We introduce a stochastic variant of the patience-constrained problem, where the patience is chosen randomly according to some known distribution and is not known until the point at which patience has been exhausted. We also consider stochastic arrival settings (i.e., online vertex arrival is determined by a known random process), which are natural settings that are able to beat the hard worst-case bounds of more pessimistic adversarial arrivals. Our approach to online matching utilizes black-box algorithms for matching on star graphs under various models of patience. In support of this, we design algorithms which solve the star graph problem optimally for patience with a constant hazard rate and yield a 1/2-approximation for any patience distribution. This 1/2-approximation also improves existing guarantees for cascade-click models in the product ranking literature, in which a user must be shown a sequence of items with various click-through-rates and the user's patience could run out at any time. We then build a framework which uses these star graph algorithms as black boxes to solve the online matching problems under different arrival settings. We show improved (or first-known) competitive ratios for these problems. Finally, we present negative results that include formalizing the concept of a stochasticity gap for LP upper bounds on these problems, bounding the worst-case performance of some popular greedy approaches, and showing the impossibility of having an adversarial patience in the product ranking setting.Comment: 43 page

Secretary Matching Meets Probing with Commitment

We consider the online bipartite matching problem within the context of stochastic probing with commitment. This is the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching. We consider the competitiveness of online algorithms in the adversarial order model (AOM) and the secretary/random order model (ROM). More specifically, we consider an unknown bipartite stochastic graph G = (U,V,E) where U is the known set of offline vertices, V is the set of online vertices, G has edge probabilities (p_{e})_{e ? E}, and G has edge weights (w_{e})_{e ? E} or vertex weights (w_u)_{u ? U}. Additionally, G has a downward-closed set of probing constraints (?_{v})_{v ? V}, where ?_v indicates which sequences of edges adjacent to an online vertex v can be probed. This model generalizes the various settings of the classical bipartite matching problem (i.e. with and without probing). Our contributions include the introduction and analysis of probing within the random order model, and our generalization of probing constraints which includes budget (i.e. knapsack) constraints. Our algorithms run in polynomial time assuming access to a membership oracle for each ?_v. In the vertex weighted setting, for adversarial order arrivals, we generalize the known 1/2 competitive ratio to our setting of ?_v constraints. For random order arrivals, we show that the same algorithm attains an asymptotic competitive ratio of 1-1/e, provided the edge probabilities vanish to 0 sufficiently fast. We also obtain a strict competitive ratio for non-vanishing edge probabilities when the probing constraints are sufficiently simple. For example, if each ?_v corresponds to a patience constraint ?_v (i.e., ?_v is the maximum number of probes of edges adjacent to v), and any one of following three conditions is satisfied (each studied in previous papers), then there is a conceptually simple greedy algorithm whose competitive ratio is 1-1/e. - When the offline vertices are unweighted. - When the online vertex probabilities are "vertex uniform"; i.e., p_{u,v} = p_v for all (u,v) ? E. - When the patience constraint ?_v satisfies ?_v ? {[1,|U|} for every online vertex; i.e., every online vertex either has unit or full patience. Finally, in the edge weighted case, we match the known optimal 1/e asymptotic competitive ratio for the classic (i.e. without probing) secretary matching problem

Improved Bounds in Stochastic Matching and Optimization

We consider two fundamental problems in stochastic optimization: approximation algorithms for stochastic matching, and sampling bounds in the black-box model. For the former, we improve the current-best bound of 3.709 due to Adamczyk et al. (2015), to 3.224; we also present improvements on Bansal et al. (2012) for hypergraph matching and for relaxed versions of the problem. In the context of stochastic optimization, we improve upon the sampling bounds of Charikar et al. (2005)

Markets, Elections, and Microbes: Data-driven Algorithms from Theory to Practice

Many modern problems in algorithms and optimization are driven by data which often carries with it an element of uncertainty. In this work, we conduct an investigation into algorithmic foundations and applications across three main areas. The first area is online matching algorithms for e-commerce applications such as online sales and advertising. The importance of e-commerce in modern business cannot be overstated and even minor algorithmic improvements can have huge impacts. In online matching problems, we generally have a known offline set of goods or advertisements while users arrive online and allocations must be made immediately and irrevocably when a user arrives. However, in the real world, there is also uncertainty about a user's true interests and this can be modeled by considering matching problems in a graph with stochastic edges that only have a probability of existing. These edges can represent the probability of a user purchasing a product or clicking on an ad. Thus, we optimize over data which only provides an estimate of what types of users will arrive and what they will prefer. We survey a broad landscape of problems in this area, gain a deeper understanding of the algorithmic challenges, and present algorithms with improved worst case performance The second area is constrained clustering where we explore classical clustering problems with additional constraints on which data points should be clustered together. Utilizing these constraints is important for many clustering problems because they can be used to ensure fairness, exploit expert advice, or capture natural properties of the data. In simplest case, this can mean some pairs of points have ``must-link'' constraints requiring that that they must be clustered together. Moving into stochastic settings, we can describe more general pairwise constraints such as bounding the probability that two points are separated into different clusters. This lets us introduce a new notion of fairness for clustering and address stochastic problems such as semi-supervised learning with advice from imperfect experts. Here, we introduce new models of constrained clustering including new notions of fairness for clustering applications. Since these problems are NP-hard, we give approximation algorithms and in some cases conduct experiments to explore how the algorithms perform in practice. Finally, we look closely at the particular clustering problem of drawing election districts and show how constraining the clusters based on past voting data can interact with voter incentives. The third area is string algorithms for bioinformatics and metagenomics specifically where the data deluge from next generation sequencing drives the necessity for new algorithms that are both fast and accurate. For metagenomic analysis, we present a tool for clustering a microbial marker gene, the 16S ribosomal RNA gene. On the more theoretical side, we present a succinct application of the Method of the Four Russians to edit distance computation as well as new algorithms and bounds for the maximum duo-preservation string mapping (MPSM) problem