ALGORITHMS FOR MARKETS: MATCHING AND PRICING

In their most basic form \emph{markets} consist of a collection of resources (goods or services) and a set of agents interested in obtaining them. This thesis is a stepping stone toward answering the most central question in the Econ/CS literature surrounding markets: How should the resources be allocated to the interested parties? The first contribution of this thesis is designing pricing algorithms for modern monetary markets (such as advertising markets) in which resources are sold via auctions. The second contribution is designing matching algorithms for markets in which money often plays little to no role (i.e., matching markets). Auctions have become the standard method of allocating resources in monetary markets, and when it comes to multi-unit auctions Vickrey–Clarke–Groves (VCG) with {\em reserve prices} is one of the most well-known and widely used auctions. A reserve price is a minimum price with which the auctioneer is willing to sell the item. In this thesis, we consider optimizing {\em personalized reserve prices} which are crucial for obtaining a high revenue. To that end, we take a \emph{data-driven} approach where given the buyers' bids in a set of auctions, the goal is to find a single vector of reserve prices (one for each buyer) that maximizes the total revenue across all these auctions. This problem is shown to be NP-hard, and the best-known algorithm for that achieves a $\frac{1}{2}$ fraction of the optimal revenue. We first present an LP-based algorithm with a $0.68$ approximation factor for single-item environments. We then show that this approach can be generalized to get a $0.63$-approximation for general multi-unit environments. To achieve these results we develop novel LP-rounding procedures which may be of independent interest. Matching markets have long held a central place in the mechanism design literature. Examples include kidney exchange, labor markets, and dating platforms. When it comes to designing algorithms for these markets, the presence of uncertainty is a common challenge. This uncertainty is often due to the stochastic nature of the data or restrictions that result in limited access to information. In this thesis, we study the {\em stochastic matching} problem in which the goal is to find a large matching of a graph whose edges are uncertain but can be accessed via queries. Particularly, we only know the existence probability of each edge but to verify their existence, we need to perform costly queries. Since these queries are costly, our goal is to find a large matching with only a few (a constant number of) queries. For instance, in labor markets, the existence of an edge between a freelancer and an employer represents their compatibility to work with one another, and a query translates to an interview between them which is often a time-consuming process. While this problem has been studied extensively, before our work, the best-known approximation ratio for unweighted graphs was almost $\frac{2}{3}$, and slightly better than $\frac{1}{2}$ for weighted graphs. In this thesis, we present algorithms that find almost optimal matchings despite the uncertainty in the graph (weighted and unweighted) by conducting only a constant number of queries per vertex

Brief Announcement: Streaming and Massively Parallel Algorithms for Edge Coloring

A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. In this paper, we revisit this problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: Massively Parallel Computation. We give a randomized MPC algorithm that w.h.p., returns a (1+o(1))Delta edge coloring in O(1) rounds using O~(n) space per machine and O(m) total space. The space per machine can also be further improved to n^{1-Omega(1)} if Delta = n^{Omega(1)}. This is, to our knowledge, the first constant round algorithm for a natural graph problem in the strongly sublinear regime of MPC. Our algorithm improves a previous result of Harvey et al. [SPAA 2018] which required n^{1+Omega(1)} space to achieve the same result. Graph Streaming. Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors. We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors w.h.p., if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors

Streaming and Massively Parallel Algorithms for Edge Coloring

A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: - Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018]. - Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors. We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors

Antimicrobial Effect of Dill Seed Oil Essence on Growth of Staphylococcus Aureus in Hamburgers

In the present study, antimicrobial effect of dill seed’s oil in 6 levels (0.003-0.006-0.0125-0.025-0.05-0.1 percents) and 4 time period of 10, 20, 30, and 40 days on the growth of staphylococcus aures and total amount of microorganisms in preserved hamburger at the temperature of -18±1 °C was considered. Also, to find out about the effect of dill seed essence addition to hamburger on sensuous characteristics of the product, sensuous evaluation based on a 5-point hedonic scale was performed in three sections of taste, odor, and total reception. Results obtained from microbial experiments showed reduced staphylococcus aureus amount and total amounting, and these changes were significant in concentrations over 0.0125 (P<0.05). Among essence concentration levels, the least and most effects were determined to be those of 0.003 and 0.1, respectively. Sensuous evaluation results revealed that as dill essence concentration increased to 0.0125 percent, it had a positive effect on sensuous characteristics of hamburger. In higher concentration, however, it reduced sensuous characteristics of the product. These variations were significant in the concentration of 0.1 compared with other treatments (P<0.05). According to obtained results, dill seed’s oil essence had a harnessing and lethal impact on staphylococcus of hamburger, and it can be used as a natural preserver in meat products especially hamburger

Antimicrobial Effect of Dill Seed Oil Essence on Growth of Staphylococcus Aureus in Hamburgers

In the present study, antimicrobial effect of dill seed’s oil in 6 levels (0.003-0.006-0.0125-0.025-0.05-0.1 percents) and 4 time period of 10, 20, 30, and 40 days on the growth of staphylococcus aures and total amount of microorganisms in preserved hamburger at the temperature of -18±1 °C was considered. Also, to find out about the effect of dill seed essence addition to hamburger on sensuous characteristics of the product, sensuous evaluation based on a 5-point hedonic scale was performed in three sections of taste, odor, and total reception. Results obtained from microbial experiments showed reduced staphylococcus aureus amount and total amounting, and these changes were significant in concentrations over 0.0125 (P<0.05). Among essence concentration levels, the least and most effects were determined to be those of 0.003 and 0.1, respectively. Sensuous evaluation results revealed that as dill essence concentration increased to 0.0125 percent, it had a positive effect on sensuous characteristics of hamburger. In higher concentration, however, it reduced sensuous characteristics of the product. These variations were significant in the concentration of 0.1 compared with other treatments (P<0.05). According to obtained results, dill seed’s oil essence had a harnessing and lethal impact on staphylococcus of hamburger, and it can be used as a natural preserver in meat products especially hamburger