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Scheduling a set of jobs over a collection of machines to optimize a certain quality-of-service measure is one of the most important research topics in both computer science theory and practice. In this thesis, we design algorithms that optimize {\em flow-time} (or delay) of jobs for scheduling problems that arise in a wide range of applications. We consider the classical model of unrelated machine scheduling and resolve several long standing open problems; we introduce new models that capture the novel algorithmic challenges in scheduling jobs in data centers or large clusters; we study the effect of selfish behavior in distributed and decentralized environments; we design algorithms that strive to balance the energy consumption and performance. The technically interesting aspect of our work is the surprising connections we establish between approximation and online algorithms, economics, game theory, and queuing theory. It is the interplay of ideas from these different areas that lies at the heart of most of the algorithms presented in this thesis.The main contributions of the thesis can be placed in one of the following categories.1. Classical Unrelated Machine Scheduling: We give the first polygorithmic approximation algorithms for minimizing the average flow-time and minimizing the maximum flow-time in the offline setting. In the online and non-clairvoyant setting, we design the first non-clairvoyant algorithm for minimizing the weighted flow-time in the resource augmentation model. Our work introduces iterated rounding technique for the offline flow-time optimization, and gives the first framework to analyze non-clairvoyant algorithms for unrelated machines.2. Polytope Scheduling Problem: To capture the multidimensional nature of the scheduling problems that arise in practice, we introduce Polytope Scheduling Problem (\psp). The \psp problem generalizes almost all classical scheduling models, and also captures hitherto unstudied scheduling problems such as routing multi-commodity flows, routing multicast (video-on-demand) trees, and multi-dimensional resource allocation. We design several competitive algorithms for the \psp problem and its variants for the objectives of minimizing the flow-time and completion time. Our work establishes many interesting connections between scheduling and market equilibrium concepts, fairness and non-clairvoyant scheduling, and queuing theoretic notion of stability and resource augmentation analysis.3. Energy Efficient Scheduling: We give the first non-clairvoyant algorithm for minimizing the total flow-time + energy in the online and resource augmentation model for the most general setting of unrelated machines.4. Selfish Scheduling: We study the effect of selfish behavior in scheduling and routing problems. We define a fairness index for scheduling policies called {\em bounded stretch}, and show that for the objective of minimizing the average (weighted) completion time, policies with small stretch lead to equilibrium outcomes with small price of anarchy. Our work gives the first linear/ convex programming duality based framework to bound the price of anarchy for general equilibrium concepts such as coarse correlated equilibrium.

We design algorithms for markets consisting of multiple items, and agents with budget constraints on the maximum amount of money they can afford to spend. This problem can be considered under two broad frameworks. (a) From the standpoint of Auction Theory, the agents valuation functions over the items are private knowledge. Here, a "truthful auction" computes the subset of items received by every agent and her payment, and ensures that no agent can manipulate the scheme to her advantage by misreporting her valuation function. The question is to design a truthful auction whose outcome can be computed in polynomial time. (b) A different, but equally important, question is to investigate if and when the market is in "equilibrium", meaning that every item is assigned a price, every agent gets her utility-maximizing subset of items under the current prices, and every unallocated item is priced at zero. First, we consider the setting of multiple heterogeneous items and present approximation algorithms for revenue-optimal truthful auctions. When the items are homogeneous, we give an efficient algorithm whose outcome defines a truthful and Pareto-optimal auction. Finally, we focus on the notion of "competitive equilibrium", which is a well known solution concept for market clearing. We present efficient algorithms for finding competitive equilibria in markets with budget constrained agents, and show that these equilibria outcomes have strong revenue guarantees.