By Jon Kleinberg; Eva Tardos

Advent: a few consultant difficulties -- fundamentals of algorithms research -- Graphs -- grasping algorithms -- Divide and triumph over -- Dynamic programming -- community circulate -- NP and computational intractability -- PSPACE: a category of difficulties past NP -- Extending the boundaries of tractability -- Approximation algorithms -- neighborhood seek -- Randomized algorithms -- Epilogue: algorithms that run ceaselessly

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**Sample text**

Does there exist a stable matching for every set of preference lists? Given a set of preference lists, can we efﬁciently construct a stable matching if there is one? Some Examples To illustrate these deﬁnitions, consider the following two very simple instances of the Stable Matching Problem. First, suppose we have a set of two men, {m, m }, and a set of two women, {w, w }. The preference lists are as follows: m prefers w to w . m prefers w to w . 1 A First Problem: Stable Matching w prefers m to m .

We begin this chapter by talking about how to put this notion on a concrete footing, as making it concrete opens the door to a rich understanding of computational tractability. Having done this, we develop the mathematical machinery needed to talk about the way in which different functions scale with increasing input size, making precise what it means for one function to grow faster than another. We then develop running-time bounds for some basic algorithms, beginning with an implementation of the Gale-Shapley algorithm from Chapter 1 and continuing to a survey of many different running times and certain characteristic types of algorithms that achieve these running times.

You have a resource— it may be a lecture room, a supercomputer, or an electron microscope—and many people request to use the resource for periods of time. A request takes the form: Can I reserve the resource starting at time s, until time f ? We will assume that the resource can be used by at most one person at a time. A scheduler wants to accept a subset of these requests, rejecting all others, so that the accepted requests do not overlap in time. The goal is to maximize the number of requests accepted.