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10 Lagrangian Duality10.1 Lagrangian Relaxation考虑如下整数规划问题（IP）：其中，Ax≤bAx\le bAx≤b表示的是复杂的约束，而“好”的约束（例如网络流约束）被包含在了x∈Xx\in Xx∈X中。它的拉格朗日松弛问题是：其中，uuu被称为价格或者对偶变量或者拉格朗日乘子。因为每个拉格朗日松弛问题都是原整数规划问题的一个上界，所以为了找到最好的（也即最小的）上界，我们需要求解如下拉格朗日对偶问题：何时拉格朗日松弛问题的最优解是原整数规划问

3 Well-Solved Problems3.1 Properties of Easy ProblemsIn examining a problem to see if it has an efficient algorithm, the following four properties often go together:In the next sections, we examine several classes of problems for which we will see t

目录4 Matchings and Assignments4.1 Augmenting Paths and Optimality4.2 Bipartite Maximum Cardinality MatchingQ:拿到一个问题问自己难不难，以后。。。max matchingmin covering哪个是np难/polynomial time4 Matchings and Assignments4.1 Augmenting Paths and Optimality在一些组合算法中会用到的

Chapter 2 explains how it is possible to prove that feasible solutions are optimal or close to optimal.目录2 Optimality, Relaxation, and Bounds2.1 Optimality and Relaxation2.2 Linear Programming Relaxations2.3 Combinatorial Relaxations2.4 Lagrangian Relaxat

Chapter 1 introduces the reader to various integer programming problems and their formulation and introduces the important distinction between good and bad formulations.目录1 Formulations1.1 Introduction1.2 What Is an Integer Program?1.3 Formulating IPs an.