Vijay V. Vazirani

Introduction to LP-Duality

A large fraction of the theory of approximation algorithms, as we know it today, is built around linear programming (LP). In Section 12.1 we will review some key concepts from this theory. In Section 12.2 we will show how the LP-duality theorem gives rise to min-max relations which have far-reaching algorithmic significance. Finally, in Section 12.3 we introduce the two fundamental algorithm design techniques of rounding and the primal-dual schema, as well as the method of dual fitting, which yield all the algorithms of Part II of this book.

Feedback Vertex Set

In this chapter we will use the technique of layering, introduced in Chapter 2, to obtain a factor 2 approximation algorithm for the following problem. Recall that the idea behind layering was to decompose the given weight function into convenient functions on a nested sequence of subgraphs of G.

Rounding Applied to Set Cover

We will introduce the technique of LP-rounding by using it to design two approximation algorithms for the set cover problem, Problem 2.1. The first is a simple rounding algorithm achieving a guarantee of f, where f is the frequency of the most frequent element. The second algorithm, achieving an approximation guarantee of O(log n), illustrates the use of randomization in rounding.

Multiway Cut and k-Cut

The theory of cuts occupies a central place in the study of exact algorithms In this chapter, we will present approximation algorithms for natural generalizations of the minimum cut problem. These generalizations are NP-hard.

Multicut and Integer Multicommodity Flow in Trees

The theory of cuts in graphs occupies a central place not only in the study of exact algorithms, but also approximation algorithms. We will present some key results in the next four chapters. This will also give us the opportunity to develop further the two fundamental algorithm design techniques introduced in Chapters 14 and 15.

Multicut in General Graphs

The importance of min-max relations to combinatorial optimization was mentioned in Chapter 1. Perhaps the most useful of these is the celebrated max-flow min-cut theorem. Indeed, much of flow theory, and the theory of cuts in graphs, has been built around this theorem. It is not surprising, therefore, that a concerted effort was made to obtain generalizations of this theorem to the case of multiple commodities.

Scheduling on Unrelated Parallel Machines

LP-rounding has yielded approximation algorithms for a large number of NP-hard problems in scheduling theory (see Section 17.6). As an illustrative example, we present a factor 2 algorithm for the problem of scheduling on unrelated parallel machines. We will apply the technique of parametric pruning, introduced in Chapter 5, together with LP-rounding, to obtain the algorithm.

Steiner Tree and TSP

In this chapter, we will present constant factor algorithms for two fundamental problems, metric Steiner tree and metric TSP. The reasons for considering the metric case of these problems are quite different. For Steiner tree, this is the core of the problem — the rest of the problem reduces to this case. For TSP, without this restriction, the problem admits no approximation factor, assuming P ≠ NP. The algorithms, and their analyses, are similar in spirit, which is the reason for presenting these problems together.

Set Cover via Dual Fitting

In this chapter we will introduce the method of dual fitting, which helps analyze combinatorial algorithms using LP-duality theory. Using this method, we will present an alternative analysis of the natural greedy algorithm (Algorithm 2.2) for the set cover problem (Problem 2.1). Recall that in Section 2.1 we deferred giving the lower bounding method on which this algorithm was based. We will provide the answer below. The power of this approach will become apparent when we show the ease with which it extends to solving several generalizations of the set cover problem (see Section 13.2).

Set Cover via the Primal-Dual Schema

As noted in Section 12.3, the primal-dual schema is the method of choice for designing approximation algorithms since it yields combinatorial algorithms with good approximation factors and good running times. We will first present the central ideas behind this schema and then use it to design a simple f factor algorithm for set cover, where f is the frequency of the most frequent element.