The simplest and most scalable type of optimization problem is one in which the objective function and constraints are formulated using the simplest type of functions – linear functions. We refer to this class of problems as linear optimization (LO) problems.

This chapter introduces robust optimization (RO), where we aim to solve a MILO in which some of the parameters/data can take multiple (possibly infinitely many) values and we want the optimal solution to perform “the best possible,” assuming that the unknown problem parameters can always turn out to be “the worst possible.”

Our aim so far has been to formulate a real-life decision problem as a (mixed integer) linear optimization problem. The reason was clear: Linear functions are simple, so problems formulated with them should also be simple. However, two questions arise. First, is the world of linear functions flexible enough to model all real-life problems? Second, are MILOs the only simple problems we can solve quickly?

In this introductory chapter, we will formally introduce the main variants of the traveling salesman problem, symmetric and asymmetric, explain a very useful graph-theoretic view based on Euler’s theorem, and describe the classical simple approximation algorithms: Christofides’ algorithm and the cycle cover algorithm.

We also introduce basic notation, in particular from graph theory, and some fundamental combinatorial optimization problems.

A major step towards the first constant-factor approximation algorithm for the Asymmetric TSP was made by Svensson. He devised a constant-factor approximation algorithm for Asymmetric Graph TSP, which is the special case of the Asymmetric TSP with c(e)=1 for all e ∈ E.

In this chapter, we present Svensson’s algorithm for the Asymmetric Graph TSP. We also incorporate some improvements, from Traub and Vygen, who gave a variant of Svensson’s algorithm with improved approximation ratio. Moreover, we present an improved algorithm for finding a graph subtour cover, which is the main subroutine of Svensson’s algorithm. Overall, we will obtain an approximation ratio of 8+ε for Asymmetric Graph TSP, for every ε>0.

Almost all techniques presented in this chapter will be used again in Chapters 7 and 8 for the general Asymmetric TSP.

The random sampling approach described in Chapter 5 for the Asymmetric TSP has also been used successfully for the Symmetric TSP. First, Oveis Gharan, Saberi, and Singh obtained the first algorithm with approximation ratio less than 3/2 for Graph TSP. More recently, Karlin, Klein, and Oveis Gharan proved that essentially the same algorithm has approximation ratio less than 3/2 for the general Symmetric TSP.

The algorithm is simple, but its analysis is very complicated. While for Graph TSP we know simpler and better algorithms today (see Chapters 12 and 13), the random sampling algorithm is still the best-known approximation algorithm for Symmetric TSP.

The algorithm samples a spanning tree from an (approximately) marginal-preserving λ-uniform distribution and then proceeds with parity correction like Christofides’ algorithm. After briefly discussing the analysis for Graph TSP, we present the first part of the analysis by Karlin, Klein, and Oveis Gharan, with some simplifications suggested by Drees. The main point is to reduce the set of relevant cuts that need to be considered to bound the cost of parity correction and obtain a nice structure that will be exploited in Chapter 11.

Traub and Vygen used recursive dynamic programming to obtain a (3/2+ε)-approximation algorithm for Path TSP for any ε>0. This approach was then improved and simplified by Zenklusen, who obtained a 3/2-approximation for Path TSP. After discussing the dynamic programming approach in a simple context, we present Zenklusen’s algorithm.

Then we present a black-box reduction from Path TSP to Symmetric TSP, similar to the one proposed by Traub, Vygen, and Zenklusen. This shows that the former is not much harder to approximate than the latter. This implies the currently best-known approximation guarantees for Path TSP and the special case Graph Path TSP. Our new proof, again based on dynamic programming, actually yields the same result even for a more general problem, which we call Multi-Path TSP.

So far, all algorithms for Symmetric TSP began with a spanning tree and then added edges to make the graph Eulerian. In contrast, Mömke and Svensson suggested to begin with a 2-connected graph; then we may also delete some edges for making it Eulerian, and this may be cheaper overall. They introduced the notion of removable pairings, which allow to control that we maintain connectivity when deleting edges.

This idea led to a substantial improvement and is still used for the best algorithm for Graph TSP that we know today (cf. Chapter 12). It also yields the ratio 4/3 for the special case of subcubic graphs.

In this short concluding chapter, we show two figures that summarize the state of the art.

Moreover, we list again the open problems that we mentioned in this book.