In Chapter 2, we have seen how to solve LPs using the simplex algorithm, an algorithm that is still widely used in practice. In Chapter 3, we discussed efficient algorithms to solve the special class of IPs describing the shortest path problem and the minimum cost perfect matching problem in bipartite graphs. In both these examples, it is sufficient to solve the LP relaxation of the problem.

Integer programming is widely believed to be a difficult problem (see Appendix A). Nonetheless, we will present algorithms that are guaranteed to solve IPs in finite time. The drawback of these algorithms is that the running time may be exponential in the worst case. However, they can be quite fast for many instances, and are capable of solving many large-scale, real-life problems.

These algorithms follow two general strategies. The first attempts to reduce IPs to LPs – this is known as the cutting plane approach and will be described in Section 6.2. The other strategy is a divide and conquer approach and is known as branch and bound and will be discussed in Section 6.3. In practice, both strategies are combined under the heading of branch and cut. This remains the preferred approach for all general purpose commercial codes.

In this chapter, in the interest of simplicity we will restrict our attention to pure IPs where all the variables are required to be integer. The theory developed here extends to mixed IPs where only some of the variables are required to be integer, but the material is beyond the scope of this book.

Possible outcomes

Consider an LP (P) with variables x1, …, xn. Recall that an assignment of values to each of x1, …, xn is a feasible solution if the constraints of (P) are satisfied. We can view a feasible solution to (P) as a vector x = (x1, …, xn)T. Given a vector x, by the value of x we mean the value of the objective function of (P) for x. Suppose (P) is a maximization problem. Then recall that we call a vector x an optimal solution if it is a feasible solution and no feasible solution has larger value. The value of the optimal solution is the optimal value. By definition, an LP has only one optimal value; however, it may have many optimal solutions. When solving an LP, we will be satisfied with finding any optimal solution. Suppose (P) is a minimization problem. Then a vector x is an optimal solution if it is a feasible solution and no feasible solution has smaller value.

If an LP (P) has a feasible solution, then it is said to be feasible, otherwise it is infeasible. Suppose (P) is a maximization problem and for every real number α there is a feasible solution to (P) which has value greater than α, then we say that (P) is unbounded. In other words, (P) is unbounded if we can find feasible solutions of arbitrarily high value. Suppose (P) is a minimization problem and for every real number α there is a feasible solution to (P) which has value smaller than α, then we say that (P) is unbounded.

In this chapter, we revisit the shortest path and minimum-cost matching problems. Both were first introduced in Chapter 1, where we discussed practical example applications. We further showed that these problems can be expressed as IPs. The focus in this chapter will be on solving instances of the shortest path and matching problems. Our starting point will be to use the IP formulation we introduced in Section 1.5. We will show that studying the two problems through the lens of linear programming duality will allow us to design efficient algorithms. We develop this theory further in Chapter 4.

The shortest path problem

Recall the shortest path problem from Section 1.4.1. We are given a graph G = (V, E), nonnegative lengths ce for all edges e ∈ E, and two distinct vertices s, t ∈ V. The length c(P) of a path P is the sum of the length of its edges, i.e. Σ(ce: e ∈ P). We wish to find among all possible st-paths one that is of minimum length.

Example 7 In the following figure, we show an instance of this problem. Each of the edges in the graph is labeled by its length. The thick black edges in the graph form an st-path P = sa, ac, cb, bt of total length 3 + 1 + 2 + 1 = 7. This st-path is of minimum length, hence is a solution to our problem.

An algorithm is a formal procedure that describes how to solve a problem. For instance, the simplex algorithm in Chapter 2 takes as input a linear program in standard equality form and either returns an optimal solution, or detects that the linear program is infeasible or unbounded. Another example is the shortest path algorithm in Chapter 3.1. It takes as input a graph with distinct vertices s, t and nonnegative integer edge lengths, and returns an st-path of shortest length (if one exists).

The two basic properties we require for an algorithm are: correctness and termination. By correctness, we mean that the algorithm is always accurate when it claims that we have a particular outcome. One way to ensure this is to require that the algorithm provides a certificate, i.e. a proof, to justify its answers. By termination, we mean that the algorithm will stop after a finite number of steps.

In Section A.1, we will define the running time of an algorithm; we will formalize the notions of slow and fast algorithms. Section A.2 reviews the algorithms presented in this book and discusses which ones are fast and which ones are slow. In Sections A.3 and A.4 we discuss the inherent complexity of various classes of optimization problems and discuss the possible existence of classes of problems for which it is unlikely that any fast algorithm exists. We explain how an understanding of computational complexity can guide us in the design of algorithms.

Broadly speaking, optimization is the problem of minimizing or maximizing a function subject to a number of constraints. Optimization problems are ubiquitous. Every chief executive officer (CEO) is faced with the problem of maximizing profit given limited resources. In general, this is too general a problem to be solved exactly; however, many aspects of decision making can be successfully tackled using optimization techniques. This includes, for instance, production, inventory, and machine-scheduling problems. Indeed, the overwhelming majority of Fortune 500 companies make use of optimization techniques. However, optimization problems are not limited to the corporate world. Every time you use your GPS, it solves an optimization problem, namely how to minimize the travel time between two different locations. Your hometown may wish to minimize the number of trucks it requires to pick up garbage by finding the most efficient route for each truck. City planners may need to decide where to build new fire stations in order to efficiently serve their citizens. Other examples include: how to construct a portfolio that maximizes its expected return while limiting volatility; how to build a resilient tele-communication network as cheaply as possible; how to schedule flights in a cost-effective way while meeting the demand for passengers; or how to schedule final exams using as few classrooms as possible.

Suppose that you are a consultant hired by the CEO of the WaterTech company to solve an optimization problem.