Polyhedral techniques in combinatorial optimization

Generally, combinatorial optimization problems are easy to formulate, but hard to solve. The most successfull approaches, cutting plane algorithms and column generation, rely on the (mixed) integer linear programming formulation of a problem. The theory of polyhedra, i.e., polyhedral combinatorics, is the foundation of these techniques. This manuscript intends to give an overview of polyhedral theory and its implications for cutting plane algorithms. We describe the major theoretical developments, like Gomory's general cutting plane algorithm, and the complexity of separation, Practical issues are illustrated on a diversity of well-known problems, like the traveling salesman problem and facility location, and on some generic problems like the knapsack problem, and vertex packing problem. The cutting plane approach extended with preprocessing techniques, and it is embedded in a branch-and-cut framework. Typical computational results are provided. Combinatorial optimization deals with maximizing or minimizing a function subject to a set of constraints and subject to the restriction that some, or all, variables should be integers. Several problems that occur in management and planning situations can be formulated as combinatorial optimIzations problems, such as the lot sizing problem, where we need to decide on which time periods to produce, and how much to produce in these periods to satisfy customers demand at minimal total production, storage and setup costs. Another well-known combinatorial optimization problem is the traveling salesman problem where we want to determine in which order a "salesman" shall visit a number of "cities" such that all cities are visited exactly once and such that the length of the tour is minimal. This problem is one of the most studied combinatorial optimization problems, not because of its importance in the planning of salesmen tours, but because onts numerous other applications, both in its own right and as substructures of more complex models, and because it is notoriously difficult to solve. The combination of being easy to state, relatively easy to formulate as a mathematical programming problem, but computationally intractable is something a majority of combinatorial optimization problems have in common. The computational intractability of most core combinatorial optimization problems has been theoretically indicated, i.e. it is possible to show that most of these problems belong to the class of NP~hard problems, see Karp (1972), and Garey and Johnson (1979). No algorithm with a worst-case running time bouded by a polynomial in the size of the input is known for any NPhard problem, and it is strongly believed that no such algorithm exists. Therefore, to solve these problems we have to use an enumerative algorithm, such as dynamic programming or branch and bound, with a worst~case running time that is exponetial in the size of the input. The computational hardness of most combinatorial optimization problems has inspired researchers to develop good formulations, and algorithms that are expected to reduce the size of the enumeration tree. To use information about the structure of the convex hull of feasible solutions, which is the basis for polyhedral techniques, has been one ofthe most successful approaches so far. The pioneering work in this direction was done by Dantzig, Fulkerson and Johnson (1954), who invented a method to solve the traveling salesman problem. They demonstrated the power of their technique on a 49-city instance, which was huge at that time. The idea behind the Dantzig-Fulkerson-Johnson method is the following. Assume we want to solve the problem min{cx subject to XES}, (1) where S is the set of feasible solutions, which in our case is the set of traveling salesman tours. Let S P n 7ln , where P == {x E 1R n : Ax S b} and Ax S b is a system of linear inequalities. Since S is difficult to characterize, we could solve the problem min{cx subject to x E P} (2) instead. Problem (2) is easy to solve, but since it is a relaxation of (1) it may us a solution x* that is nota tour. More precisely, the following two things can happen if we solve (2): either the optimal soution x* is a tour which means that x* is also optimal for (1), or x* is not a tour in which case it is not feasible for (1). If the solution x* is not feasible for (1) it lies outside the convex hull of S which means we can cut off x* by identifying a hyperplane separating x* from the convex hull of S, i.e. a hyperplane that is satisfied by all tours, but violated by x*. An