Section: Scientific Foundations

Polyhedral Approaches

Adding valid inequalities to the polyhedral description of an MIP allows one to improve the resulting LP bound and hence to better prune the enumeration tree. In a cutting plane procedure, one attemps to identify valid inequalities that are violated by the LP solution of the current formulation and adds them to the formulation. This can be done at each node of the branch-and-bound tree giving rise to a so-called branch-and-cut algorithm [137] . Introduced by Edmonds in 1965 [74] , the polyhedral approach has turned out to be one of the main sources of progress in solving NP-hard combinatorial optimization problems in the last two decades. A benchmark problem, in this regard, is the traveling salesman problem [112] . In the early 80's, the best algorithm was able to solve instances with around 300 cities. A recent paper [50] reports that branch-and-cut algorithms are able to solve instances with nearly 25,000 cities. Similar significant improvements have been observed for instance for network design problems arising in telecommunication (see Kerivin and Mahjoub, 2005 [109] ) or vehicle routing problems in logistic (see Letchford and Salazar-Gonzalez, 2006 [114] ).

The goal of these approaches is to reduce the resolution of an integer program to that of a linear program by deriving a linear description of the convex hull of the feasible solutions, conv(X) , where X is the discrete set of solutions to the combinatorial problem on hand. A fundamental result in this field is the equivalence of complexity between solving the combinatorial optimization problem and solving the separation problem over the associated polyhedron: if , find a linear inequality x₀ satisfied by all points in conv(X) but violated by [93] . Hence, for NP-hard problems, one cannot hope to define either a closed-form description of conv(X) or a polynomial time exact separation routine. Nevertheless, one does not need to know such a description to take advantage of the polyhedral approach. Only a subset of the inequalities can already yield a good approximation of the ideal polytope. Moreover, non-exact separation, using heuristic procedures, turns out to be quite efficient for practical purposes.

Polyhedral theory provides ways to derive automatically new inequalities from an initial polyhedral description P of the problem. For instance, it is known [134] that any valid inequalities for an IP can be obtained by iteratively taking linear combinations of existing constraints and rounding their coefficients. Such general purposes cuts have only recently made their way as practical tools: for instance, Gomory fractional cuts are now generated by default into commercial MIP solvers. Recent work [62] , [80] has consisted in numerically testing the strength of the formulation obtained by application of a single round of such general purpose cuts (called the first-closure): the separation problem being set as an MIP problem which is solved with a commercial MIP solver. Cornuéjols (2006) [65] provides a comparative review of general purpose cuts such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts, and intersection cuts, as well as their practical contributions to dual bound improvements. However, the most promising results have often been obtained with so-called template cuts , i.e. family of valid inequalities derived in an application specific context: the close form expression of these additional inequalities is a template from which specific cuts are generated dynamically. To prove validity, one can show that such inequalities can be obtained as a special case of general purpose procedures. If it can be shown that the inequalities define so-called facets of conv(X) , these inequalities are needed for its description. In practice, one needs to develop efficient procedures (exact or heuristic) to separate these inequalities. Then, numerical evaluation can show the impact of the additional inequalities not only on the strength of the resulting dual bound but also in yielding solutions more likely to satisfy integrality restrictions (which is good for primal heuristics).

The connections between polyhedral structure and graph theory are deep. Many facets of various polyhedra are directly related to special classes of graphs. The literature is rich with such examples of facet-defining systems described by exhibiting a bijection to a collection of subgraphs of the studied graph: forest polytopes (Edmonds, 1961), the matching polytope (Edmonds, 1965, [74] ), and many others. There are even results showing that the structure of a specific polyhedron itself is closely related to the structure of a related graph. For instance, Chvátal has shown that the property of adjacency in the stable set polytope of a graph G (i.e., the fact that two solutions satisfy the same facet defining constraint at equality) is characterized as a connectivity property of G itself. A number of researchers have recently described analogous interpretations of other facets of the stable set polytope. For example, Lipták and Lovász (1999) [116] exhibited such a relationship; in [138] we describe another and use it to generate a new set of facets for the stable set polytope of webs.