## 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 x_{0} 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.