## Section: Scientific Foundations

### Polyhedral Combinatorics and Graph Theory

Many fundamental combinatorial optimization
problems can be modeled as the search for a specific structure in a
graph. For example, ensuring connectivity in a network amounts to
building a *tree* that spans all the nodes. Inquiring about its resistance
to failure amounts to searching for a minimum cardinality *cut*
that partitions the graph. Selecting disjoint pairs of objects is
represented by a so-called *matching*. Disjunctive choices can be modeled by edges in a
so-called *conflict graph* where one searches for *stable
sets* – a set of nodes that are not incident to one another.
Polyhedral combinatorics is the study of combinatorial algorithms involving polyhedral considerations. Not only it leads to efficient algorithms, but also,
conversely, efficient algorithms often imply polyhedral
characterizations and related min-max relations. Developments of
polyhedral properties of a fundamental problem will typically provide us with more
interesting inequalities well suited for a branch-and-cut algorithm to
more general problems. Furthermore, one can use the fundamental problems as
new building bricks to decompose the more general problem
at hand. For problem that let themselves easily be formulated in
a graph setting, the graph theory and in particular graph
decomposition theorem might help.