## Section: Research Program

### 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.