## Section: Research Program

### Approximation Algorithms

In some contexts, obtaining an exact solution to an optimization
problem is not feasible: when instances are too large, or when
decisions need to be taken rapidly. Since most of the combinatorial
optimization problems are NP-hard, another direction to obtain good
quality solutions in reasonable time is to focus on
**approximation algorithms**. The definition of approximation
algorithms is based on the notion of input set $\mathcal{I}$ and each
$I\in \mathcal{I}$ defines a solution space ${\mathcal{S}}_{I}$. For a
minimization problem ${min}_{x\in {\mathcal{S}}_{I}}f\left(x\right)$, an algorithm
$\mathcal{A}$ is an $\alpha $-approximation algorithm if it provides a
solution within $\alpha $ of the optimal solution for all instances in
the input set:

The objective is to search for polynomial algorithms, with
approximation ratios as close to 1 as possible. Such algorithms are
called *worst-case* approximation algorithms, because the
performance guarantee is expressed over all possible inputs of the
problem. The design of these algorithms have strong links with the
enumeration techniques described above: since computing ${f}^{*}\left(I\right)$ is an
NP-hard problem, it is often required to derive **strong a
priori bounds** on the optimal solution value which can afterward be
compared to estimations of the value of the solution produced. In many
cases, it is also possible to build $\alpha $-approximate solutions by
a careful rounding of a solution obtained from the linear relaxation
of an integer formulation of the problem. Members of the team have
expertise in designing and evaluating approximation algorithms for
resource allocation in computer
systems, using a
variety of techniques, such as dual approximation (where a guess of
the optimal value ${f}^{*}$ is provided, and $\mathcal{A}$ either provides
a solution within $\alpha {f}^{*}$, or guarantees that no solution of
value ${f}^{*}$ or less exists), or resource augmentation (where an
approximation is obtained by relaxing some of the constraints of the
problem).