December 7, Wednesday 14:15, Room 303, Jacobs Building
Linear programming (LP) and semidefinite programming (SDP) are
powerful convex optimization tools which have become ubiquitous in the
field of approximation algorithms in the last few decades. Given a
combinatorial optimization problem (e.g. Maximum Independent Set), a
standard approach to obtain an approximation algorithm is as follows:
formulate the problem as an integer program (IP), relax this
formulation to an LP or SDP, solve the relaxation, and then "round"
the solution back to an integer solution.
This approach is limited by how well the convex program (LP or SDP)
approximates the original IP formulation, i.e. the integrality gap.
One way to circumvent this limitation is through hierarchies of convex
programs, which give a systematic way of iteratively strengthening any
relaxation (at the cost of increased running time to solve it), so
that the integrality gap gradually decreases.
While initially, most of the literature on hierarchies in the context
of approximation algorithms had focused on impossibility results,
there has been a surprising surge of recent positive results. I will
survey this recent development, by describing a number of
combinatorial optimization problems for which we have been able to
achieve improved approximations using hierarchies.