3 research outputs found
Extended Formulation Lower Bounds via Hypergraph Coloring?
Exploring the power of linear programming for combinatorial optimization
problems has been recently receiving renewed attention after a series of
breakthrough impossibility results. From an algorithmic perspective, the
related questions concern whether there are compact formulations even for
problems that are known to admit polynomial-time algorithms.
We propose a framework for proving lower bounds on the size of extended
formulations. We do so by introducing a specific type of extended relaxations
that we call product relaxations and is motivated by the study of the
Sherali-Adams (SA) hierarchy. Then we show that for every approximate
relaxation of a polytope P, there is a product relaxation that has the same
size and is at least as strong. We provide a methodology for proving lower
bounds on the size of approximate product relaxations by lower bounding the
chromatic number of an underlying hypergraph, whose vertices correspond to
gap-inducing vectors.
We extend the definition of product relaxations and our methodology to mixed
integer sets. However in this case we are able to show that mixed product
relaxations are at least as powerful as a special family of extended
formulations. As an application of our method we show an exponential lower
bound on the size of approximate mixed product formulations for the metric
capacitated facility location problem, a problem which seems to be intractable
for linear programming as far as constant-gap compact formulations are
concerned
Extended Formulation Lower Bounds via Hypergraph Coloring?
Exploring the power of linear programming for combinatorial optimization
problems has been recently receiving renewed attention after a series of
breakthrough impossibility results. From an algorithmic perspective, the
related questions concern whether there are compact formulations even
for problems that are known to admit polynomial-time algorithms.
We propose a framework for proving lower bounds on the size of extended
formulations. We do so by introducing a specific type of extended
relaxations that we call product relaxations and is motivated by the
study of the Sherali-Adams (SA) hierarchy. Then we show that for every
approximate extended formulation of a polytope P, there is a product
relaxation that has the same size and is at least as strong. We provide
a methodology for proving lower bounds on the size of approximate
product relaxations by lower bounding the chromatic number of an
underlying hypergraph, whose vertices correspond to gap-inducing
vectors.
We extend the definition of product relaxations and our methodology to
mixed integer sets. However in this case we are able to show that mixed
product relaxations are at least as powerful as a special family of
extended formulations. As an application of our method we show an
exponential lower bound on the size of approximate mixed product
relaxations for the metric capacitated facility location problem (CFL),
a problem which seems to be intractable for linear programming as far as
constant-gap compact formulations are concerned. Our lower bound implies
an unbounded integrality gap for CFL at Theta(N) levels of the universal
SA hierarchy which is independent of the starting relaxation; we only
require that the starting relaxation has size 2 degrees((N)), where N is
the number of facilities in the instance. This proof yields the first
such tradeoff for an SA procedure that is independent of the initial
relaxation