31 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
New Dependencies of Hierarchies in Polynomial Optimization
We compare four key hierarchies for solving Constrained Polynomial
Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant
Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams
(SA) hierarchies. We prove a collection of dependencies among these hierarchies
both for general CPOPs and for optimization problems on the Boolean hypercube.
Key results include for the general case that the SONC and SOS hierarchy are
polynomially incomparable, while SDSOS is contained in SONC. A direct
consequence is the non-existence of a Putinar-like Positivstellensatz for
SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like
versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent.
Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that
provides a O(n) degree bound.Comment: 26 pages, 4 figure
Lift & Project Systems Performing on the Partial-Vertex-Cover Polytope
We study integrality gap (IG) lower bounds on strong LP and SDP relaxations
derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and
Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the
t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover
problem in which only t edges need to be covered. t-PVC admits a
2-approximation using various algorithmic techniques, all relying on a natural
LP relaxation. Starting from this LP relaxation, our main results assert that
for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P
systems that have been used for positive algorithmic results (but the Lasserre
hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices
of the input graph. Our lower bounds are nearly tight.
Our results show that restricted yet powerful models of computation derived
by many L&P systems fail to witness c-approximate solutions to t-PVC for any
constant c, and for t = O(n). This is one of the very few known examples of an
intractable combinatorial optimization problem for which LP-based algorithms
induce a constant approximation ratio, still lift-and-project LP and SDP
tightenings of the same LP have unbounded IGs.
We also show that the SDP that has given the best algorithm known for t-PVC
has integrality gap n/t on instances that can be solved by the level-1 LP
relaxation derived by the LS system. This constitutes another rare phenomenon
where (even in specific instances) a static LP outperforms an SDP that has been
used for the best approximation guarantee for the problem at hand. Finally, one
of our main contributions is that we make explicit of a new and simple
methodology of constructing solutions to LP relaxations that almost trivially
satisfy constraints derived by all SDP L&P systems known to be useful for
algorithmic positive results (except the La system).Comment: 26 page