15 research outputs found
Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre
hierarchy. For binary polynomial optimization problems of degree and an
odd number of variables , we prove that levels of the
SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This
matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear
representation of a set with no integral points. She showed that the
Sherali-Adams hierarchy requires levels to detect the empty integer hull,
and conjectured that the SoS/Lasserre rank for the same problem is . We
disprove this conjecture and derive lower and upper bounds for the rank
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
Approximation Limits of Linear Programs (Beyond Hierarchies)
We develop a framework for approximation limits of polynomial-size linear
programs from lower bounds on the nonnegative ranks of suitably defined
matrices. This framework yields unconditional impossibility results that are
applicable to any linear program as opposed to only programs generated by
hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations
for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound
applies to linear programs using a certain encoding of CLIQUE as a linear
optimization problem.) Moreover, we establish a similar result for
approximations of semidefinite programs by linear programs. Our main ingredient
is a quantitative improvement of Razborov's rectangle corruption lemma for the
high error regime, which gives strong lower bounds on the nonnegative rank of
certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure
Complexity of optimizing over the integers
In the first part of this paper, we present a unified framework for analyzing
the algorithmic complexity of any optimization problem, whether it be
continuous or discrete in nature. This helps to formalize notions like "input",
"size" and "complexity" in the context of general mathematical optimization,
avoiding context dependent definitions which is one of the sources of
difference in the treatment of complexity within continuous and discrete
optimization. In the second part of the paper, we employ the language developed
in the first part to study information theoretic and algorithmic complexity of
{\em mixed-integer convex optimization}, which contains as a special case
continuous convex optimization on the one hand and pure integer optimization on
the other. We strive for the maximum possible generality in our exposition.
We hope that this paper contains material that both continuous optimizers and
discrete optimizers find new and interesting, even though almost all of the
material presented is common knowledge in one or the other community. We see
the main merit of this paper as bringing together all of this information under
one unifying umbrella with the hope that this will act as yet another catalyst
for more interaction across the continuous-discrete divide. In fact, our
motivation behind Part I of the paper is to provide a common language for both
communities
Stable Set Polytopes with High Lift-and-Project Ranks for the Lov\'asz-Schrijver SDP Operator
We study the lift-and-project rank of the stable set polytopes of graphs with
respect to the Lov{\'a}sz--Schrijver SDP operator , with a
particular focus on a search for relatively small graphs with high
-rank (the least number of iterations of the
operator on the fractional stable set polytope to compute the stable set
polytope). We provide families of graphs whose -rank is
asymptotically a linear function of its number of vertices, which is the best
possible up to improvements in the constant factor (previous best result in
this direction, from 1999, yielded graphs whose -rank only grew
with the square root of the number of vertices). We also provide several new
-minimal graphs, most notably a -vertex graph with
-rank , and study the properties of a vertex-stretching
operation that appears to be promising in generating -minimal
graphs