373 research outputs found
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
Hierarchies of Relaxations for Online Prediction Problems with Evolving Constraints
We study online prediction where regret of the algorithm is measured against
a benchmark defined via evolving constraints. This framework captures online
prediction on graphs, as well as other prediction problems with combinatorial
structure. A key aspect here is that finding the optimal benchmark predictor
(even in hindsight, given all the data) might be computationally hard due to
the combinatorial nature of the constraints. Despite this, we provide
polynomial-time \emph{prediction} algorithms that achieve low regret against
combinatorial benchmark sets. We do so by building improper learning algorithms
based on two ideas that work together. The first is to alleviate part of the
computational burden through random playout, and the second is to employ
Lasserre semidefinite hierarchies to approximate the resulting integer program.
Interestingly, for our prediction algorithms, we only need to compute the
values of the semidefinite programs and not the rounded solutions. However, the
integrality gap for Lasserre hierarchy \emph{does} enter the generic regret
bound in terms of Rademacher complexity of the benchmark set. This establishes
a trade-off between the computation time and the regret bound of the algorithm
Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups
The little Grothendieck problem consists of maximizing
over binary variables , where C is a
positive semidefinite matrix. In this paper we focus on a natural
generalization of this problem, the little Grothendieck problem over the
orthogonal group. Given C a dn x dn positive semidefinite matrix, the objective
is to maximize restricting to take
values in the group of orthogonal matrices, where denotes the (ij)-th
d x d block of C. We propose an approximation algorithm, which we refer to as
Orthogonal-Cut, to solve this problem and show a constant approximation ratio.
Our method is based on semidefinite programming. For a given , we show
a constant approximation ratio of , where is
the expected average singular value of a d x d matrix with random Gaussian
i.i.d. entries. For d=1 we recover the known
approximation guarantee for the classical little Grothendieck problem. Our
algorithm and analysis naturally extends to the complex valued case also
providing a constant approximation ratio for the analogous problem over the
Unitary Group.
Orthogonal-Cut also serves as an approximation algorithm for several
applications, including the Procrustes problem where it improves over the best
previously known approximation ratio of~. The little
Grothendieck problem falls under the class of problems approximated by a recent
algorithm proposed in the context of the non-commutative Grothendieck
inequality. Nonetheless, our approach is simpler and it provides a more
efficient algorithm with better approximation ratios and matching integrality
gaps.
Finally, we also provide an improved approximation algorithm for the more
general little Grothendieck problem over the orthogonal (or unitary) group with
rank constraints.Comment: Updates in version 2: extension to the complex valued (unitary group)
case, sharper lower bounds on the approximation ratios, matching integrality
gap, and a generalized rank constrained version of the problem. Updates in
version 3: Improvement on the expositio
Subsampling Mathematical Relaxations and Average-case Complexity
We initiate a study of when the value of mathematical relaxations such as
linear and semidefinite programs for constraint satisfaction problems (CSPs) is
approximately preserved when restricting the instance to a sub-instance induced
by a small random subsample of the variables. Let be a family of CSPs such
as 3SAT, Max-Cut, etc., and let be a relaxation for , in the sense
that for every instance , is an upper bound the maximum
fraction of satisfiable constraints of . Loosely speaking, we say that
subsampling holds for and if for every sufficiently dense instance and every , if we let be the instance obtained by
restricting to a sufficiently large constant number of variables, then
. We say that weak subsampling holds if the
above guarantee is replaced with whenever
. We show: 1. Subsampling holds for the BasicLP and BasicSDP
programs. BasicSDP is a variant of the relaxation considered by Raghavendra
(2008), who showed it gives an optimal approximation factor for every CSP under
the unique games conjecture. BasicLP is the linear programming analog of
BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional
constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of
unique games type. 3. There are non-unique CSPs for which even weak subsampling
fails for the above tighter semidefinite programs. Also there are unique CSPs
for which subsampling fails for the Sherali-Adams linear programming hierarchy.
As a corollary of our weak subsampling for strong semidefinite programs, we
obtain a polynomial-time algorithm to certify that random geometric graphs (of
the type considered by Feige and Schechtman, 2002) of max-cut value
have a cut value at most .Comment: Includes several more general results that subsume the previous
version of the paper
Rounding Sum-of-Squares Relaxations
We present a general approach to rounding semidefinite programming
relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our
approach is based on using the connection between these relaxations and the
Sum-of-Squares proof system to transform a *combining algorithm* -- an
algorithm that maps a distribution over solutions into a (possibly weaker)
solution -- into a *rounding algorithm* that maps a solution of the relaxation
to a solution of the original problem.
Using this approach, we obtain algorithms that yield improved results for
natural variants of three well-known problems:
1) We give a quasipolynomial-time algorithm that approximates the maximum of
a low degree multivariate polynomial with non-negative coefficients over the
Euclidean unit sphere. Beyond being of interest in its own right, this is
related to an open question in quantum information theory, and our techniques
have already led to improved results in this area (Brand\~{a}o and Harrow, STOC
'13).
2) We give a polynomial-time algorithm that, given a d dimensional subspace
of R^n that (almost) contains the characteristic function of a set of size n/k,
finds a vector in the subspace satisfying ,
where . Aside from being a natural relaxation, this
is also motivated by a connection to the Small Set Expansion problem shown by
Barak et al. (STOC 2012) and our results yield a certain improvement for that
problem.
3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time
algorithm with substantially improved guarantees for recovering a planted
-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n
nonzero coordinates, we can recover it with high probability whenever , improving for prior methods which
intrinsically required
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