6,808 research outputs found
Sum-of-squares proofs and the quest toward optimal algorithms
In order to obtain the best-known guarantees, algorithms are traditionally
tailored to the particular problem we want to solve. Two recent developments,
the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method,
surprisingly suggest that this tailoring is not necessary and that a single
efficient algorithm could achieve best possible guarantees for a wide range of
different problems.
The Unique Games Conjecture (UGC) is a tantalizing conjecture in
computational complexity, which, if true, will shed light on the complexity of
a great many problems. In particular this conjecture predicts that a single
concrete algorithm provides optimal guarantees among all efficient algorithms
for a large class of computational problems.
The Sum-of-Squares (SOS) method is a general approach for solving systems of
polynomial constraints. This approach is studied in several scientific
disciplines, including real algebraic geometry, proof complexity, control
theory, and mathematical programming, and has found applications in fields as
diverse as quantum information theory, formal verification, game theory and
many others.
We survey some connections that were recently uncovered between the Unique
Games Conjecture and the Sum-of-Squares method. In particular, we discuss new
tools to rigorously bound the running time of the SOS method for obtaining
approximate solutions to hard optimization problems, and how these tools give
the potential for the sum-of-squares method to provide new guarantees for many
problems of interest, and possibly to even refute the UGC.Comment: Survey. To appear in proceedings of ICM 201
A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut
We give improved pseudorandom generators (PRGs) for Lipschitz functions of
low-degree polynomials over the hypercube. These are functions of the form
psi(P(x)), where P is a low-degree polynomial and psi is a function with small
Lipschitz constant. PRGs for smooth functions of low-degree polynomials have
received a lot of attention recently and play an important role in constructing
PRGs for the natural class of polynomial threshold functions. In spite of the
recent progress, no nontrivial PRGs were known for fooling Lipschitz functions
of degree O(log n) polynomials even for constant error rate. In this work, we
give the first such generator obtaining a seed-length of (log
n)\tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps.
Previous generators had an exponential dependence on the degree.
We use our PRG to get better integrality gap instances for sparsest cut, a
fundamental problem in graph theory with many applications in graph
optimization. We give an instance of uniform sparsest cut for which a powerful
semi-definite relaxation (SDP) first introduced by Goemans and Linial and
studied in the seminal work of Arora, Rao and Vazirani has an integrality gap
of exp(\Omega((log log n)^{1/2})). Understanding the performance of the
Goemans-Linial SDP for uniform sparsest cut is an important open problem in
approximation algorithms and metric embeddings and our work gives a
near-exponential improvement over previous lower bounds which achieved a gap of
\Omega(log log n)
Propositional computability logic I
In the same sense as classical logic is a formal theory of truth, the
recently initiated approach called computability logic is a formal theory of
computability. It understands (interactive) computational problems as games
played by a machine against the environment, their computability as existence
of a machine that always wins the game, logical operators as operations on
computational problems, and validity of a logical formula as being a scheme of
"always computable" problems. The present contribution gives a detailed
exposition of a soundness and completeness proof for an axiomatization of one
of the most basic fragments of computability logic. The logical vocabulary of
this fragment contains operators for the so called parallel and choice
operations, and its atoms represent elementary problems, i.e. predicates in the
standard sense. This article is self-contained as it explains all relevant
concepts. While not technically necessary, however, familiarity with the
foundational paper "Introduction to computability logic" [Annals of Pure and
Applied Logic 123 (2003), pp.1-99] would greatly help the reader in
understanding the philosophy, underlying motivations, potential and utility of
computability logic, -- the context that determines the value of the present
results. Online introduction to the subject is available at
http://www.cis.upenn.edu/~giorgi/cl.html and
http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi
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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