776 research outputs found
Approximability and proof complexity
This work is concerned with the proof-complexity of certifying that
optimization problems do \emph{not} have good solutions. Specifically we
consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic
proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor,
Lasserre, and Parrilo shows that this proof system is automatizable using
semidefinite programming (SDP), meaning that any -variable degree- proof
can be found in time . Furthermore, the SDP is dual to the well-known
Lasserre SDP hierarchy, meaning that the "-round Lasserre value" of an
optimization problem is equal to the best bound provable using a degree- SOS
proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC
2012) which shows that the known "hard instances" for the Unique-Games problem
are in fact solved close to optimally by a constant level of the Lasserre SDP
hierarchy.
We continue the study of the power of SOS proofs in the context of difficult
optimization problems. In particular, we show that the Balanced-Separator
integrality gap instances proposed by Devanur et al.\ can have their optimal
value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of
the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi
Max-Cut integrality gap instances can have their optimum value certified by an
SOS proof. We show they can be certified to within a factor .952 ()
using a constant-degree proof. These investigations also raise an interesting
mathematical question: is there a constant-degree SOS proof of the Central
Limit Theorem?Comment: 34 page
Sum-of-squares lower bounds for planted clique
Finding cliques in random graphs and the closely related "planted" clique
variant, where a clique of size k is planted in a random G(n, 1/2) graph, have
been the focus of substantial study in algorithm design. Despite much effort,
the best known polynomial-time algorithms only solve the problem for k ~
sqrt(n).
In this paper we study the complexity of the planted clique problem under
algorithms from the Sum-of-squares hierarchy. We prove the first average case
lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the
SOS hierarchy cannot find a planted k-clique unless k > n^{1/2r} (up to
logarithmic factors). Thus, for any constant number of rounds planted cliques
of size n^{o(1)} cannot be found by this powerful class of algorithms. This is
shown via an integrability gap for the natural formulation of maximum clique
problem on random graphs for SOS and Lasserre hierarchies, which in turn follow
from degree lower bounds for the Positivestellensatz proof system.
We follow the usual recipe for such proofs. First, we introduce a natural
"dual certificate" (also known as a "vector-solution" or "pseudo-expectation")
for the given system of polynomial equations representing the problem for every
fixed input graph. Then we show that the matrix associated with this dual
certificate is PSD (positive semi-definite) with high probability over the
choice of the input graph.This requires the use of certain tools. One is the
theory of association schemes, and in particular the eigenspaces and
eigenvalues of the Johnson scheme. Another is a combinatorial method we develop
to compute (via traces) norm bounds for certain random matrices whose entries
are highly dependent; we hope this method will be useful elsewhere
Reconstruction of Support of a Measure From Its Moments
In this paper, we address the problem of reconstruction of support of a
measure from its moments. More precisely, given a finite subset of the moments
of a measure, we develop a semidefinite program for approximating the support
of measure using level sets of polynomials. To solve this problem, a sequence
of convex relaxations is provided, whose optimal solution is shown to converge
to the support of measure of interest. Moreover, the provided approach is
modified to improve the results for uniform measures. Numerical examples are
presented to illustrate the performance of the proposed approach.Comment: This has been submitted to the 53rd IEEE Conference on Decision and
Contro
A Complete Characterization of the Gap between Convexity and SOS-Convexity
Our first contribution in this paper is to prove that three natural sum of
squares (sos) based sufficient conditions for convexity of polynomials, via the
definition of convexity, its first order characterization, and its second order
characterization, are equivalent. These three equivalent algebraic conditions,
henceforth referred to as sos-convexity, can be checked by semidefinite
programming whereas deciding convexity is NP-hard. If we denote the set of
convex and sos-convex polynomials in variables of degree with
and respectively, then our main
contribution is to prove that if and
only if or or . We also present a complete
characterization for forms (homogeneous polynomials) except for the case
which is joint work with G. Blekherman and is to be published
elsewhere. Our result states that the set of convex forms in
variables of degree equals the set of sos-convex forms if
and only if or or . To prove these results, we present
in particular explicit examples of polynomials in
and
and forms in
and , and a
general procedure for constructing forms in from nonnegative but not sos forms in variables and degree .
Although for disparate reasons, the remarkable outcome is that convex
polynomials (resp. forms) are sos-convex exactly in cases where nonnegative
polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for
computer assisted proofs of the paper added to arXi
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
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