51 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
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
Limitations of Algebraic Approaches to Graph Isomorphism Testing
We investigate the power of graph isomorphism algorithms based on algebraic
reasoning techniques like Gr\"obner basis computation. The idea of these
algorithms is to encode two graphs into a system of equations that are
satisfiable if and only if if the graphs are isomorphic, and then to (try to)
decide satisfiability of the system using, for example, the Gr\"obner basis
algorithm. In some cases this can be done in polynomial time, in particular, if
the equations admit a bounded degree refutation in an algebraic proof systems
such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on
the polynomial calculus degree over all fields of characteristic different from
2 and also linear lower bounds for the degree of Positivstellensatz calculus
derivations.
We compare this approach to recently studied linear and semidefinite
programming approaches to isomorphism testing, which are known to be related to
the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the
power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system
that lies between degree-k Nullstellensatz and degree-k polynomial calculus
Tight Size-Degree Bounds for Sums-of-Squares Proofs
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively
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