49 research outputs found
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
201
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
Algorithms for group isomorphism via group extensions and cohomology
The isomorphism problem for finite groups of order n (GpI) has long been
known to be solvable in time, but only recently were
polynomial-time algorithms designed for several interesting group classes.
Inspired by recent progress, we revisit the strategy for GpI via the extension
theory of groups.
The extension theory describes how a normal subgroup N is related to G/N via
G, and this naturally leads to a divide-and-conquer strategy that splits GpI
into two subproblems: one regarding group actions on other groups, and one
regarding group cohomology. When the normal subgroup N is abelian, this
strategy is well-known. Our first contribution is to extend this strategy to
handle the case when N is not necessarily abelian. This allows us to provide a
unified explanation of all recent polynomial-time algorithms for special group
classes.
Guided by this strategy, to make further progress on GpI, we consider
central-radical groups, proposed in Babai et al. (SODA 2011): the class of
groups such that G mod its center has no abelian normal subgroups. This class
is a natural extension of the group class considered by Babai et al. (ICALP
2012), namely those groups with no abelian normal subgroups. Following the
above strategy, we solve GpI in time for central-radical
groups, and in polynomial time for several prominent subclasses of
central-radical groups. We also solve GpI in time for
groups whose solvable normal subgroups are elementary abelian but not
necessarily central. As far as we are aware, this is the first time there have
been worst-case guarantees on a -time algorithm that tackles
both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation,
with some new result
Polynomial Identity Testing and the Ideal Proof System: PIT is in NP if and only if IPS can be p-simulated by a Cook-Reckhow proof system
The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018)
is an algebraic proof system that uses algebraic circuits to refute the
solvability of unsatisfiable systems of polynomial equations. One potential
drawback of IPS is that verifying an IPS proof is only known to be doable using
Polynomial Identity Testing (PIT), which is solvable by a randomized algorithm,
but whose derandomization, even into NSUBEXP, is equivalent to strong lower
bounds. However, the circuits that are used in IPS proofs are not arbitrary,
and it is conceivable that one could get around general PIT by leveraging some
structure in these circuits. This proposal may be even more tempting when IPS
is used as a proof system for Boolean Unsatisfiability, where the equations
themselves have additional structure.
Our main result is that, on the contrary, one cannot get around PIT as above:
we show that IPS, even as a proof system for Boolean Unsatisfiability, can be
p-simulated by a deterministically verifiable (Cook-Reckhow) proof system if
and only if PIT is in NP. We use our main result to propose a potentially new
approach to derandomizing PIT into NP