6 research outputs found
A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits
In this paper we study the complexity of constructing a hitting set for the
closure of VP, the class of polynomials that can be infinitesimally
approximated by polynomials that are computed by polynomial sized algebraic
circuits, over the real or complex numbers. Specifically, we show that there is
a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the
rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all
n-variate polynomials of degree-r that are the limit of size-s algebraic
circuits. Previously it was known that a random set of this size is a hitting
set, but a construction that is certified to work was only known in EXPSPACE
(or EXPH assuming the generalized Riemann hypothesis). As a corollary we get
that a host of other algebraic problems such as Noether Normalization Lemma,
can also be solved in PSPACE deterministically, where earlier only randomized
algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann
hypothesis) were known.
The proof relies on the new notion of a robust hitting set which is a set of
inputs such that any nonzero polynomial that can be computed by a polynomial
size algebraic circuit, evaluates to a not too small value on at least one
element of the set. Proving the existence of such a robust hitting set is the
main technical difficulty in the proof.
Our proof uses anti-concentration results for polynomials, basic tools from
algebraic geometry and the existential theory of the reals
The Complexity of the Distributed Constraint Satisfaction Problem
We study the complexity of the Distributed Constraint Satisfaction Problem
(DCSP) on a synchronous, anonymous network from a theoretical standpoint. In
this setting, variables and constraints are controlled by agents which
communicate with each other by sending messages through fixed communication
channels. Our results endorse the well-known fact from classical CSPs that the
complexity of fixed-template computational problems depends on the template's
invariance under certain operations. Specifically, we show that DCSP()
is polynomial-time tractable if and only if is invariant under
symmetric polymorphisms of all arities. Otherwise, there are no algorithms that
solve DCSP() in finite time. We also show that the same condition holds
for the search variant of DCSP. Collaterally, our results unveil a feature of
the processes' neighbourhood in a distributed network, its iterated degree,
which plays a major role in the analysis. We explore this notion establishing a
tight connection with the basic linear programming relaxation of a CSP.Comment: Full version of a STACS'21 pape
Know your audience
Distributed function computation is the problem, for a networked system of
autonomous agents, to collectively compute the value
of some input values, each initially private to one agent in the network. Here,
we study and organize results pertaining to distributed function computation in
anonymous networks, both for the static and the dynamic case, under a
communication model of directed and synchronous message exchanges, but with
varying assumptions in the degree of awareness or control that a single agent
has over its outneighbors.
Our main argument is three-fold. First, in the "blind broadcast" model, where
in each round an agent merely casts out a unique message without any knowledge
or control over its addressees, the computable functions are those that only
depend on the set of the input values, but not on their multiplicities or
relative frequencies in the input. Second, in contrast, when we assume either
that a) in each round, the agents know how many outneighbors they have; b) all
communications links in the network are bidirectional; or c) the agents may
address each of their outneighbors individually, then the set of computable
functions grows to contain all functions that depend on the relative
frequencies of each value in the input - such as the average - but not on their
multiplicities - thus, not the sum. Third, however, if one or several agents
are distinguished as leaders, or if the cardinality of the network is known,
then under any of the above three assumptions it becomes possible to recover
the complete multiset of the input values, and thus compute any function of the
distributed input as long as it is invariant under permutation of its
arguments. In the case of dynamic networks, we also discuss the impact of
multiple connectivity assumptions
Hitting Sets for Orbits of Circuit Classes and Polynomial Families
The orbit of an n-variate polynomial f(?) over a field ? is the set {f(A?+?) : A ? GL(n,?) and ? ? ??}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of:
1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials.
2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ? ?}, which is complete for arithmetic formulas.
3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials.
4) Polynomials computable by occur-once formulas