9 research outputs found
Polynomials that Sign Represent Parity and Descartes' Rule of Signs
A real polynomial sign represents if
for every , the sign of equals
. Such sign representations are well-studied in computer
science and have applications to computational complexity and computational
learning theory. In this work, we present a systematic study of tradeoffs
between degree and sparsity of sign representations through the lens of the
parity function. We attempt to prove bounds that hold for any choice of set
. We show that sign representing parity over with the
degree in each variable at most requires sparsity at least . We show
that a tradeoff exists between sparsity and degree, by exhibiting a sign
representation that has higher degree but lower sparsity. We show a lower bound
of on the sparsity of polynomials of any degree representing
parity over . We prove exact bounds on the sparsity of such
polynomials for any two element subset . The main tool used is Descartes'
Rule of Signs, a classical result in algebra, relating the sparsity of a
polynomial to its number of real roots. As an application, we use bounds on
sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with
a Threshold Gate at the top. We use this to give a simple proof that such
circuits need size to compute parity, which improves the previous bound
of due to Goldmann (1997). We show a tight lower bound of
for the inner product function over .Comment: To appear in Computational Complexit
Entanglement-assisted zero-error source-channel coding
We study the use of quantum entanglement in the zero-error source-channel
coding problem. Here, Alice and Bob are connected by a noisy classical one-way
channel, and are given correlated inputs from a random source. Their goal is
for Bob to learn Alice's input while using the channel as little as possible.
In the zero-error regime, the optimal rates of source codes and channel codes
are given by graph parameters known as the Witsenhausen rate and Shannon
capacity, respectively. The Lov\'asz theta number, a graph parameter defined by
a semidefinite program, gives the best efficiently-computable upper bound on
the Shannon capacity and it also upper bounds its entanglement-assisted
counterpart. At the same time it was recently shown that the Shannon capacity
can be increased if Alice and Bob may use entanglement.
Here we partially extend these results to the source-coding problem and to
the more general source-channel coding problem. We prove a lower bound on the
rate of entanglement-assisted source-codes in terms Szegedy's number (a
strengthening of the theta number). This result implies that the theta number
lower bounds the entangled variant of the Witsenhausen rate. We also show that
entanglement can allow for an unbounded improvement of the asymptotic rate of
both classical source codes and classical source-channel codes. Our separation
results use low-degree polynomials due to Barrington, Beigel and Rudich,
Hadamard matrices due to Xia and Liu and a new application of remote state
preparation.Comment: Title has been changed. Previous title was 'Zero-error source-channel
coding with entanglement'. Corrected an error in Lemma 1.
Violating Constant Degree Hypothesis Requires Breaking Symmetry
The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to
study some extensions of -groups by nilpotent groups and the power of these
groups in a certain computational model. In its simplest formulation, it
establishes exponential lower bounds for circuits computing AND of unbounded arity (for
constant integers and a prime ). While it has been proved in some
special cases (including ), it remains wide open in its general form for
over 30 years.
In this paper we prove that the hypothesis holds when we restrict our
attention to symmetric circuits with being a prime. While we build upon
techniques by Grolmusz and Tardos (2000), we have to prove a new symmetric
version of their Degree Decreasing Lemma and apply it in a highly non-trivial
way. Moreover, to establish the result we perform a careful analysis of
automorphism groups of subcircuits and
study the periodic behaviour of the computed functions.
Finally, our methods also yield lower bounds when is treated as a
function of
Constructing Ramsey Graphs from Boolean Function Representations
Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdős’ probabilistic argument shows the existence of graphs on 2^n vertices with no clique or independent set of size 2n, the best explicit constructions achieve a far weaker bound. Constructing Ramsey graphs is closely related to polynomial representations of Boolean functions; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all ¦¨§�©��¥� points in except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [17] and Alon [2] are captured by this framework; they can all be derived from variou
Constructing Ramsey graphs from Boolean function representations
Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdős ’ probabilistic argument shows the existence of graphs on ¡£ ¢ vertices with no clique or independent set of size ¡¥¤, the best explicit constructions achieve a far weaker bound. Constructing Ramsey graphs is closely related to polynomial representations of Boolean functions; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in except the origin. We give a simple construction of a Ramsey graph using such polynomials. Fur-thermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [17] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials. � Supported by NSF grant CCR-3606B64
Quantum entanglement: insights via graph parameters and conic optimization
In this PhD thesis we study the effects of quantum entanglement, one of quantum mechanics most peculiar features, in nonlocal games and communication problems in zero-error information theory. A nonlocal game is a thought experiment in which two cooperating players, who are forbidden to communicate, want to perform a certain task.
Zero-error information theory is the mathematical field that studies communication problems where no error is tolerated. The unifying link among the various scenarios we consider is their combinatorial nature and in particular their reformulations as graph theoretical problems, mainly concerning the chromatic and stability numbers and some quantum generalizations thereof.
In this thesis we propose a novel approach to the study of these quantum graph parameters using the paradigm of conic optimization. For that, we introduce and study the completely positive semidefinite cone, a new matrix cone consisting of all symmetric matrices that admit a Gram representation by positive semidefinite matrices.
Furthermore, we investigate whether entanglement allows for better-than-classical communication schemes in some well-known problems from zero-error information theory. For example we study the channel coding problem, which asks a sender to transmit data reliably to a receiver in the presence of noise, as well as some of its generalizations