58,062 research outputs found
The Complexity of Weighted Boolean #CSP with Mixed Signs
We give a complexity dichotomy for the problem of computing the partition
function of a weighted Boolean constraint satisfaction problem. Such a problem
is parameterized by a set of rational-valued functions, which generalize
constraints. Each function assigns a weight to every assignment to a set of
Boolean variables. Our dichotomy extends previous work in which the weight
functions were restricted to being non-negative. We represent a weight function
as a product of the form (-1)^s g, where the polynomial s determines the sign
of the weight and the non-negative function g determines its magnitude. We show
that the problem of computing the partition function (the sum of the weights of
all possible variable assignments) is in polynomial time if either every weight
function can be defined by a "pure affine" magnitude with a quadratic sign
polynomial or every function can be defined by a magnitude of "product type"
with a linear sign polynomial. In all other cases, computing the partition
function is FP^#P-complete.Comment: 24 page
The Dual Polynomial of Bipartite Perfect Matching
We obtain a description of the Boolean dual function of the Bipartite Perfect
Matching decision problem, as a multilinear polynomial over the Reals. We show
that in this polynomial, both the number of monomials and the magnitude of
their coefficients are at most exponential in . As an
application, we obtain a new upper bound of on the approximate degree of the bipartite perfect matching function,
improving the previous best known bound of . We deduce
that, beyond a factor, the polynomial method
cannot be used to improve the lower bound on the bounded-error quantum query
complexity of bipartite perfect matching
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
Low-Sensitivity Functions from Unambiguous Certificates
We provide new query complexity separations against sensitivity for total
Boolean functions: a power separation between deterministic (and even
randomized or quantum) query complexity and sensitivity, and a power
separation between certificate complexity and sensitivity. We get these
separations by using a new connection between sensitivity and a seemingly
unrelated measure called one-sided unambiguous certificate complexity
(). We also show that is lower-bounded by fractional block
sensitivity, which means we cannot use these techniques to get a
super-quadratic separation between and . We also provide a
quadratic separation between the tree-sensitivity and decision tree complexity
of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and
Wigderson (CCC 2016).
Along the way, we give a power separation between certificate
complexity and one-sided unambiguous certificate complexity, improving the
power separation due to G\"o\"os (FOCS 2015). As a consequence, we
obtain an improved lower-bound on the
co-nondeterministic communication complexity of the Clique vs. Independent Set
problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and
Avishay Tal as author
On Symmetric Circuits and Fixed-Point Logics
We study properties of relational structures such as graphs that are decided
by families of Boolean circuits. Circuits that decide such properties are
necessarily invariant to permutations of the elements of the input structures.
We focus on families of circuits that are symmetric, i.e., circuits whose
invariance is witnessed by automorphisms of the circuit induced by the
permutation of the input structure. We show that the expressive power of such
families is closely tied to definability in logic. In particular, we show that
the queries defined on structures by uniform families of symmetric Boolean
circuits with majority gates are exactly those definable in fixed-point logic
with counting. This shows that inexpressibility results in the latter logic
lead to lower bounds against polynomial-size families of symmetric circuits.Comment: 22 pages. Full version of a paper to appear in STACS 201
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