255 research outputs found
A Dichotomy Theorem for Homomorphism Polynomials
In the present paper we show a dichotomy theorem for the complexity of
polynomial evaluation. We associate to each graph H a polynomial that encodes
all graphs of a fixed size homomorphic to H. We show that this family is
computable by arithmetic circuits in constant depth if H has a loop or no edge
and that it is hard otherwise (i.e., complete for VNP, the arithmetic class
related to #P). We also demonstrate the hardness over the rational field of cut
eliminator, a polynomial defined by B\"urgisser which is known to be neither VP
nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is
the class of polynomials computable by arithmetic circuit of polynomial size)
Separations of Matroid Freeness Properties
Properties of Boolean functions on the hypercube invariant with respect to
linear transformations of the domain are among the most well-studied properties
in the context of property testing. In this paper, we study the fundamental
class of linear-invariant properties called matroid freeness properties. These
properties have been conjectured to essentially coincide with all testable
linear-invariant properties, and a recent sequence of works has established
testability for increasingly larger subclasses. One question left open,
however, is whether the infinitely many syntactically different properties
recently shown testable in fact correspond to new, semantically distinct ones.
This is a crucial issue since it has also been shown that there exist
subclasses of these properties for which an infinite set of syntactically
different representations collapse into one of a small, finite set of
properties, all previously known to be testable.
An important question is therefore to understand the semantics of matroid
freeness properties, and in particular when two syntactically different
properties are truly distinct. We shed light on this problem by developing a
method for determining the relation between two matroid freeness properties P
and Q. Furthermore, we show that there is a natural subclass of matroid
freeness properties such that for any two properties P and Q from this
subclass, a strong dichotomy must hold: either P is contained in Q or the two
properties are "well separated." As an application of this method, we exhibit
new, infinite hierarchies of testable matroid freeness properties such that at
each level of the hierarchy, there are functions that are far from all
functions lying in lower levels of the hierarchy. Our key technical tool is an
apparently new notion of maps between linear matroids, called matroid
homomorphisms, that might be of independent interest
On the expressive power of homomorphism counts
A classical result by Lovász asserts that two graphs G and H are isomorphic if and only if they have the same left profile, that is, for every graph F, the number of homomorphisms from F to G coincides with the number of homomorphisms from F to H. Dvorák and later on Dell, Grohe, and Rattan showed that restrictions of the left profile to a class of graphs can capture several different relaxations of isomorphism, including equivalence in counting logics with a fixed number of variables (which contains fractional isomorphism as a special case) and co-spectrality (i.e., two graphs having the same characteristic polynomial). On the other side, a result by Chaudhuri and Vardi asserts that isomorphism is also captured by the right profile, that is, two graphs G and H are isomorphic if and only if for every graph F, the number of homomorphisms from G to F coincides with the number of homomorphisms from H to F. In this paper, we embark on a study of the restrictions of the right profile by investigating relaxations of isomorphism that can or cannot be captured by restricting the right profile to a fixed class of graphs. Our results unveil striking differences between the expressive power of the left profile and the right profile. We show that fractional isomorphism, equivalence in counting logics with a fixed number of variables, and co-spectrality cannot be captured by restricting the right profile to a class of graphs. In the opposite direction, we show that chromatic equivalence cannot be captured by restricting the left profile to a class of graphs, while, clearly, it can be captured by restricting the right profile to the class of all cliques.The research of Albert Atserias was partially supported by MICIN project PID2019-109137GBC22 (PROOFS). The research of Phokion Kolaitis and Wei-Lin Wu was partially supported by NSF Grant 1814152.Peer ReviewedPostprint (author's final draft
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