1,839 research outputs found
The Complexity of Surjective Homomorphism Problems -- a Survey
We survey known results about the complexity of surjective homomorphism
problems, studied in the context of related problems in the literature such as
list homomorphism, retraction and compaction. In comparison with these
problems, surjective homomorphism problems seem to be harder to classify and we
examine especially three concrete problems that have arisen from the
literature, two of which remain of open complexity
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
We prove an algebraic preservation theorem for positive Horn definability in
aleph-zero categorical structures. In particular, we define and study a
construction which we call the periodic power of a structure, and define a
periomorphism of a structure to be a homomorphism from the periodic power of
the structure to the structure itself. Our preservation theorem states that,
over an aleph-zero categorical structure, a relation is positive Horn definable
if and only if it is preserved by all periomorphisms of the structure. We give
applications of this theorem, including a new proof of the known complexity
classification of quantified constraint satisfaction on equality templates
Exact Algorithm for Graph Homomorphism and Locally Injective Graph Homomorphism
For graphs and , a homomorphism from to is a function , which maps vertices adjacent in to adjacent vertices
of . A homomorphism is locally injective if no two vertices with a common
neighbor are mapped to a single vertex in . Many cases of graph homomorphism
and locally injective graph homomorphism are NP-complete, so there is little
hope to design polynomial-time algorithms for them. In this paper we present an
algorithm for graph homomorphism and locally injective homomorphism working in
time , where is the bandwidth of the
complement of
Topological Birkhoff
One of the most fundamental mathematical contributions of Garrett Birkhoff is
the HSP theorem, which implies that a finite algebra B satisfies all equations
that hold in a finite algebra A of the same signature if and only if B is a
homomorphic image of a subalgebra of a finite power of A. On the other hand, if
A is infinite, then in general one needs to take an infinite power in order to
obtain a representation of B in terms of A, even if B is finite.
We show that by considering the natural topology on the functions of A and B
in addition to the equations that hold between them, one can do with finite
powers even for many interesting infinite algebras A. More precisely, we prove
that if A and B are at most countable algebras which are oligomorphic, then the
mapping which sends each function from A to the corresponding function in B
preserves equations and is continuous if and only if B is a homomorphic image
of a subalgebra of a finite power of A.
Our result has the following consequences in model theory and in theoretical
computer science: two \omega-categorical structures are primitive positive
bi-interpretable if and only if their topological polymorphism clones are
isomorphic. In particular, the complexity of the constraint satisfaction
problem of an \omega-categorical structure only depends on its topological
polymorphism clone.Comment: 21 page
Locally constrained homomorphisms on graphs of bounded treewidth and bounded degree.
A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even when G has pathwidth at most 5, 4 or 2, respectively, or when both G and H have maximum degree 3. We complement these hardness results by showing that the three problems are polynomial-time solvable if G has bounded treewidth and in addition G or H has bounded maximum degree
Approximation for Maximum Surjective Constraint Satisfaction Problems
Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are
computational problems where we are given a set of variables denoting values
from a finite domain B and a set of constraints on the variables. A solution to
such a problem is a surjective mapping from the set of variables to B such that
the number of satisfied constraints is maximized. We study the approximation
performance that can be acccchieved by algorithms for these problems, mainly by
investigating their relation with Max-CSPs (which are the corresponding
problems without the surjectivity requirement). Our work gives a complexity
dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the
assumption that there is a complexity dichotomy for Max-CSP(B) between PO and
APX-complete, which has already been proved on the Boolean domain and 3-element
domains
Twisted Alexander polynomials and a partial order on the set of prime knots
We give a survey of some recent papers by the authors and Masaaki Wada
relating the twisted Alexander polynomial with a partial order on the set of
prime knots. We also give examples and pose open problems.Comment: This is the version published by Geometry & Topology Monographs on 25
February 200
Relations Between Graphs
Given two graphs G and H, we ask under which conditions there is a relation R
that generates the edges of H given the structure of graph G. This construction
can be seen as a form of multihomomorphism. It generalizes surjective
homomorphisms of graphs and naturally leads to notions of R-retractions,
R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are
unique up to isomorphism and can be computed in polynomial time.Comment: accepted by Ars Mathematica Contemporane
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