2,083 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
Curious properties of free hypergraph C*-algebras
A finite hypergraph consists of a finite set of vertices and a
collection of subsets which we consider as partition
of unity relations between projection operators. These partition of unity
relations freely generate a universal C*-algebra, which we call the "free
hypergraph C*-algebra" . General free hypergraph C*-algebras were first
studied in the context of quantum contextuality. As special cases, the class of
free hypergraph C*-algebras comprises quantum permutation groups, maximal group
C*-algebras of graph products of finite cyclic groups, and the C*-algebras
associated to quantum graph homomorphism, isomorphism, and colouring.
Here, we conduct the first systematic study of aspects of free hypergraph
C*-algebras. We show that they coincide with the class of finite colimits of
finite-dimensional commutative C*-algebras, and also with the class of
C*-algebras associated to synchronous nonlocal games. We had previously shown
that it is undecidable to determine whether is nonzero for given .
We now show that it is also undecidable to determine whether a given
is residually finite-dimensional, and similarly whether it only has
infinite-dimensional representations, and whether it has a tracial state. It
follows that for each one of these properties, there is such that the
question whether has this property is independent of the ZFC axioms,
assuming that these are consistent. We clarify some of the subtleties
associated with such independence results in an appendix.Comment: 19 pages. v2: minor clarifications. v3: terminology 'free hypergraph
C*-algebra', added Remark 2.2
The complexity of the list homomorphism problem for graphs
We completely classify the computational complexity of the list H-colouring
problem for graphs (with possible loops) in combinatorial and algebraic terms:
for every graph H the problem is either NP-complete, NL-complete, L-complete or
is first-order definable; descriptive complexity equivalents are given as well
via Datalog and its fragments. Our algebraic characterisations match important
conjectures in the study of constraint satisfaction problems.Comment: 12 pages, STACS 201
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