Barnaby Martin

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 whose complexity remains open.

Towards a trichotomy for quantified H -coloring

Hell and Nesetřil proved that the H-colouring problem is NP-complete if, and only if, H is bipartite. In this paper, we investigate the complexity of the quantified H-colouring problem (a restriction of the quantified constraint satisfaction problem to undirected graphs). We introduce this problem using a new two player colouring game. We prove that the quantified H-colouring problem is: 1. tractable, if H is bipartite; 2. NP-complete, if H is not bipartite and not connected; and, 3. Pspace-complete, if H is connected and has a unique cycle, which is of odd length. We conjecture that the last case extends to all non-bipartite connected graphs.

The computational complexity of disconnected cut and 2K 2 -partition

For a connected graph G = (V,E), a subset U ⊆ V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is NP-complete. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K2-partition, testing if a graph allows a vertex-surjective homomorphism to the reflexive 4-cycle and testing if a graph has a spanning subgraph that consists of at most two bicliques. Hence, as an immediate consequence, these three decision problems are NP-complete as well. This settles an open problem frequently posed in each of the four settings.

On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction

The universal-algebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates. Our first result is an exact characterization of those CSPs that can be formulated with (a finite or) an omega-categorical template. The universal-algebraic approach relies on the fact that in finite or omega-categorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. In this paper, we present results that can be used to study the computational complexity of CSPs with arbitrary infinite templates. Specifically, we prove that every CSP can be formulated with a template A such that a relation is primitive positive definable in A if and only if it is first-order definable on A and preserved by the infinitary polymorphisms of A. We present applications of our general results to the description and analysis of the computational complexity of CSPs. In particular, we present a polymorphism-based description of those CSPs that are first-order definable (and therefore can be solved in polynomial-time), and give general hardness criteria based on the absence of polymorphisms that depend on more than one argument.

Constraint Satisfaction Problems over the Integers with Successor

A distance constraint satisfaction problem is a constraint satisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over \(({\mathbb Z};{{\mathrm{succ}}})\), i.e., over the integers with the successor function. Our main result says that every distance CSP is in P or NP-complete, unless it can be formulated as a finite domain CSP in which case the computational complexity is not known in general.

First-Order Model Checking Problems Parameterized by the Model

We study the complexity of the model checking problem, for fixed models A, over certain fragments

Finding vertex-surjective graph homomorphisms

\mathcal{L}

The packing chromatic number of the infinite square lattice is between 13 and 15

of first-order logic, obtained by restricting which of the quantifiers and boolean connectives we permit. These are sometimes known as the expression complexities of

The computational complexity of disconnected cut and 2 K 2 -partition

\mathcal{L}

QCSP on partially reflexive forests

. We obtain various full and partial complexity classification theorems for these logics