Flexible constraint satisfiability and a problem in semigroup theory

We examine some flexible notions of constraint satisfaction, observing some relationships between model theoretic notions of universal Horn class membership and robust satisfiability. We show the \texttt{NP}-completeness of $2$-robust monotone 1-in-3 3SAT in order to give very small examples of finite algebras with \texttt{NP}-hard variety membership problem. In particular we give a $3$-element algebra with this property, and solve a widely stated problem by showing that the $6$-element Brandt monoid has \texttt{NP}-hard variety membership problem. These are the smallest possible sizes for a general algebra and a semigroup to exhibit \texttt{NP}-hardness for the membership problem of finite algebras in finitely generated varieties

On the complexity of k-rainbow cycle colouring problems

An edge-coloured cycle is $rainbow$ if all edges of the cycle have distinct colours. For $k\geq 1$, let $\mathcal{F}_{k}$ denote the family of all graphs with the property that any $k$ vertices lie on a cycle. For $G\in \mathcal{F}_{k}$, a $k$-$rainbow$ $cycle$ $colouring$ of $G$ is an edge-colouring such that any $k$ vertices of $G$ lie on a rainbow cycle in $G$. The $k$-$rainbow$ $cycle$ $index$ of $G$, denoted by $crx_{k}(G)$, is the minimum number of colours needed in a $k$-rainbow cycle colouring of $G$. In this paper, we restrict our attention to the computational aspects of $k$-rainbow cycle colouring. First, we prove that the problem of deciding whether $crx_1=3$ can be solved in polynomial time, but that of deciding whether $crx_1 \leq k$ is NP-Complete, where $k\geq 4$. Then we show that the problem of deciding whether $crx_2=3$ can be solved in polynomial time, but those of deciding whether $crx_2 \leq 4$ or $5$ are NP-Complete. Furthermore, we also consider the cases of $crx_3=3$ and $crx_3 \leq 4$. Finally, We prove that the problem of deciding whether a given edge-colouring (with an unbounded number of colours) of a graph is a $k$-rainbow cycle colouring, is NP-Complete for $k=1$, $2$ and $3$, respectively. Some open problems for further study are mentioned.Comment: 18 pages, to appear in Discrete Applied Mathematic

Role colouring graphs in hereditary classes

We study the computational complexity of computing role colourings of graphs in hereditary classes. We are interested in describing the family of hereditary classes on which a role colouring with k colours can be computed in polynomial time. In particular, we wish to describe the boundary between the "hard" and "easy" classes. The notion of a boundary class has been introduced by Alekseev in order to study such boundaries. Our main results are a boundary class for the k-role colouring problem and the related k-coupon colouring problem which has recently received a lot of attention in the literature. The latter result makes use of a technique for generating regular graphs of arbitrary girth which may be of independent interest

Algebraic approach to promise constraint satisfaction

The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the last 20 years. A new version of the CSP, the promise CSP (PCSP) has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms --- high-dimensional symmetries of solution spaces --- to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases. The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this paper we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem, and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a "measure of symmetry" that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by improving the state-of-the-art in approximate graph colouring: we show that, for any $k\geq 3$, it is NP-hard to find a $(2k-1)$-colouring of a given $k$-colourable graph.Comment: Extended version (73 pages). Preliminary versions of parts of this paper were published in the proceedings of STOC 2019 and LICS 201

Rainbow Colouring of Split Graphs

A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one rainbow path. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Between them, Chakraborty et al. [J. Comb. Optim., 2011] and Ananth et al. [FSTTCS, 2012] have shown that for every integer k, k \geq 2, it is NP-complete to decide whether a given graph can be rainbow coloured using k colours. A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. Chandran and Rajendraprasad have shown that the problem of deciding whether a given split graph G can be rainbow coloured using 3 colours is NP-complete and further have described a linear time algorithm to rainbow colour any split graph using at most one colour more than the optimum [COCOON, 2012]. In this article, we settle the computational complexity of the problem on split graphs and thereby discover an interesting dichotomy. Specifically, we show that the problem of deciding whether a given split graph can be rainbow coloured using k colours is NP-complete for k \in {2,3}, but can be solved in polynomial time for all other values of k.Comment: This is the full version of a paper to be presented at ICGT 2014. This complements the results in arXiv:1205.1670 (which were presented in COCOON 2013), and both will be merged into a single journal submissio

Critical Vertices and Edges in HH-free Graphs

A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by 1. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively. We give a complexity dichotomy for both problems restricted to $H$-free graphs, that is, graphs with no induced subgraph isomorphic to $H$. Moreover, we show that an edge is critical if and only if its contraction reduces the chromatic number by 1. Hence, we also obtain a complexity dichotomy for the problem of deciding if a graph has an edge whose contraction reduces the chromatic number by 1

Colouring and Covering Nowhere Dense Graphs

It was shown by Grohe et al. that nowhere dense classes of graphs admit sparse neighbourhood covers of small degree. We show that a monotone graph class admits sparse neighbourhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak colouring numbers. The core results of this paper are various lower and upper bounds on the weak colouring numbers and other, closely related generalised colouring numbers. We prove tight bounds for these numbers on graphs of bounded tree width. We clarify and tighten the relation between the expansion (in the sense of "bounded expansion" as defined by Nesetril and Ossona de Mendez) and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the generalised colouring numbers. Finally, we show that computing weak r-colouring numbers is NP-complete for all r>2

The complexity of signed graph and edge-coloured graph homomorphisms

We study homomorphism problems of signed graphs from a computational point of view. A signed graph $(G,\Sigma)$ is a graph $G$ where each edge is given a sign, positive or negative; $\Sigma\subseteq E(G)$ denotes the set of negative edges. Thus, $(G, \Sigma)$ is a $2$-edge-coloured graph with the property that the edge-colours, $\{+, -\}$, form a group under multiplication. Central to the study of signed graphs is the operation of switching at a vertex, that results in changing the sign of each incident edge. We study two types of homomorphisms of a signed graph $(G,\Sigma)$ to a signed graph $(H,\Pi)$: ec-homomorphisms and s-homomorphisms. Each is a standard graph homomorphism of $G$ to $H$ with some additional constraint. In the former, edge-signs are preserved. In the latter, edge-signs are preserved after the switching operation has been applied to a subset of vertices of $G$. We prove a dichotomy theorem for s-homomorphism problems for a large class of (fixed) target signed graphs $(H,\Pi)$. Specifically, as long as $(H,\Pi)$ does not contain a negative (respectively a positive) loop, the problem is polynomial-time solvable if the core of $(H,\Pi)$ has at most two edges, and is NP-complete otherwise. (Note that this covers all simple signed graphs.) The same dichotomy holds if $(H,\Pi)$ has no negative digons, and we conjecture that it holds always. In our proofs, we reduce s-homomorphism problems to certain ec-homomorphism problems, for which we are able to show a dichotomy. In contrast, we prove that a dichotomy theorem for ec-homomorphism problems (even when restricted to bipartite target signed graphs) would settle the dichotomy conjecture of Feder and Vardi.Comment: 21 pages; 6 figures. In this version, we have adopted some changes in terminology and notatio

Nauty in Macaulay2

We introduce a new Macaulay2 package, Nauty, which gives access to powerful methods on graphs provided by the software nauty by Brendan McKay. The primary motivation for accessing nauty is to determine if two graphs are isomorphic. We also implement methods to generate families of graphs restricted in various ways using tools provided with the software nauty.Comment: 4 pages; second updated for clarit

Colourings, Homomorphisms, and Partitions of Transitive Digraphs

We investigate the complexity of generalizations of colourings (acyclic colourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), for the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.Comment: 13 pages, 3 figure