Semi-Transitive Orientations and Word-Representable Graphs

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.Comment: arXiv admin note: text overlap with arXiv:0810.031

Double Threshold Digraphs

A semiorder is a model of preference relations where each element x is associated with a utility value alpha(x), and there is a threshold t such that y is preferred to x iff alpha(y) - alpha(x) > t. These are motivated by the notion that there is some uncertainty in the utility values we assign an object or that a subject may be unable to distinguish a preference between objects whose values are close. However, they fail to model the well-known phenomenon that preferences are not always transitive. Also, if we are uncertain of the utility values, it is not logical that preference is determined absolutely by a comparison of them with an exact threshold. We propose a new model in which there are two thresholds, t_1 and t_2; if the difference alpha(y) - alpha(x) is less than t_1, then y is not preferred to x; if the difference is greater than t_2 then y is preferred to x; if it is between t_1 and t_2, then y may or may not be preferred to x. We call such a relation a (t_1,t_2) double-threshold semiorder, and the corresponding directed graph G = (V,E) a (t_1,t_2) double-threshold digraph. Every directed acyclic graph is a double-threshold digraph; increasing bounds on t_2/t_1 give a nested hierarchy of subclasses of the directed acyclic graphs. In this paper we characterize the subclasses in terms of forbidden subgraphs, and give algorithms for finding an assignment of utility values that explains the relation in terms of a given (t_1,t_2) or else produces a forbidden subgraph, and finding the minimum value lambda of t_2/t_1 that is satisfiable for a given directed acyclic graph. We show that lambda gives a useful measure of the complexity of a directed acyclic graph with respect to several optimization problems that are NP-hard on arbitrary directed acyclic graphs

On multivariate quantiles under partial orders

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, generalize the concept of efficient frontier, and can measure dispersion from the partial order perspective. We also study several statistical aspects of partial quantiles. We provide estimators, associated rates of convergence, and asymptotic distributions that hold uniformly over a continuum of quantile indices. Furthermore, we provide procedures that can restore monotonicity properties that might have been disturbed by estimation error, establish computational complexity bounds, and point out a concentration of measure phenomenon (the latter under independence and the componentwise natural order). Finally, we illustrate the concepts by discussing several theoretical examples and simulations. Empirical applications to compare intake nutrients within diets, to evaluate the performance of investment funds, and to study the impact of policies on tobacco awareness are also presented to illustrate the concepts and their use.Comment: Published in at http://dx.doi.org/10.1214/10-AOS863 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Satisfiability of Temporal Logics with Concrete Domains

Temporal logics are a very popular family of logical languages, used to specify properties of abstracted systems. In the last few years, many extensions of temporal logics have been proposed, in order to address the need to express more than just abstract properties. In our work we study temporal logics extended by local constraints, which allow to express quantitative properties on data values from an arbitrary relational structure called the concrete domain. An example of concrete domain can be (Z, <, =), where the integers are considered as a relational structure over the binary order relation and the equality relation. Formulas of temporal logics with constraints are evaluated on data-words or data-trees, in which each node or position is labeled by a vector of data from the concrete domain. We call the constraints local because they can only compare values at a fixed distance inside such models. Several positive results regarding the satisfiability of LTL (linear temporal logic) with constraints over the integers have been established in the past years, while the corresponding results for branching time logics were only partial. In this work we prove that satisfiability of CTL* (computation tree logic) with constraints over the integers is decidable and also lift this result to ECTL*, a proper extension of CTL*. We also consider other classes of concrete domains, particularly ones that are \"tree-like\". We consider semi-linear orders, ordinal trees and trees of a fixed height, and prove decidability in this framework as well. At the same time we prove that our method cannot be applied in the case of the infinite binary tree or the infinitely branching infinite tree. We also look into extending the expressiveness of our logic adding non-local constraints, and find that this leads to undecidability of the satisfiability problem, even on very simple domains like (Z, <, =). We then find a way to restrict the power of the non-local constraints to regain decidability

Byzantine Approximate Agreement on Graphs

Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors\u27 input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures

RiskStructures : A Design Algebra for Risk-Aware Machines

Machines, such as mobile robots and delivery drones, incorporate controllers responsible for a task while handling risk (e.g. anticipating and mitigating hazards; and preventing and alleviating accidents). We refer to machines with this capability as risk-aware machines. Risk awareness includes robustness and resilience, and complicates monitoring (i.e., introspection, sensing, prediction), decision making, and control. From an engineering perspective, risk awareness adds a range of dependability requirements to system assurance. Such assurance mandates a correct-by-construction approach to controller design, based on mathematical theory. We introduce RiskStructures, an algebraic framework for risk modelling intended to support the design of safety controllers for risk-aware machines. Using the concept of a risk factor as a modelling primitive, this framework provides facilities to construct, examine, and assure these controllers. We prove desirable algebraic properties of these facilities, and demonstrate their applicability by using them to specify key aspects of safety controllers for risk-aware automated driving and collaborative robots

On the perfect orderability of unions of two graphs

A graph G is perfectly orderable if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph if it contains no P4 , 2K2 or C4 as induced subgraph. A theorem of Chvatal, Hoang, Mahadev and de Werra states that a graph is perfectly orderable if it can be written as the union of two threshold graphs. In this thesis, we investigate possible generalizations of the above theorem. We conjecture that if G is the union of two graphs G1 and G2 then G is perfectly orderable whenever (i) G1 and G2 are both P4 -free and 2K2-free, or (ii) G1 is P4-free, 2K2-free and G2 is P4 -free, C4 -free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter-example to our conjecture and we prove, jointly with Hoang, a special case of (i): if G1 and G2 are two edge disjoint graphs that are P4 -free and 2K2 -free then the union of G1 and G2 is perfectly orderable