Recognizing Visibility Graphs of Polygons with Holes and Internal-External Visibility Graphs of Polygons

Visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge between their corresponding vertices in the graph. Two vertices of a polygon see each other if and only if their connecting line segment completely lies inside the polygon, and they are externally visible if and only if this line segment completely lies outside the polygon. Recognizing visibility graphs is the problem of deciding whether there is a simple polygon whose visibility graph is isomorphic to a given input graph. This problem is well-known and well-studied, but yet widely open in geometric graphs and computational geometry. Existential Theory of the Reals is the complexity class of problems that can be reduced to the problem of deciding whether there exists a solution to a quantifier-free formula F(X1,X2,...,Xn), involving equalities and inequalities of real polynomials with real variables. The complete problems for this complexity class are called Existential Theory of the Reals Complete. In this paper we show that recognizing visibility graphs of polygons with holes is Existential Theory of the Reals Complete. Moreover, we show that recognizing visibility graphs of simple polygons when we have the internal and external visibility graphs, is also Existential Theory of the Reals Complete.Comment: Sumbitted to COCOON2018 Conferenc

An algorithmic proof for the completeness of two-dimensional Ising model

We show that the two dimensional Ising model is complete, in the sense that the partition function of any lattice model on any graph is equal to the partition function of the 2D Ising model with complex coupling. The latter model has all its spin-spin coupling equal to i\pi/4 and all the parameters of the original model are contained in the local magnetic fields of the Ising model. This result has already been derived by using techniques from quantum information theory and by exploiting the universality of cluster states. Here we do not use the quantum formalism and hence make the completeness result accessible to a wide audience. Furthermore our method has the advantage of being algorithmic in nature so that by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D lattice.Comment: 18 pages, 15 figures, Accepted for publication in Physical Review

To Greener Pastures: An Action Research Study on the Environmental Sustainability of Humanitarian Supply Chains

Purpose: While humanitarian supply chains (HSCs) inherently contribute to social sustainability by alleviating the suffering of afflicted communities, their unintended adverse environmental impact has been overlooked hitherto. This paper draws upon contingency theory to synthesize green practices for HSCs, identify the contingency factors that impact on greening HSCs and explore how focal humanitarian organizations (HOs) can cope with such contingency factors. Design/methodology/approach: Deploying an action research methodology, two-and-a-half cycles of collaboration between researchers and a United Nations agency were completed. The first half-cycle developed a deductive greening framework, synthesizing extant green practices from the literature. In the second and third cycles, green practices were adopted/customized/developed reflecting organizational and contextual contingency factors. Action steps were implemented in the HSC for prophylactics, involving an operational mix of disaster relief and development programs. Findings: First, the study presents a greening framework that synthesizes extant green practices in a suitable form for HOs. Second, it identifies the contingency factors associated with greening HSCs regarding funding environment, stakeholders, field of activity and organizational management. Third, it outlines the mechanisms for coping with the contingency factors identified, inter alia, improving the visibility of headquarters over field operations, promoting collaboration and resource sharing with other HOs as well as among different implementing partners in each country, and working with suppliers for greener packaging. The study advances a set of actionable propositions for greening HSCs. Practical implications: Using an action research methodology, the study makes strong practical contributions. Humanitarian practitioners can adopt the greening framework and the lessons learnt from the implementation cycles presented in this study. Originality/value: This is one of the first empirical studies to integrate environmental sustainability and HSCs using an action research methodology