Coalition structure generation over graphs

We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph

The Holst Spin Foam Model via Cubulations

Spin foam models are an attempt for a covariant, or path integral formulation of canonical loop quantum gravity. The construction of such models usually rely on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional "Regge-like" constraints arising from simplicial triangulations, and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a "cubulation" of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: first, the structure of its amplitudes differs from the standard spin foam models. Second, the tetrad n-point functions admit a "Wick-like" structure. Third, the restriction to simple representations does not automatically occur -- unless one makes use of the time gauge, just as in the classical theory.Comment: 25 pages, 1 figure; v3: published version. arXiv admin note: substantial text overlap with arXiv:0911.213

Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime

We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its ``purification''. As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein-Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its ``purification'' induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III_1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.Comment: 42 pages, LaTeX. The Def. 3.3 was incomplete and this has been corrected. Several misprints have been removed. All results and proofs remain unchange

From treebank resources to LFG F-structures

We present two methods for automatically annotating treebank resources with functional structures. Both methods define systematic patterns of correspondence between partial PS configurations and functional structures. These are applied to PS rules extracted from treebanks, or directly to constraint set encodings of treebank PS trees