A Dichotomy Theorem for Homomorphism Polynomials

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over the rational field of cut eliminator, a polynomial defined by B\"urgisser which is known to be neither VP nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size)

Arithmetic Branching Programs with Memory

We extend the well known characterization of VPws as the class of polynomials computed by polynomial size arithmetic branching programs to other complexity classes. In order to do so we add additional memory to the computation of branching programs to make them more expressive. We show that allowing different types of memory in branching programs increases the computational power even for constant width programs. In particular, this leads to very natural and robust characterizations of VP and VNP by branching programs with memory. 1

Fast computation of zeros of polynomial systems with - See more at: Fast computation of zeros of polynomial systems with bounded degree under finite-precision

A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite precision. In this paper we describe a finite-precision version of this algorithm. Our main result shows that this version works within the same time bounds and requires a precision which, on the average, amounts to a polynomial amount of bits in the mantissa of the intervening floating-point numbers

Revisiting Graph Width Measures for CNF-Encodings

International audienceWe consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors

Rural–Urban Regions: A Spatial Approach to Define Urban–Rural Relationships in Europe

The previous chapter introduced the range of issues associated with the peri-urban, the subject of this book. The peri-urban as a specific morphological type was defined and the different dimensions of its dynamics were explored. This periurban zone is intimately associated with the transition from a dense urban structure to that of a rural character and since it also involves movements into, out of and across it from both these extremes, it is difficult to consider it properly without understanding the broader regional context and dynamics across the urban–rural gradient. Therefore, this chapter will focus on the broader context of urban–rural relationships. Based on recent scientific debates concerning the concept of functional regions and urban–rural relationships, both current and previous definitions and their political implementations are introduced before presenting a new typology to represent Rural–urban Regions (RUR) spatially. Covering the territory of European Union (EU), this typology classifies regions into different types, considering city size, degree of regionalmono- and poly-centricity, as well as their urban, peri-urban or rural predominance. The development of the typology includes a further delineation of regions into urban, peri-urban and rural sub-regions, all based on land use patterns and population distribution and density. The typology was subsequently used throughout the PLUREL project and each of the case studies presented in Part Two refers to one of these types, although not all are represented there, since the case studies were unavoidably selected before the typology was developed

Notes on Etruscan cosmology: the case of the Tumulus of the Crosses at Cerveteri

The aim of this contribution is to test the possibility of the use of cosmological principles connected with Etruscan religion, for composing an inscription which is incised on the wall of a passageway running beneath a ramp attached to the northeast side of the Tumulus of the Crosses, in the Banditaccia necropolis at Cerveteri. The ramp features a stairway leading to a \ufb02at ceremonial platform. On the basis of the letterforms the inscription may be dated to the end of the seventh or the beginning of the sixth century BCE. It is a rare example of a monumental inscription of the Orientalizing period of Etruscan Civilization. Directly beneath the inscription is a sign (siglum) formed by a cross inscribed in a circle. This sign has been recognized as the representation of the Etruscan concept of sacred space, whose crucial attributes are delimitation, division and orientation. A recent new reading of the inscription points out four theonymic elements, which recall divinities that, in the Etruscan cosmology, it may be argued, occupied the northeastern quadrant of the sky. Any ampli\ufb01cation of this recent new reading must take into account interdisciplinary research focused on a possible relationship, in the \ufb01eld of archaeoastronomy, between the theonymic elements and the physical space that they occupy on the wall of the passageway, since the ramp is a crucial element of Etruscan funerary cultic practices