The complexity of weighted and unweighted #CSP

We give some reductions among problems in (nonnegative) weighted #CSP which restrict the class of functions that needs to be considered in computational complexity studies. Our reductions can be applied to both exact and approximate computation. In particular, we show that a recent dichotomy for unweighted #CSP can be extended to rational-weighted #CSP.Comment: 11 page

Matrix norms and rapid mixing for spin systems

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

An approximation trichotomy for Boolean #CSP

We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the co-clone IM_2 from Post's lattice, then the problem of counting satisfying assignments is complete with respect to approximation-preserving reductions in the complexity class #RH\Pi_1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respect to approximation-preserving reductions, meaning that there is no fully polynomial randomised approximation scheme for counting satisfying assignments unless NP=RP

Central Saharan trade in the early Islamic centuries (7th - 9th centuries A.D.)

African Studies Center Working Paper No. 1

Markov chain comparison

This is an expository paper, focussing on the following scenario. We have two Markov chains, $\mathcal {M}$ and $\mathcal {M}'$. By some means, we have obtained a bound on the mixing time of $\mathcal {M}'$. We wish to compare $\mathcal {M}$ with $\mathcal {M}'$ in order to derive a corresponding bound on the mixing time of $\mathcal {M}$. We investigate the application of the comparison method of Diaconis and Saloff-Coste to this scenario, giving a number of theorems which characterize the applicability of the method. We focus particularly on the case in which the chains are not reversible. The purpose of the paper is to provide a catalogue of theorems which can be easily applied to bound mixing times.Comment: Published at http://dx.doi.org/10.1214/154957806000000041 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Food Habits of the Steelcolor Shiner, Notropis whipplei (Girard)

Specimens of the steelcolor shiner, Notropis whipplei (Girard) were seined at irregular intervals from Polecat Creek (Coles County, Illinois) from October 1974 to September 1975. Drift and benthic samples were taken in conjunction with fish collections. The stomach contents of the fish were analyzed and compared to organisms found in the drift and benthos. More food taxa were represented in the fish collected during the summer. Aquatic insect forms, primarily chironomid larvae, amounted to 75.6% of the total percentage of foods eaten during the winter as compared to 18.8% for the summer. Adult insects, mainly coleopterans and dipterans, were the major components of the diet in the summer samples. More fish had empty stomachs in the winter (67.3%) than during the summer (8.9%). Food volumes of different sized fish were found to vary linearly with little difference in percent fullness. The smaller fish ate the smallest foods (ants and chironomids). Medium sized fish ate more algae and aquatic insect larvae while the largest fish ate more of the larger coleopterans and some crayfish chelicerae. Males and females were found to have differences in their diets. Males ate some different food taxa during the winter. Females had the greatest food volume during the winter. Varying amounts of different food taxa were eaten by each sex during the summer. Great differences were found between foods eaten and foods available. Ivlev\u27s Index of Electivity was applied to the summer collections and it was determined that the fish selected for terrestrial insect forms. Hymenopterans and coleopterans were found to have high positive Index values. Dipterans (primarily aquatic) were consistently selected against and had low Index values. Electivity was consistent between sexes. Differences in Index values between sexes indicate the possible preference of some food types by each sex. Food consumption may possibly be influenced by availability and density. Most foods were selected in the same general proportion as they were found in the drift. Evidence exists showing the fish to be surface feeders. Digestion of soft foods was found to be rapid and complete for fish in vitro

Fish, feather, fur and forest: exploitation of wild mammals in medieval Novgorod and its territory

The city of Novgorod the Great in north-west Russia has been subjected to extensive excavation of its superbly preserved medieval anaerobic deposits for many decades. Situated on the River Volkhov near Lake Ilmen and surrounded by mixed boreal and deciduous woodland and seasonally flooded meadowlands, Novgorod was well-placed for the exploitation of local wild fauna and flora. It was also the focus for international trade in furs obtained from a much broader catchment area. Although its inhabitants relied heavily on domestic animals and crops for their food, evidence for the acquisition of wild resources is also provided by animal bones, plant macrofossils, birch-bark documents and other archaeological finds. Pollen analysis has also provided information about landscape history of its hinterland. Even from a limited programme of sieved sampling, it is clear that a very large number of fish bones were present in the deposits. The main taxa are cyprinids, pike and zander, whereas birch-bark documents, largely concerned with tribute, mention salmonids and sturgeon, rare amongst the excavated remains. The wild bird assemblage is dominated by various species of ducks. Other waterfowl were utilised as well as large game birds such as capercaillie. Birds of prey are also present and other remains such as jackdaws reveal the local bird life in town. Wild mammals contributed little to the Novgorodian diet: bones of hare, beaver and elk being the most frequently found. Very few bones of fur-bearing mammals were recovered. The few remains include bear claws and bones of squirrel, marten, otter, and fox, in addition to beaver. Their paucity can be explained by the fact that most would have arrived from the hunting grounds to the north as prepared pelts without bones. Evidence for the hunting for these species is provided on sites deep within the forest zone such as Minino. Most of the wild animals eaten in Novgorod itself were obtained from the land and waters of its near hinterland. The paper discusses the character of the local forest based on pollen and other evidence. Models are being developed to provide a more detailed understanding of the changes in the composition of the forest during the medieval period and the consequences this had for wildlife. The paper demonstrates the benefits of developing a multi-disciplinary approach comparing urban assemblages with contemporary sites in its hinterland and further afield, to understand more fully how wild species were exploited in complex societies

The complexity of approximating conservative counting CSPs

We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem of counting independent sets in bipartite graphs. #BIS is complete with respect to approximation-preserving reductions for a logically-defined complexity class #RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in difficulty to #SAT, the problem of approximately counting the satisfying assignments of a Boolean formula in conjunctive normal form. We also discuss the algorithmic aspects of our classification.Comment: Minor revisio

05201 Abstracts Collection -- Design and Analysis of Randomized and Approximation Algorithms

From 15.05.05 to 20.05.05, the Dagstuhl Seminar 05201 ``Design and Analysis of Randomized and Approximation Algorithms\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available