Exact arborescences, matchings and cycles

AbstractSuppose we are given a graph in which edge has an integral weight. An ‘exact’ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a ‘continuity’ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary

A Simple Algorithm for Local Conversion of Pure States

We describe an algorithm for converting one bipartite quantum state into another using only local operations and classical communication, which is much simpler than the original algorithm given by Nielsen [Phys. Rev. Lett. 83, 436 (1999)]. Our algorithm uses only a single measurement by one of the parties, followed by local unitary operations which are permutations in the local Schmidt bases.Comment: 5 pages, LaTeX, reference adde

FACES OF MATCHING POLYHEDRA

Let G = (V, E, ~) be a finite loopless graph, let b=(bi:ieV) be a vector of positive integers. A feasible matching is a vector X = (x.: j e: E) J of nonnegative integers such that for each node i of G, the sum of the over the edges j of G incident with i is no greater than bi. The matching polyhedron P(G, b) is the convex hull of the set of feasible matchings. In Chapter 3 we describe a version of Edmonds' blossom algorithm which solves the problem of maximizing C • X over P (G, b) where c =. (c.: j e: E) J is an arbitrary real vector. This algorithm proves a theorem of Edmonds which gives a set of linear inequalities sufficient to define P(G, b). In Chapter 4 we prescribe the unique subset of these inequalities which are necessary to define P(G, b), that is, we characterize the facets of P(G, b). We also characterize the vertices of P(G, b), thus describing the structure possessed by the members of the minimal set X of feasible matchings of G such that for any real vector c = (c.: j e: E), c • x is maximized over P(G, b) J member of X. by a In Chapter 5 we present a generalization of the blossom algorithm which solves the problem: maximize c • x over a face F of P(G, b) for any real vector c = (c.: j e: E). J In other words, we find a feasible matching x of G which satisfies the constraints obtained by replacing an arbitrary subset of the inequalities which define P(G, b) by equations and which maximizes c • x subject to this restriction. We also describe an application of this algorithm to matching problems having a hierarchy of objective functions, so called ''multi-optimization'' problems. In Chapter 6 we show how the blossom algorithm can be combined with relatively simple initialization algorithms to give an algorithm which solves the following postoptimality problem. Given that we know a matching 0 x £ P(G, b) maximizes c · x over P(G, b), we wish to utilize 0 X which to find a feasible matching x' £ P(G, b') which maximizes c • x over P(G, b'), where b' = (b!: i £ V) ]_ vector of positive integers and arbitrary real vector. c=(c.:j£E) J is a is an In Chapter 7 we describe a computer implementation of the blossom algorithm described herein

Biostratigraphic Evidence Relating to the Age-Old Question of Hannibal's Invasion of Italy, II: Chemical Biomarkers and Microbial Signatures

Open access article. Creative Commons Attribution 4.0 International License (CC BY 4.0) appliesAs discussed in Part I, a large accumulation of mammalian faeces at the mire site in the upper Guil Valley near Mt. Viso, dated to 2168 cal 14C yr., provides the first evidence of the passage of substantial but indeterminate numbers of mammals within the time frame of the Punic invasion of Italia. Specialized organic biomarkers bound up in a highly convoluted and bioturbated bed constitute an unusual anomaly in a histosol comprised of fibric and hemist horizons that are usually expected to display horizontal bedding. The presence of deoxycholic acid and ethylcoprostanol derived from faecal matter, coupled with high relative numbers of Clostridia 16S rRNA genes, suggests a substantial accumulation of mammalian faeces at the site over 2000 years ago. The results reported here constitute the first chemical and biological evidence of the passage of large numbers of mammals, possibly indicating the route of the Hannibalic army at this time. Combined with the geological analysis reported in Part I, these data provide a background supporting the need for further historical archaeological exploration in this area.Ye

On the exactness of the cavity method for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs

We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming relaxation of the problem has no fractional solutions, then the cavity or belief propagation equations converge to the correct solution both for synchronous and asynchronous updating

Biostratigraphic Evidence Relating to the Age-Old Question of Hannibal's Invasion of Italy, I: History and Geological Reconstruction

Controversy over the alpine route that Hannibal of Carthage followed from the Rhône Basin into Italia has raged amongst classicists and ancient historians for over two millennia. The motivation for identifying the route taken by the Punic Army through the Alps lies in its potential for identifying sites of historical archaeological significance and for the resolution of one of history's most enduring quandaries. Here, we present stratigraphic, geochemical and microbiological evidence recovered from an alluvial floodplain mire located below the Col de la Traversette (~3000 m asl—above sea level) on the French/Italian border that potentially identifies the invasion route as the one originally proposed by Sir Gavin de Beer (de Beer 1974). The dated layer is termed the MAD bed (mass animal deposition) based on disrupted bedding, greatly increased organic carbon and key/specialized biological components/compounds, the latter reported in Part II of this paper. We propose that the highly abnormal churned up (bioturbated) bed was contaminated by the passage of Hannibal's animals, possibly thousands, feeding and watering at the site, during the early stage of Hannibal's invasion of Italia (218 bc)

Helical Chirality: a Link between Local Interactions and Global Topology in DNA

DNA supercoiling plays a major role in many cellular functions. The global DNA conformation is however intimately linked to local DNA-DNA interactions influencing both the physical properties and the biological functions of the supercoiled molecule. Juxtaposition of DNA double helices in ubiquitous crossover arrangements participates in multiple functions such as recombination, gene regulation and DNA packaging. However, little is currently known about how the structure and stability of direct DNA-DNA interactions influence the topological state of DNA. Here, a crystallographic analysis shows that due to the intrinsic helical chirality of DNA, crossovers of opposite handedness exhibit markedly different geometries. While right-handed crossovers are self-fitted by sequence-specific groove-backbone interaction and bridging Mg2+ sites, left-handed crossovers are juxtaposed by groove-groove interaction. Our previous calculations have shown that the different geometries result in differential stabilisation in solution, in the presence of divalent cations. The present study reveals that the various topological states of the cell are associated with different inter-segmental interactions. While the unstable left-handed crossovers are exclusively formed in negatively supercoiled DNA, stable right-handed crossovers constitute the local signature of an unusual topological state in the cell, such as the positively supercoiled or relaxed DNA. These findings not only provide a simple mechanism for locally sensing the DNA topology but also lead to the prediction that, due to their different tertiary intra-molecular interactions, supercoiled molecules of opposite signs must display markedly different physical properties. Sticky inter-segmental interactions in positively supercoiled or relaxed DNA are expected to greatly slow down the slithering dynamics of DNA. We therefore suggest that the intrinsic helical chirality of DNA may have oriented the early evolutionary choices for DNA topology

Theoretical Analysis of Competing Conformational Transitions in Superhelical DNA

We develop a statistical mechanical model to analyze the competitive behavior of transitions to multiple alternate conformations in a negatively supercoiled DNA molecule of kilobase length and specified base sequence. Since DNA superhelicity topologically couples together the transition behaviors of all base pairs, a unified model is required to analyze all the transitions to which the DNA sequence is susceptible. Here we present a first model of this type. Our numerical approach generalizes the strategy of previously developed algorithms, which studied superhelical transitions to a single alternate conformation. We apply our multi-state model to study the competition between strand separation and B-Z transitions in superhelical DNA. We show this competition to be highly sensitive to temperature and to the imposed level of supercoiling. Comparison of our results with experimental data shows that, when the energetics appropriate to the experimental conditions are used, the competition between these two transitions is accurately captured by our algorithm. We analyze the superhelical competition between B-Z transitions and denaturation around the c-myc oncogene, where both transitions are known to occur when this gene is transcribing. We apply our model to explore the correlation between stress-induced transitions and transcriptional activity in various organisms. In higher eukaryotes we find a strong enhancement of Z-forming regions immediately 5′ to their transcription start sites (TSS), and a depletion of strand separating sites in a broad region around the TSS. The opposite patterns occur around transcript end locations. We also show that susceptibility to each type of transition is different in eukaryotes and prokaryotes. By analyzing a set of untranscribed pseudogenes we show that the Z-susceptibility just downstream of the TSS is not preserved, suggesting it may be under selection pressure