Approximations for the Random Minimal Spanning Tree With Application to Network Provisioning

This paper considers the problem of determining the mean and distribution of the length of a minimal spanning tree (MST) on an undirected graph whose arc lengths are independently distributed random variables. We obtain bounds and approximations for the MST length and show that our upper bound is much tighter than the naive bound obtained by computing the MST length of the deterministic graph with the respective means as arc lengths. We analyze the asymptotic properties of our approximations and establish conditions under which our bounds are asymptotically optimal. We apply these results to a network provisioning problem and show that the relative error induced by using our approximations tends to zero as the graph grows large

The Minimum Spanning Tree Constant in Geometrical Probability and Under the Independent Model; A Unified Approach

Given n uniformly and independently points in the d dimensional cube of unit volume, it is well established that the length of the minimum spanning tree on these n points is asymptotic to /3MsT(d)n(d-l)/d,where the constant PMST(d) depends only on the dimension d. It has been a major open problem to determine the constant 3MST(d). In this paper we obtain an exact expression of the constant MST(d) as a series expansion. Truncating the expansion after a finite number of terms yields a sequence of lower bounds; the first 3 terms give a lower bound which is already very close to the empirically estimated value of the constant. Our proof technique unifies the derivation for the MST asymptotic behavior for the Euclidean and the independent model

The Probabilistic Minimum Spanning Tree, Part II: Probabilistic Analysis and Asymptotic Results

In this paper, which is a sequel to [3], we perform probabilistic analysis under the random Euclidean and the random length models of the probabilistic minimum spanning tree (PMST) problem and the two re-optimization strategies, in which we find the MST or the Steiner tree respectively among the points that are present at a particular instance. Under the random Euclidean model we prove that with probability 1, as the number of points goes to infinity, the expected length of the PMST is the same with the expectation of the MST re-optimization strategy and within a constant of the Steiner re-optimization strategy. In the random length model, using a result of Frieze [6], we prove that with probability 1 the expected length of the PMST is asymptotically smaller than the expectation of the MST re-optimization strategy. These results add evidence that a priori strategies may offer a useful and practical method for resolving combinatorial optimization problems on modified instances. Key words: Probabilistic analysis, combinatorial optimization, minimum spanning tree, Steiner tree

A priori optimization

Includes bibliographical references.Partially supported by the National Science Foundation. ECS-8717970Dimitris J. Bertsimas, Patrick Jaillet and Amedeo R. Odoni

Analysis of Linear Programming Relaxations for a Class of Connectivity Problems

We consider the analysis of linear programming (LP) relaxations for a class of connectivity problems. The central problem in the class is the survivable network design problem - the problem of designing a minimum cost undirected network satisfying prespecified connectivity requirements between every pair of vertices. This class includes a number of classical combinatorial optimization problems as special cases such as the Steiner tree problem, the traveling salesman problem, the k-person traveling salesman problem and the k-edge-connected network problem. We analyze a classical linear programming relaxation for this class of problems under three perspectives: structural, worst-case and probabilistic. Our analysis rests mainly upon a deep structural property, the parsimonious property, of this LP relaxation. Roughly stated, the parsimonious property says that, if the cost function satisfies the triangle inequality, there exists an optimal solution to the LP relaxation for which the degree of each vertex is the smallest it can possibly be. The numerous consequences of the parsimonious property make it particularly important. First, several special cases of the parsimonious property are interesting properties by themselves. For example, we derive the monotonicity of the Held-Karp lower bound for the traveling salesman problem and the fact that this bound is a relaxation on the 2-connected network problem. Another consequence is the fact that vertices with no connectivity requirement, such as Steiner vertices in the undirected Steiner tree problem, are unnecessary for the LP relaxation under consideration. From the parsimonious property, it also follows that the LP relaxation bounds corresponding to the Steiner tree problem, the kedge-connected network problem or even the Steiner k-edge-connected network problem can be computed a la Held and Karp.Secondly, we use the parsimonious property to perform worst-case analyses of the duality gap corresponding to these LP relaxations. For this purpose, we introduce two heuristics for the survivable network design problem and present bounds dependent on the actual connectivity requirements. Among other results, we show that the value of the LP relaxation of the Steiner tree problem is within twice the value of the minimum spanning tree heuristic and that several generalizations of the Steiner tree problem, including the k-edge-connected network problem, can also be approximated within a factor of 2 (in some cases, even smaller than 2). We also introduce a new relaxation a la Held and Karp for the k-person traveling salesman problem and show that a variation of an existing heuristic is within times the value of this relaxation. We show that most of our bounds are tight and we investigate whether the bound of 3 for the Held-Karp lower bound is tight.We also perform a probabilistic analysis of the duality gap of these LP relaxations. The model we consider is the Euclidean model. We generalize Steele's theorem on the asymptotic behavior of Euclidean functionals in a way that is particularly convenient for the analysis of LP relaxations. We show that, under the Euclidean model, the duality gap is almost surely a constant and we provide theoretical and empirical bounds on these constants for different problems. From this analysis, we conclude that the undirected LP relaxation for the Steiner tree problem is fairly loose. Finally, we consider the use of directed relaxations for undirected problems. We establish in which settings a related parsimonious property holds and show that, for the Steiner tree problem, the directed relaxation strictly improves upon the undirected relaxation in the worst-case. This latter result uses an elementary but powerful property of linear programs

Trajan's column : the sculpting and relief content of a Roman propaganda monument

PhD ThesisThis thesis examines the reliefs of Trajan's Column in Rome (dedicated A. D. 113). It explores sculptural processes and provides a full and critical commentary on the relief content. Section 1 reviews prior work on the column and explains how the present research was conducted whilst taking advantage of scaffolding erected in conjunction with conservation studies. Section 2 examines the role of the column as a propaganda monument, exploring the value of the depictions of Trajan's wars as a source of historical information. This runs parallel to an enquiry into the imperial propaganda functions of the project. These two lines of approach are set against the column's immediate architectural environment which suggests how the reliefs were observed by the public audience. Section 3 is a technical enquiry into the methods by which the column was fabricated, and the sculptures were planned and executed, based on minute observation and computer-assisted recording of the reliefs. Section 4 deals with each of the potential sources of information concerning historical events, warfare, architecture and the Roman army available to, and employed by the sculptors working in Rome. It concludes that input from the war zone on the Danube was minimal in comparison with models and verbal information available in the capital. In Section 5, the sixteen categories of human figures on the spiral frieze ('Figure Types') are dealt with in turn and examined in the light of comparative textual, artefactual and representational evidence with regard to their accuracy. Relationships with other contemporary monuments are also examined. The last Section reviews the place of Trajan's Column in Roman monumental art, examining its innovative features and-its influence on later works

Volo non Valeo quia Nequeo quod Desidero : antithetic aristocrat : George Howard, ninth Earl of Carlisle (1843 - 1911), artist and patron

The current thesis is concerned with the artistic life of George Howard, and with his role as patron-participant. Howard was bom in 1843, eventual heir to the earldom of Carlisle and the vast estates pertaining to Castle Howard in Yorkshire and Naworth Castle near Brampton in Cumberland. He sought to live the life of an artist, eschewing, to a greater or lesser degree, both political and ancestral responsibilities. On his accession to the title Ninth Earl of Carlisle in 1889 he agreed, at his wife's request, to place the lion's share of the management of the family estates in her hands; by this date Rosalind Howard had already proved herself a highly competent and rigorous administrator. Having overcome familial opposition to his desire to become an artist, Howard faced a lifelong struggle for self-determination. Howard's intentions were epitomised by the studio-house which he commissioned Philip Webb to design during the early years of his London life in 1868. This marked the beginning of a lifelong relationship and, literally, cemented his place at the heart of the former Oxford Set. However, his vacillating self-belief, his wife's misplaced political ambitions on his behalf, and a social climate which distinguished amateur artist from professional more usually on an economic basis rather than by merit, combined to place him in an anomalous position. The artistic road along which Howard doggedly travelled was a circuitous one, punctated by crises of confidence succeeded by renewed endeavour and altered direction. It took him through Pre-Raphaelitism under the tutelage of Burne-Jones, to French realism under Alphonse Legros, and beyond. He finally opted to follow the teachings of Giovanni Costa, whom he visited in Italy on an almost annual basis from 1866. Howard, with a small band of Costa's acolytes from England, America and Italy, comprised the Etruscan School. Members' work aimed at the expression of landscape's latent sentiment, characterised by a panoramic format. Italian subjects predominated in Howard's work for many years, being augmented by those depicting India, Egypt and other countries to which he travelled later in his career. Throughout his life Howard remained unimpressed by social distinctions, and his role as patron to Legros, Costa, Webb and many other fellow artists whose economic wellbeing he underpinned, is inseparable from his friendship with them. Particularly noteworthy is his role as facilitator which, in several instances, resulted in some of the principal works associated with those concerned. Howard was widely acknowledged by his contemporaries as being an influential figure in cultural matters. He stood for Parliament three times, with varying degrees of unwillingness, and was returned as Liberal MP for East Cumberland twice. He was a Trustee of the National Gallery for thirty years and expended much energy working with galleries and museums both in Britain and abroad, as well as other reform- and education-based bodies such as the Society for the Protection of Ancient Buildings and the National Association for the Advancement of Art and its Application to Industry.EThOS - Electronic Theses Online ServiceGBUnited Kingdo