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

Lehman's Theorem and the Directed Steiner Tree Problem

In the directed Steiner tree problem, we are given a digraph, nonnegative arc weights, a subset of vertices called terminals, and a special terminal called the root. The goal is to compute a minimum weight directed tree that connects each terminal to the root. We study the classical directed cut linear programming (LP) formulation which has a variable for every arc, and a constraint for every cut that separates a terminal from the root. For what instances is the directed cut LP integral? In this paper we demonstrate how the celebrated theorem of Lehman [Math. Program., 17 (1979), pp. 403-417] on minimally nonideal clutters provides a framework for deriving answers to this question. Specifically, we show that this framework yields short proofs of the optimum arborescences theorem and the integrality result for series-parallel digraphs. Furthermore, we use this framework to show that the directed cut linear program is integral for digraphs that are acyclic and have at most two nonterminal vertices

Texas Law Review

Journal containing articles, notes, book reviews, and other analyses of law and legal cases. Topics covered in this issue are regulatory arbitrage, Criminal Law Theory, death penalty provisions of the Model Penal Code, report to the ALI concerning Capital Punishment, and fixing the False Marking Statute

History of Psychology

Openly licensed anthology focused on the theme of the History of Psychology. Contains: The Mind and the Brain by Alfred Binet; Dream Psychology: Psychoanalysis for Beginners by Sigmund Freud; The Principles of Psychology, Volume 1 (of 2) by William James; The Principles of Psychology, Volume 2 (of 2) by William James; Collected Papers on Analytical Psychology by C. G. Jung; Memoirs of Extraordinary Popular Delusions and the Madness of Crowds by Charles Mackay; The Psychology of Arithmetic by Edward L. Thorndike

The drivers of Corporate Social Responsibility in the supply chain. A case study.

Purpose: The paper studies the way in which a SME integrates CSR into its corporate strategy, the practices it puts in place and how its CSR strategies reflect on its suppliers and customers relations. Methodology/Research limitations: A qualitative case study methodology is used. The use of a single case study limits the generalizing capacity of these findings. Findings: The entrepreneur’s ethical beliefs and value system play a fundamental role in shaping sustainable corporate strategy. Furthermore, the type of competitive strategy selected based on innovation, quality and responsibility clearly emerges both in terms of well defined management procedures and supply chain relations as a whole aimed at involving partners in the process of sustainable innovation. Originality/value: The paper presents a SME that has devised an original innovative business model. The study pivots on the issues of innovation and eco-sustainability in a context of drivers for CRS and business ethics. These values are considered fundamental at International level; the United Nations has declared 2011 the “International Year of Forestry”