The travelling salesman problem: new solvable cases and linkages with the development of approximation algorithms

We identify new solvable cases of the travelling salesman problem (TSP) by an indirect analysis that has useful consequences. First, we develop new procedures for the TSP that require only linear time to execute and yield TSP tours that are better than an exponential number of alternative tours. We then identify special subgraphs, easily generated, so that our method yields these outcomes for every instance of these subgraphs. Finally, when the associated costs satisfy prescribed conditions, we show the solutions produced by these algorithms are optimal and thus we have new solvable cases of the TSP. Besides possible practical applications to problems that may exhibit these cost conditions, our algorithms may also be applied as subroutines within more complex metaheuristics. Our methods extend in a natural way to bottleneck TSP formulations, and their underlying results raise new theoretical questions about the analysis of heuristics for hard combinatorial problems.     

Solving the maximum edge weight clique problem via unconstrained quadratic programming

The unconstrained quadratic binary program (UQP) is proving to be a successful modeling and solution framework for a variety of combinatorial optimization problems. Experience reported in the literature with several problem classes has demonstrated that this approach works surprisingly well in terms of solution quality and computational times, often rivaling and sometimes surpassing more traditional methods. In this paper we report on the application of UQP to the maximum edge-weighted clique problem. Computational experience is reported illustrating the attractiveness of the approach.     

A Fritz John optimality condition using the approximate subdifferential

A Fritz John type first-order optimality condition is derived for infinite-dimensional programming problems involving the approximate subdifferential. A discussion of the important properties of the approximate subdifferential for locally Lipschitz functions is included. In addition, an upper semicontinuity condition is obtained for an approximate subdifferential multifunction related to the class of locally compactly Lipschitzian mappings.The authors would like to thank two anonymous referees for their detailed comments which have significantly improved the presentation of this paper. In particular, they thank the first referee for pointing out a number of important references on the approximate subdifferential and the second referee for various corrections and for bringing an important recent reference (Ref. 17) to their attention.     

A network-related nuclear power plant model with an intelligent branch-and-bound solution approach

This paper presents a network model with discrete requirements for a nuclear power plant. The model determines the batch size and timing for nuclear unit refueling and how much energy should be produced by nuclear and non-nuclear units for each time period to satisfy forecasted demand with minimum total operating costs over the planning horizon. Efficient modeling and solution strategies are developed which constitute a merger of operations research and artificial intelligence. A branch-and-bound solution approach is combined with a pattern recognition component, involving non-parametric discrimination analyses, to select branching variables and directions. By coupling this approach with network optimization techniques to exploit the underlying network structure of the problem, substantial improvements are obtained both in solution quality and solution efficiency.This research was supported in part by the Center for Business Decision Analysis, the Hugh Roy Cullen Centennial Chair in Business Administration, and the Office of Naval Research under contract N00014-87-K-0190. Reproduction in whole or in part is permitted for any purpose of the U.S. Government. CENTER FOR BUSINESS DECISION ANALYSIS Darwin Klingman, Director The University of Texas at Austin Austin, TX 78712, U.S.A.     

The heat of fusion of polybutene-1

Through x-ray diffraction, the changes in the unit-cell parameters of crystal forms I and II of polybutene-1 were followed from room temperature to the melting point. These data were combined with the results of other measurements to computer the heats and entropies of fusion by use of the Clapeyron equation.     

Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions

We use surrogate analysis and constraint pairing in multidimensional knapsack problems to fix some variables to zero and to separate the rest into two groups – those that tend to be zero and those that tend to be one, in an optimal integer solution. Using an initial feasible integer solution, we generate logic cuts based on our analysis before solving the problem with branch and bound. Computational testing, including the set of problems in the OR-library and our own set of difficult problems, shows our approach helps to solve difficult problems in a reasonable amount of time and, in most cases, with a fewer number of nodes in the search tree than leading commercial software.     

Interactive decision software and computer graphics for architectural and space planning

We describe the development and successful implementation of a decision support system now being used by several leading firms in the architecture and space planning industries. The system, which we call SPDS (spatial programming design system) has the following characteristics: (i) user-friendly convenience features permitting architects and space planners to operate the system without being experienced programmers; (ii) interactive capabilities allowing the user to control and to manipulate relevant parameters, orchestrating conditions to which his or her intuition provides valuable input; (iii) informative and understandable graphics, providing visual displays of interconnections that the computer itself treats in a more abstract methematical form; (iv) convenient ways to change configurations, and to carry out ‘what if’ analyses calling on the system’s decision support capabilities; (v) a collection of new methods, invisible to the user, capable of generating good solutions to the mathematical programming problems that underlie each major design component. These new methods succeed in generating high quality solutions to a collection of complex discrete, highly nonlinear problems. While these problems could only be solved in hours, or not at all, with previously existing software, the new methods obtain answers in seconds to minutes on a minicomputer. Major users, including Dalton, Dalton, Newport, and Marshal Erdwin, report numerous advantages of the system over traditional architectural design methods.     

Trifluoroethanol and 19F magic angle spinning nuclear magnetic resonance as a basic surface hydroxyl reactivity probe for zirconium(IV) hydroxide structures

A novel technique for determining the relative accessibility and reactivity of basic surface hydroxyl sites by reacting various zirconium(IV) hydroxide materials with 2,2,2-trifluoroethanol (TFE) and characterizing the resulting material using (19)F magic angle spinning (MAS) nuclear magnetic resonance (NMR) is presented here. Studied here are three zirconium hydroxide samples, two unperturbed commercial materials, and one commercial material that is crushed by a pellet press. Factors, such as the ratio of bridging/terminal hydroxyls, surface area, and pore size distribution, are examined and found to affect the ability of the zirconium(IV) hydroxide to react with TFE. X-ray diffraction, nitrogen isotherms, and (1)H MAS NMR were used to characterize the unperturbed materials, while thermogravitric analysis with gas chromatography and mass spectrometry along with the (19)F MAS NMR were used to characterize the materials that were reacted with TFE. Zirconium hydroxide materials with a high surface area and a low bridging/terminal hydroxyl ratio were found to react TFE in the greatest amounts.     

Simulating the formation of sodium:electron tight-contact pairs: watching the solvation of atoms in liquids one molecule at a time

The motions of solvent molecules during a chemical transformation often dictate both the dynamics and the outcome of solution-phase reactions. However, a microscopic picture of solvation dynamics is often obscured by the concerted motions of numerous solvent molecules that make up a condensed-phase environment. In this study, we use mixed quantum/classical molecular dynamics simulations to furnish the molecular details of the solvation dynamics that leads to the formation of a sodium cation-solvated electron contact pair, (Na(+), e(-)), in liquid tetrahydrofuran following electron photodetachment from sodide (Na(-)). Our simulations reveal that the dominant solvent response is comprised of a series of discrete solvent molecular events that work sequentially to build up a shell of coordinating THF oxygen sites around the sodium cation end of the contact pair. With the solvent response described in terms of the sequential motion of single molecules, we are then able to compare the calculated transient absorption spectroscopy of the sodium species to experiment, providing a clear microscopic interpretation of ultrafast pump-probe experiments on this system. Our findings suggest that for solute-solvent interactions similar to the ones present in our study, the solvation dynamics is best understood as a series of kinetic events consisting of reactions between chemically distinct local structures in which key solvent molecules must be considered to be part of the identity of the reacting species.