Lower Bounds from State Space Relaxations for Concave Cost Network Flow Problems

In this paper we obtain Lower Bounds (LBs) to concave cost network flow problems. The LBs are derived from state space relaxations of a dynamic programming formulation, which involve the use of non-injective mapping functions guaranteing a reduction on the cardinality of the state space. The general state space relaxation procedure is extended to address problems involving transitions that go across several stages, as is the case of network flow problems. Applications for these LBs include: estimation of the quality of heuristic solutions; local search methods that use information of the LB solution structure to find initial solutions to restart the search (Fontes et al., 2003, Networks, 41, 221–228); and branch-and-bound (BB) methods having as a bounding procedure a modified version of the LB algorithm developed here, (see Fontes et al., 2005a). These LBs are iteratively improved by penalizing, in a Lagrangian fashion, customers not exactly satisfied or by performing state space modifications. Both the penalties and the state space are updated by using the subgradient method. Additional constraints are developed to improve further the LBs by reducing the searchable space. The computational results provided show that very good bounds can be obtained for concave cost network flow problems, particularly for fixed-charge problems.     

Reactions of dimethyldivinylsilane,dimethyldivinyltin and allyltrimethyltin with diethylene (tertiary) phosphine)platinum complexes

The compounds [Pt(C₂H₄)₂(PR₃)] [PR₃ = P-tBu₂Me, P(C₆H₁₁)₃, PPh₃] react dimethyldivinylsilane or dimethyldivinyltin to give chelate complexes [Pt{(CH₂CH)₂MMe₂} (PR₃)] (M = Si or Sn). allyltrimethyltin reacts with various diethylene (tertiary phosphine)platinum compounds with cleavage of the allyl group to afford complexes [Pt(SnMe₃)(η³-C₃H₅)(PR₂)]. The NMR spectra (¹³C, ¹H and ³¹P) of the new compounds have been recorded, and the data are discussed in terms of the structures proposed.     

Thermal wave photopyroelectric characterization of advanced materials: State of the art

This article reviews the history, background, theoretical basis, development, attempts to optimize, and experimental performance of the photopyroelectric technique for the optothermal characterization of advanced materials such as semiconductors, superconductors, pure metals and alloys, quantum wells, liquid crystals, paramagnetic and ferromagnetic materials, as well as solar cells. The state of the art in the experimental processes in this field is also reviewed. This new photothermal technique can be used after a careful optimization, as a highly sensitive method for photopyroelectric spectroscopy and general thermal wave measurements. It has been shown to be a highly sensitive spectroscopic method for the nondestructive evaluation of advanced materials. This review presents the main photopyroelectric theoretical models that have been used for the extraction of some important optoelectronic properties such as the optical absorption coefficient and the nonradiative quantum efficiency spectra, as well as some thermal properties such as the thermal diffusivity, thermal conductivity, and specific heat. The applicability of the general basic theoretical model with its many special cases is also described in detail. This review demonstrates how photopyroelectric spectroscopy can be complementary to the conventional spectroscopic methods. The different experimental modes of the technique are also discussed. Moreover, some ideas concerning future perspectives of applying the technique to other scientific fields are outlined. This article does not aspire to an in-depth analysis of the experimental results in the field; rather, it focuses on the technique itself.     

The thresholds for diameter 2 in random Cayley graphs

Given a group G, the model denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any and any family of groups G_k of order n_k for which , a graph with high probability has diameter at most 2 if and with high probability has diameter greater than 2 if . We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erd?s‐Renyi random graphs. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 218–235, 2014     

Algorithms for large scale set covering problems

This paper is concerned with the set covering problem (SCP), and in particular with the development of a new algorithm capable of solving large-scale SCPs of the size found in real-life situations. The set covering problem has a wide variety of practical applications which, in general, yield large and sparse instances, normally with hundreds of rows and thousands of columns. In this paper, we present an algorithm capable of solving problems of this size and test problems up to 400 rows and 4000 columns are solved. The method developed in this paper consists of a tree-search procedure based on a combination of decomposition and state space relaxation which is a technique developed for obtaining lower bounds on the dynamic program associated with a combinatorial optimization problem. The large size SCPs are decomposed into many smaller SCPs which are then solved or bounded by state space relaxation (SSR). Before using the decomposition and SSR, reductions both in the number of columns and the number of rows of the problem are made by applying Lagrangian relaxation, linear programming and heuristic methods.     

Extensions to a Lagrangean relaxation approach for the capacitated warehouse location problem

In this paper we present a lower bound for the capacitated warehouse location problem based upon the Lagrangean relaxation of a mixed-integer formulation of the problem, where we use subgradient optimisation in an attempt to maximise this lower bound. Problem reduction tests based upon this lower bound and the original problem are given. Incorporating this bound and the reduction tests into a tree search procedure enables us to solve problems involving up to 50 warehouses and 150 customers.     

A restricted Lagrangean approach to the traveling salesman problem

We describe an algorithm for the asymmetric traveling salesman problem (TSP) using a new, restricted Lagrangean relaxation based on the assignment problem (AP). The Lagrange multipliers are constrained so as to guarantee the continued optimality of the initial AP solution, thus eliminating the need for repeatedly solving AP in the process of computing multipliers. We give several polynomially bounded procedures for generating valid inequalities and taking them into the Lagrangean function with a positive multiplier without violating the constraints, so as to strengthen the current lower bound. Upper bounds are generated by a fast tour-building heuristic. When the bound-strengthening techniques are exhausted without matching the upper with the lower bound, we branch by using two different rules, according to the situation: the usual subtour breaking disjunction, and a new disjunction based on conditional bounds. We discuss computational experience on 120 randomly generated asymmetric TSP's with up to 325 cities, the maximum time used for any single problem being 82 seconds. This is a considerable improvement upon earlier methods. Though the algorithm discussed here is for the asymmetric TSP, the approach can be adapted to the symmetric TSP by using the 2-matching problem instead of AP.Research supported by the National Science Foundation through grant no. MCS76-12026 A02 and the U.S. Office of Naval Research through contract no. N0014-75-C-0621 NR 047-048.     

Induced lines in Hales-Jewett cubes

A line in d[n] is a set {x(1),…,x(n)} of n elements of d[n] such that for each 1?i?d, the sequence is either strictly increasing from 1 to n, or strictly decreasing from n to 1, or constant. How many lines can a set S⊆d[n] of a given size contain?One of our aims in this paper is to give a counterexample to the Ratio Conjecture of Patashnik, which states that the greatest average degree is attained when S=d[n]. Our other main aim is to prove the result (which would have been strongly suggested by the Ratio Conjecture) that the number of lines contained in S is at most |S|^2−ε for some ε>0.We also prove similar results for combinatorial, or Hales-Jewett, lines, i.e. lines such that only strictly increasing or constant sequences are allowed.     

On an Approximation to the Mean Response Times of Priority Classes in a Stable G/G/c/PR Queue

Analytic approximations are proposed for the mean response-times of R(≥ 2) priority classes in a stable G/G/c/PR queue with general class interarrival and service time distributions and c(≥ 2) parallel servers under pre-emptive resume (PR) scheduling. The generalized exponential (GE) distributional model is used to represent general distributions with known first two moments per class. The analysis is based on the extension of known heuristic arguments and earlier results regarding the study of the stable GE/GE/c/FCFS (c ≥ 1, single class) and GE/G/1/PR queues. Numerical examples illustrate the accuracy of the proposed approximations in relation to simulations involving different interarrival and service time distributions per class. Moreover, GE-type performance bounds on the system response time per class are defined. Comments on the role of the new mean response time expressions towards the approximation of the joint and marginal queue length distributions of a stable G/G/c/PR queue are included.