Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1

This paper looks at decoding complexity.

This paper presents an informal discussion of issues that arise when one attempts to analyze algorithms experimentally. It is based on lessons learned by the author over the course of more than a decade of experimentation, survey paper writing, refereeing, and lively discussions with other experimentalists. Although written from the perspective of a theoretical computer scientist, it is intended to be of use to researchers from all fields who want to study algorithms experimentally. It has two goals: first, to provide a useful guide to new experimentalists about how such work can best be performed and written up, and second, to challenge current researchers to think about whether their own work might be improved from a scientific point of view. With the latter purpose in mind, the author hopes that at least a few of his recommendations will be considered controversial.

: The goal of languages like Fortran D or High Performance Fortran (HPF) is to provide a simple yet efficient machine-independent parallel programming model. By shifting much of the burden of machine-dependent optimization to the compiler, the programmer is able to write data-parallel programs that can be compiled and executed with good performance on many different architectures. However, the choice of a good data layout is still left to the programmer. Even the most sophisticated compiler may not be able to compensate for a poorly chosen data layout since many compiler decisions are driven by the data layout specified in the program. The choice of a good data layout depends on many factors, including the target machine architecture, the compilation system, the problem size, and the number of processors available. The option of remapping arrays at specific points in the program makes the choice even harder. Current programming tools provide little or no support for this difficult sele...

In this and the following chapter, we consider what approaches one should take when one is confronted with a real-world application of the TSP. What algorithms should be used under which circumstances? We

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Abstract. The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps. We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and firststage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0 % of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time. Keywords: salesman stochastic optimization, stochastic programming, stochastic routing, shortest path, traveling 1.

The Traveling Tournament Problem is a sports timetabling problem that abstracts the important issues in creating timetables where team travel is an important issue. Instances of this problem seem to be very difficult to solve even for very small cases. Given the practical importance of solving instances similar to these, this makes this problem an interesting challenge for combinatorial optimization techniques. We introduce this problem, give some interesting classes of instances and give some base computational results.

Dantzig, Fulkerson, and Johnson (1954) introduced the cutting-plane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et al.'s method that is suitable for TSP instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the general cutting-plane method in large-scale applications.

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