Algorithms For Combinatorial Optimization Problems Research Articles

A New Way of Using Semidefinite Programming with Applications to Linear Equations mod p

We introduce a new method of constructing approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as a constellation of vectors in the semidefinite program. When we apply this technique to systems of linear equations mod p with at most two variables in each equation, we can show that the problem is approximable within (1−κ(p))p, where κ(p)>0 for all p. Using standard techniques, we also show that it is NP-hard to approximate the problem within a constant ratio, independent of p.

On metaheuristic algorithms for combinatorial optimization problems

Metaheuristic algorithms are widely recognized as one of the most practical approaches for combinatorial optimization problems. Among representative metaheuristics are genetic algorithms, simulated annealing, tabu search, and so on. In this paper, we explain essential ideas used in such metaheuristic algorithms within a generalized framework of local search. We then conduct numerical experiments of metaheuristic algorithms using rather simple implementations, to observe general tendencies of their performance. From these results, we propose a few recommendations about the use of metaheuristics as simple optimization tools. We also mention some advanced techniques to enhance the ability of metaheuristics. Finally, we summarize some theoretical results on metaheuristic algorithms. © 2001 Scripta Technica, Syst Comp Jpn, 32(3): 33–55, 2001

A genetic model: analysis and application to MAXSAT.

In this paper, a genetic model based on the operations of recombination and mutation is studied and applied to combinatorial optimization problems. Results are: 1. The equations of the deterministic dynamics in the thermodynamic limit (infinite populations) are derived and, for a sufficiently small mutation rate, the attractors are characterized; 2. A general approximation algorithm for combinatorial optimization problems is designed. The algorithm is applied to the Max Ek-Sat problem, and the quality of the solution is analyzed. It is proved to be optimal for k > or = 3 with respect to the worst case analysis; for Max E3-Sat the average case performances are experimentally compared with other optimization techniques.

Interactive and probabilistic proof-checking

The notion of efficient proof-checking has always been central to complexity theory, and it gave rise to the definition of the class NP. In the last 15 years there has been a number of exciting, unexpected and deep developments in complexity theory that exploited the notion of randomized and interactive proof-checking. Results developed along this line of research have diverse and powerful applications in complexity theory, cryptography, and the theory of approximation algorithms for combinatorial optimization problems. In this paper we survey the main lines of developments in interactive and probabilistic proof-checking, with an emphasis on open questions.

The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization

The development of new mathematical theory and its application in software systems for the solution of hard optimization problems have a long tradition in mathematical programming. In this tradition we implemented ABACUS, an object-oriented software framework for branch-and-cut-and-price algorithms for the solution of mixed integer and combinatorial optimization problems. This paper discusses some difficulties in the implementation of branch-and-cut-and-price algorithms for combinatorial optimization problems and shows how they are managed by ABACUS. Copyright © 2000 John Wiley & Sons, Ltd.

Critical phenomena in a collective computation algorithm for combinatorial optimization problems

This paper discusses the critical temperature (control parameter) of an annealed neural network, a typical collective computation for combinatorial optimization problems. It is shown that the theoretical critical temperatures determined by our estimation agree with those derived by computational experiments in graph partitioning problems and in the travelling salesman problem.

Combinatorial optimization by iterative partial transcription

A procedure is presented which considerably improves the performance of local search based heuristic algorithms for combinatorial optimization problems. It increases the average `gain' of the individual local searches by merging pairs of solutions: certain parts of either solution are transcribed by the related parts of the respective other solution, corresponding to flipping clusters of a spin glass. This iterative partial transcription acts as a local search in the subspace spanned by the differing components of both solutions. Embedding it in the simple multi-start-local-search algorithm and in the thermal-cycling method, we demonstrate its effectiveness for several instances of the traveling salesman problem. The obtained results indicate that, for this task, such approaches are far superior to simulated annealing.

An Algorithm For Combinatorial Optimization Problems With Normal Distributed Variables

An Algorithm For Combinatorial Optimization Problems With Normal Distributed Variables

Adaptive annealing for chaotic optimization

The chaotic simulated annealing algorithm for combinatorial optimization problems is examined in the light of the global bifurcation structure of the chaotic neural networks. We show that the result of the chaotic simulated annealing algorithm is primarily dependent upon the global bifurcation structure of the chaotic neural networks and unlike the stochastic simulated annealing infinitely slow chaotic annealing does not necessarily provide an optimum result. As an improved algorithm, the adaptive chaotic simulated annealing algorithm is introduced. Using several instances of 20- and 40-city traveling salesman problems, efficiency of the adaptive algorithm is demonstrated. @S1063-651X~98!15510-1#

Parallel Approximation Algorithms by Positive Linear Programming

Several sequential approximation algorithms for combinatorial optimization problems are based on the following paradigm: solve a linear or semidefinite programming relaxation, then use randomized rounding to convert fractional solutions of the relaxation into integer solutions for the original combinatorial problem. We demonstrate that such a paradigm can also yield parallel approximation algorithms by showing how to convert certain linear programming relaxations into essentially equivalent positive linear programming [LN] relaxations that can be near-optimally solved in NC. Building on this technique, and finding some new linear programming relaxations, we develop improved parallel approximation algorithms for Max Sat, Max Directed Cut, and Maxk CSP. The Max Sat algorithm essentially matches the best approximation obtainable with sequential algorithms and has a fast sequential version. The Maxk CSP algorithm improves even over previous sequential algorithms. We also show a connection between probabilistic proof checking and a restricted version of Maxk CSP. This implies that our approximation algorithm for Maxk CSP can be used to prove inclusion in P for certain PCP classes.

A parallel simulated annealing algorithm

In this paper we present a generalization of the sequential simulated annealing algorithm for combinatorial optimization problems. By performing a parallel study of the current solution neighbourhood we obtain an algorithm that can be very efficiently implemented on a massively parallel computer. We test the convergence and the quality of our algorithm by comparing it to the sequential algorithm for two classical problems: the minimization of an unconstrained 0–1 quadratic function and the quadratic sum assignment problem.

Toward a Programming Environment for Combinatorial Optimization: A Case Study Oriented to Max-Flow Computations

In this paper, we try to move a few steps toward the definition of a programming environment oriented to the production and updating of algorithms for combinatorial optimization problems. Such an environment contains a certain number of basic blocks (data structures and routines) together with a set of rules for their use, which allow us to assemble the blocks into large and complex algorithmic architectures. The max-flow problem has been chosen for a case study which illustrates the main issues arising in the design of the programming environment. In order to develop such a study, a new general algorithm paradigm for the max-flow problem is proposed, from which most of the known algorithms can be derived by instantiation. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A new dominance procedure for combinatorial optimization problems

It is known that the effectiveness of the branch and bound algorithms for combinatorial optimization problems can be improved through dominance criteria which allow fathomings of large solution subsets. We describe a new dominance procedure which overcomes some of the drawbacks of the commonly used dominance criteria. An application to the Multiple Knapsack Problem and some computational results are also reported.