Introduction to the theory of complexity
Introduction to the theory of complexity
Performance Guarantees for Approximation Algorithms Depending on Parametrized Triangle Inequalities
SIAM Journal on Discrete Mathematics
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Some optimal inapproximability results
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Performance Guarantees for the TSP with a Parameterized Triangle Inequality
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
Nearly linear time approximation schemes for Euclidean TSP and other geometric problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Approximation algorithms for combinatorial problems
Journal of Computer and System Sciences
Reoptimization of the metric deadline TSP
Journal of Discrete Algorithms
On the hardness of reoptimization
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
On the approximation hardness of some generalizations of TSP
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Algorithmics – is there hope for a unified theory?
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
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To specify the set of tractable (practically solvable) computing problems is one of the few main research tasks of theoretical computer science. Because of this the investigation of the possibility or the impossibility to efficiently compute approximations of hard optimization problems becomes one of the central and most fruitful areas of recent algorithm and complexity theory. The current point of view is that optimization problems are considered to be tractable if there exist polynomial-time randomized approximation algorithms that solve them with a reasonable approximation ratio. If a optimization problem does not admit such a polynomial-time algorithm, then the problem is considered to be not tractable.The main goal of this paper is to relativize this specification of tractability. The main reason for this attempt is that we consider the requirement for the tractability to be strong because of the definition of the complexity as the "worst-case" complexity. This definition is also related to the approximation ratio of approximation algorithms and then an optimization problem is considered to be intractable because some subset of problem instances is hard. But in the practice we often have the situation that the hard problem instances do not occur. The general idea of this paper is to try to partition the set of all problem instances of a hard optimization problem into a (possibly infinite) spectrum of subclasses according to their polynomial-time approximability. Searching for a method enabling such a fine problem analysis (classification of problem instances) we introduce the concept of stability of approximation. To show that the application of this concept may lead to a "fine" characterization of the hardness of particular problem instances we consider the traveling salesperson problem and the knapsack problem.