# Performance Guarantees for Approximation Algorithms Depending on Parametrized Triangle Inequalities

## Abstract

The worst-case analyses of heuristics in combinatorial optimization are often far too pessimistic when confronted with performance on real-world problems. One approach to partially overcome this discrepancy is to resort to average-case analyses by stipulating realistic distributions of input data. Another way is to incorporate a priori information on the potential domain of the input data, for instance, assuming the triangle inequality for input matrices is in some cases instrumental for establishing approximation algorithms with fixed performance guarantee. Now, a parametrized form of the triangle inequality has a considerably larger range of applicability and allows the prediction of the heuristics performance, where otherwise no bound could be provided. For example, it is interesting to observe that two well-known approximation algorithms for the Traveling Salesman Problem (TSP), assuming the triangle inequality, behave differently when one relaxes the imposed triangle inequality. The double-spanning-tree heuristic can be adjusted (by suitably extracting a Hamilton circuit from a Eulerian walk) to yield an approximation algorithm with performance guarantee increasing quadratically with the parameter governing the relaxed triangle inequality. The Christofides algorithm cannot be modified in this way and hence does not tolerate a relaxation of the standard triangle inequality without loosing the bound on its relative performance.

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