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Certain general results relating approximation ratio and combinatorial dominance guarantees for optimization problems over subsets are established.
Although approximation ratio analysis has been highly successful in increasing our understanding of heuristics, it does not paint a complete picture of their performance in practice.
The issue is not whether the combinatorial dominance measure is better than approximation ratio, as this measure is complementary to the approximation ratio measure.
Note that it is not clear how combinatorial dominance compares to approximation ratio as a measure of the quality of a heuristic.
Certain general results relating approximation ratio and combinatorial dominance guarantees, particularly for optimization problems over subsets, are established.
We can also give general bounds on the domination number as a function of the approximation ratio of certain heuristics.
The assumption regarding the approximation ratio of [H.
A heuristic guaranteed to have an approximation ratio of k may well select a set of size k[n/(k + 1)], which will dominate (up to a constant factor) exactly the number of solutions asserted by the theorem.
We have proposed a model to better study the quality of heuristics for problems which are provably inapproximable, or heuristic techniques which do not lend themselves to approximation ratio analysis.
This elegant algorithm gives the same running time and approximation ratio as the algorithm presented in this paper.
We expect the runtime and the approximation ratio to depend on the block size.
Another open problem is whether one can improve the approximation ratio by looking at a larger neighborhood of the leftmost vertex of the uppermost row.