Dominance guarantees for above-average solutions

Gutin et al. [G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1–2) (2002) 107–116] proved that, in the ATSP problem, a tour of weight not exceeding the weight of an average tour is of dominance ratio at least 1/(n−1)1/(n−1) for all n≠6n≠6. (Tours with this property can be easily obtained.) In [N. Alon, G. Gutin, M. Krivelevich, Algorithms with large domination ratio, Journal on Algorithms 50 (2004) 118–131; G. Gutin, A. Vainshtein, A. Yeo, Domination analysis of combinatorial optimization problems, Discrete Applied Mathematics 129 (2–3) (2003) 513–520; G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1–2) (2002) 107–116], algorithms with large dominance ratio were provided for Max Cut, Maxrr-Sat, ATSP, and other problems. All these algorithms share a common property — they provide solutions of quality guaranteed to be not worse than the average solution value. In this paper we show that, in general, this property by itself does not necessarily ensure a good performance in terms of dominance. Specifically, we show that for the MaxSat problem, algorithms with this property might perform poorly in terms of dominance.