A Greedy Algorithm for the Social Golfer and the Oberwolfach Problem
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Inspired by the increasing popularity of Swiss-system tournaments in sports, we study the problem of predetermining the total number of rounds that can at least be guaranteed in a Swiss-system tournament. For tournaments with n participants, we prove that we can always guarantee n/2 rounds under the constraint that no player faces the same opponent more than once. We show that this bound is tight. We generalize our results to two combinatorial problems, namely the social golfer problem and the Oberwolfach problem. The social golfer problem can be seen as a tournament with match sizes of k≥2 players in which no pair of players meets each other more than once. For a natural greedy algorithm, we show that it calculates at least ⌊ n/(k(k-1)) ⌋ rounds and that this bound is tight. This gives rise to a simple polynomial time 1/k-approximation algorithm for k-somes in golf tournaments. Up to our knowledge, this is the first analysis of an approximation algorithm for the social golfer problem. In the Oberwolfach problem, a match corresponds to a group of k players positioned in a cycle and the constraint is that no participant should meet the same neighbor more than once. We show that the simple greedy approach guarantees at least ⌊ (n+4)/6 ⌋ rounds for the Oberwolfach problem. Assuming that El-Zahar's conjecture is true, we improve the bound to be essentially tight.
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