A tabu-search for minimising the carry-over effects value of a round-robin tournament

Abstract

A player b in a round-robin sports tournament receives a carry-over effect from another player a if some third player opposes a in round i and b in round i + 1. Let γab denote the number of times player b receives a carry-over effect from player a during a tournament. Then the carry-over effects value of the entire tournament T on n players is given by Γ(T) = ?n i=1?n j=1 γ2 ij. Furthermore, let Γ(n) denote the minimum carry-over effects value over all round-robin tournaments on n players. A strict lower bound on Γ(n) is n(n − 1) (in which case there exists a round-robin tournament of order n such that each player receives a carry- over effect from each other player exactly once), and it is known that this bound is attained for n = 2r or n = 20, 22. It is also known that round-robin tournaments can be constructed from so-called starters; round-robin tournaments constructed in this way are called cyclic. It has previously been shown that cyclic round-robin tournaments have the potential of admitting small values for Γ(T), and in this paper a tabu-search is used to find starters which produce cyclic tournaments with small carry-over effects values. The best solutions in the literature are matched for n ≤ 22, and new upper bounds are established on Γ(n) for 24 ≤ n ≤ 40.