Inspired by the increasing popularity of Swiss-system tournaments in sports, we study the problem of predetermining the number of rounds that can be guaranteed in a Swiss-system tournament. Matches of these tournaments are usually determined in a myopic round-based way dependent on the results of previous rounds. Together with the hard constraint that no two players meet more than once during the tournament, at some point it might become infeasible to schedule a next round. For tournaments with $n$ players and match sizes of $k\ge 2$ players, we prove that we can always guarantee $\lfloor \frac{n}{k(k-1)}\rfloor$ rounds. We show that this bound is tight. This provides a simple polynomial time constant factor approximation algorithm for the social golfer problem.

We extend the results to the Oberwolfach problem. We show that a simple greedy approach guarantees at least $\lfloor \frac{n+4}{6}\rfloor$ rounds for the Oberwolfach problem. This yields a polynomial time $\frac{1}{3+\epsilon }$-approximation algorithm for any fixed $\epsilon >0$ for the Oberwolfach problem. Assuming that El-Zahar’s conjecture is true, we improve the bound on the number of rounds to be essentially tight.

受瑞士制锦标赛在体育运动中日益普及的启发，我们研究了预先确定在瑞士制锦标赛中可以保证的回合数的问题。这些锦标赛的比赛通常以近视回合的方式确定，具体取决于前几轮的结果。再加上在比赛期间没有两名球员会面超过一次的硬性限制，在某些时候安排下一轮比赛可能变得不可行。对于与$n$球员和比赛规模$ķ\ge 2$玩家们，我们证明了我们永远可以保证$\lfloor \frac{n}{ķ(ķ-1)}\rfloor$回合。我们证明这个界限是紧密的。这为社交高尔夫球手问题提供了一个简单的多项式时间常数因子近似算法。

我们将结果扩展到 Oberwolfach 问题。我们证明了一个简单的贪心方法至少可以保证$\lfloor \frac{n+4}{6}\rfloor$Oberwolfach 问题的轮次。这产生一个多项式时间$\frac{1}{3+\epsilon }$- 任何固定的近似算法$\epsilon >0$对于奥伯沃尔法赫问题。假设 El-Zahar 的猜想是正确的，我们改进了对轮数的限制，使其本质上是严格的。