Challenge the champ tournaments are one of the simplest forms of competition, where a (initially selected) champ is repeatedly challenged by other players. If a player beats the champ, then that player is considered the new (current) champ. Each player in the competition challenges the current champ once in a fixed order. The champ of the last round is considered the winner of the tournament. We investigate a setting where players can be bribed to lower their winning probability against the initial champ. The goal is to maximize the probability of the initial champ winning the tournament by bribing the other players, while not exceeding a given budget for the bribes. Mattei et al. [Journal of Applied Logic, 2015] showed that the problem can be solved in pseudo-polynomial time, and that it is in XP when parameterized by the number of players. We show that the problem is weakly NP-hard and W[1]-hard when parameterized by the number of players. On the algorithmic side, we show that the problem is fixed-parameter tractable when parameterized either by the number of different bribe values or the number of different probability values. To this end, we establish several results that are of independent interest. In particular, we show that the product knapsack problem is W[1]-hard when parameterized by the number of items in the knapsack, and that constructive bribery for cup tournaments is W[1]-hard when parameterized by the number of players. Furthermore, we present a novel way of designing mixed integer linear programs, ensuring optimal solutions where all variables are integers.

翻译：战冠锦标赛是最简单的竞赛形式之一，其中（最初选定的）冠军会反复受到其他选手的挑战。若某选手击败冠军，则该选手成为新的（当前）冠军。每位参赛者按固定顺序挑战当前冠军一次，最后一轮的冠军被视为锦标赛的胜者。我们研究一种场景：可通过贿赂降低其他选手对阵初始冠军的获胜概率。目标是贿赂其他选手以最大化初始冠军赢得锦标赛的概率，同时不超过给定的贿赂预算。Mattei等人[Journal of Applied Logic, 2015] 证明该问题可以在伪多项式时间内求解，并在以选手数量为参数时属于XP类。我们证明该问题是弱NP难问题，且以选手数量为参数时属于W[1]-难问题。在算法方面，我们证明当参数分别为不同贿赂值的数量或不同概率值的数量时，该问题具有固定参数可解性。为此，我们建立了若干具有独立意义的结果。特别地，我们证明：以背包中物品数量为参数时，乘积背包问题是W[1]-难问题；以选手数量为参数时，杯赛中的建设性贿赂问题是W[1]-难问题。此外，我们提出了一种设计混合整数线性规划的新方法，可确保所有变量均为整数的最优解。