Knockout tournaments, also known as single-elimination or cup tournaments, are a popular form of sports competitions. In the standard probabilistic setting, for each pairing of players, one of the players wins the game with a certain (a priory known) probability. Due to their competitive nature, tournaments are prone to manipulation. We investigate the computational problem of determining whether, for a given tournament, a coalition has a manipulation strategy that increases the winning probability of a designated player above a given threshold. More precisely, in every round of the tournament, coalition players can strategically decide which games to throw based on the advancement of other players to the current round. We call this setting adaptive constructive coalition manipulation. To the best of our knowledge, while coalition manipulation has been studied in the literature, this is the first work to introduce adaptiveness to this context. We show that the above problem is hard for every complexity class in the polynomial hierarchy. On the algorithmic side, we show that the problem is solvable in polynomial time when the coalition size is a constant. Furthermore, we show that the problem is fixed-parameter tractable when parameterized by the coalition size and the size of a minimum player set that must include at least one player from each non-deterministic game. Lastly, we investigate a generalized setting where the tournament tree can be imbalanced.

Given a graph G , a matching is a subset of edges of G that do not share an endpoint. A matching M is uniquely restricted if the subgraph induced by the endpoints of the edges of M has exactly one perfect matching. Given a graph G and a positive integer ℓ , Uniquely Restricted Matching asks whether G has a uniquely restricted matching of size at least ℓ . In this paper, we study the parameterized complexity of Uniquely Restricted Matching under various parameters. Specifically, we show that Uniquely Restricted Matching admits a fixed-parameter tractable (FPT) algorithm on line graphs when parameterized by the solution size. We also establish that the problem is FPT when parameterized by the treewidth of the input graph. Furthermore, we show that Uniquely Restricted Matching does not admit a polynomial kernel with respect to the vertex cover number plus the size of the matching unless NP ⊆ coNP/poly.

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.

Abstract. A matching is a subset of edges in a graph [Formula: see text] that do not share an endpoint. A matching [Formula: see text] is a [Formula: see text] -matching if the subgraph of [Formula: see text] induced by the endpoints of the edges of [Formula: see text] satisfies property [Formula: see text]. For example, if the property [Formula: see text] is that of being a matching, being acyclic, or being disconnected, then we obtain an induced matching, an acyclic matching, and a disconnected matching, respectively. Given a graph [Formula: see text] and a positive integer [Formula: see text], the [Formula: see text] Matching problem asks whether [Formula: see text] has a [Formula: see text]-matching of size at least [Formula: see text]. In this paper, we analyze the [Formula: see text] Matching problems from the viewpoint of Parameterized Complexity with respect to the parameter treewidth. In particular, we present a deterministic algorithm solving Induced Matching in [Formula: see text] time and a randomized algorithm solving Acyclic Matching in [Formula: see text] time. For any fixed [Formula: see text], [Formula: see text]-Disconnected Matching can be solved in [Formula: see text] time by a deterministic algorithm. Additionally, assuming the Exponential Time Hypothesis, we show that Disconnected Matching has no [Formula: see text]-time algorithm.

Given a mapping from a set of players to the leaves of a complete binary tree (called a seeding), a knockout tournament is conducted as follows: every round, every two players with a common parent compete against each other, and the winner is promoted to the common parent; then, the leaves are deleted. When only one player remains, it is declared the winner. This is a popular competition format in sports, elections, and decision-making. Over the past decade, it has been studied intensively from both theoretical and practical points of view. Most frequently, the objective is to seed the tournament in a way that ``assists'' (or even guarantees) some particular player to win the competition. We introduce a new objective, which is very sensible from the perspective of the directors of the competition: maximize the profit or popularity of the tournament. Specifically, we associate a ``score'' with every possible match, and aim to seed the tournament to maximize the sum of the scores of the matches that take place. We focus on the case where we assume a total order on the players' strengths, and provide a wide spectrum of results on the computational complexity of the problem.