On Controlling Knockout Tournaments Without Perfect Information

TL;DR

The paper studies seedings for single-elimination tournaments under imperfect information, introducing Probabilistic Tournament Fixing (PTF) and its deterministic surrogate Simultaneous Tournament Fixing (STF). It develops an FPT algorithm for STF parameterized by the combined parameter $k$ (the sum of the shared FAS size and the number of private arcs) via blueprint templates and an ILP-Feas feasibility reduction, and shows para-NP-hardness when either component is fixed alone. It then applies STF to obtain an FPT algorithm for PTF parameterized by $c_1$ and $c_2$, by enumerating outcomes of fractional matches and solving STF on each completion; TF is recovered as the $m=1$ case. The results bridge a gap in the literature on probabilistic SE tournament design and suggest directions for future work on other parameterizations and robust design objectives.

Abstract

Over the last decade, extensive research has been conducted on the algorithmic aspects of designing single-elimination (SE) tournaments. Addressing natural questions of algorithmic tractability, we identify key properties of input instances that enable the tournament designer to efficiently schedule the tournament in a way that maximizes the chances of a preferred player winning. Much of the prior algorithmic work on this topic focuses on the perfect (complete and deterministic) information scenario, especially in the context of fixed-parameter algorithm design. Our contributions constitute the first fixed-parameter tractability results applicable to more general settings of SE tournament design with potential imperfect information.

Figures (4)

• Figure 1: An example of PTF. Thickness represents probability of a player winning and probabilities that influence a match are marked by shapes. Here, player 4 wins with probability $6/10$.
• Figure 2: The set $\mathsf{Types}\xspace = (1,a_1,2,a_2,3,a_3,4,a_4,5,a_5,6)$ sorted according to $\prec$. The flexible types are depicted containing the players of that type; note $\tau^{-1}(4)=\emptyset$. The back arcs are depicted above while the remaining arcs go "right", below the vertices we depict some of the remaining arcs for illustration.
• Figure 3: Example of a blueprint subtree on a bracket with $n=16$ and $|V_A|=5$. The blueprint subtree consists of solid vertices and edges. Empty vertices and dotted edges are in the bracket, but not in the blueprint. The colored thick edges mark the blueprint partition into paths $P_j$ for $j \in [5]$.
• Figure 4: Depiction of $\ell_i$ in cases during creation of the ILP instance. Arrows depict match where the player at arrow tail wins. Red bold path depicts the player that gets to $v$ in $\ell_i$. (\ref{['fig:case_a']})$u$ beats $w$ so all leaves necessarily lose to $u$. (\ref{['fig:case_b']}) There is a leaf $z$ that beats $u$ and it also beats all other leaves.

Theorems & Definitions (16)

• Definition 1: Brackets
• Definition 2
• Lemma 3
• Proposition 4: Lenstra83Kannan87FrankTardos87
• Theorem 5
• Lemma 6
• Definition 10
• Lemma 12
• Lemma 13
• Lemma 14