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An Exercise in Tournament Design: When Some Matches Must Be Scheduled

Sushmita Gupta, M. S. Ramanujan, Peter Strulo

TL;DR

This work studies Demand-TF, a deterministic SE-tournament design problem where a bracket must realize a given set of demanded matches. It reveals a dichotomy: NP-hard in general, yet solvable in polynomial time when the tournament has bounded feedback arc set, with broader parameterized results (FPT/XP) under additional constraints. The authors develop a DP framework around binomial arborescences, augmented by partial structures (PBA) and a packing subroutine, achieving an $n^{O(k)}$-time algorithm for FAS size $k$, plus an exact $3^n$-time method and ETH-backed lower bounds. They extend the approach to variants with demanded upset edges and specified rounds, and discuss extensions to weighted demands and edge-constrained SI, underscoring practical implications for bracket design and entertainment-driven scheduling.

Abstract

Single-elimination (SE) tournaments are a popular format used in competitive environments and decision making. Algorithms for SE tournament manipulation have been an active topic of research in recent years. In this paper, we initiate the algorithmic study of a novel variant of SE tournament manipulation that aims to model the fact that certain matchups are highly desired in a sporting context, incentivizing an organizer to manipulate the bracket to make such matchups take place. We obtain both hardness and tractability results. We show that while the problem of computing a bracket enforcing a given set of matches in an SE tournament is NP-hard, there are natural restrictions that lead to polynomial-time solvability. In particular, we show polynomial-time solvability if there is a linear ordering on the ability of players with only a constant number of exceptions where a player with lower ability beats a player with higher ability.

An Exercise in Tournament Design: When Some Matches Must Be Scheduled

TL;DR

This work studies Demand-TF, a deterministic SE-tournament design problem where a bracket must realize a given set of demanded matches. It reveals a dichotomy: NP-hard in general, yet solvable in polynomial time when the tournament has bounded feedback arc set, with broader parameterized results (FPT/XP) under additional constraints. The authors develop a DP framework around binomial arborescences, augmented by partial structures (PBA) and a packing subroutine, achieving an -time algorithm for FAS size , plus an exact -time method and ETH-backed lower bounds. They extend the approach to variants with demanded upset edges and specified rounds, and discuss extensions to weighted demands and edge-constrained SI, underscoring practical implications for bracket design and entertainment-driven scheduling.

Abstract

Single-elimination (SE) tournaments are a popular format used in competitive environments and decision making. Algorithms for SE tournament manipulation have been an active topic of research in recent years. In this paper, we initiate the algorithmic study of a novel variant of SE tournament manipulation that aims to model the fact that certain matchups are highly desired in a sporting context, incentivizing an organizer to manipulate the bracket to make such matchups take place. We obtain both hardness and tractability results. We show that while the problem of computing a bracket enforcing a given set of matches in an SE tournament is NP-hard, there are natural restrictions that lead to polynomial-time solvability. In particular, we show polynomial-time solvability if there is a linear ordering on the ability of players with only a constant number of exceptions where a player with lower ability beats a player with higher ability.
Paper Structure (17 sections, 12 theorems, 10 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 12 theorems, 10 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Organization of the paper.
  4. Preliminaries
  5. Binomial arborescences.
  6. Hardness Results for Demand-TF
  7. Going beyond Theorem \ref{['thm:exactExponentialTimeAlgorithm']}
  8. Demand-TF on Graphs of Bounded Feedback Arc Set Number
  9. Guaranteed compactness.
  10. Partial Binomial Arborescences.
  11. Guessed size.
  12. The subroutine Pack.
  13. Correctness
  14. FPT Algorithm for Demand-TF When Upsets Are Demanded
  15. XP Algorithm for Demand-TF With Specified Rounds for Demands
  16. ...and 2 more sections

Key Result

Proposition 1

Let $T$ be a tournament digraph and let ${\mathcal{S}}\subseteq A(T)$. Then, there is a seeding of $V(T)$ such that the resulting SE tournament has every match in $\mathcal{S}$ if and only if $T$ has an $\mathsf{SBA}$$H$ such that $A(H) \supseteq \mathcal{S}$.

Figures (2)

  • Figure 2: A $\mathsf{BA}$$H$ and the feedback descendants of its root. Feedback vertices are in red and demand arcs are in blue. $\textsf{ht}^*_g$ (where $g$ is $\textsf{ht}_H$ restricted to $V(F)$) is noted next to each vertex. $\textsf{ht}^*_g(v)=3$ due to its demand siblings. The green arcs are those that would be added during the inner loop of the algorithm when $v_{n-i}=v$ and $j=2$. The call to Pack would use $P=\{w,x,y\}$, add the arcs $(w,x)$ and $(x,y)$ and output $w$. Finally Step \ref{['step:edge']} would add the arc $(v, w)$.
  • Figure 3: An example of the operation of Lemma \ref{['lem:converseexchange']}. The numbers represent $\textsf{sz}_K$. Note that in this case $y$ beats $u$ so has taken the place of $u$ as the child of $v_{n-i}$ of size 16. Both $v_{n-i}$ and $w$ have many other children not pictured here. Note also that it may be the case that $v_{n-i} = w$.

Theorems & Definitions (35)

  • Definition 1: Williams10 Williams10
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Claim 1
  • ...and 25 more