arXiv subjects

Roel Lambers

Publications and source records attributed to Roel Lambers.

6 recordsLinked to original sources

arXiv2025

The uncertainty of a tournament draw: Insights from the Champions League

The group draw of major sports tournaments implies some uncertainty, with lucky teams often enjoying a substantial unfair advantage. First in the literature, we propose a technique to quantify this draw uncertainty, which, arguably, has an optimal level of zero. Our simulation-based approach requires generating a representative set of random draws to compute the variance of qualifying probabilities for each team. The method is applied to compare draw uncertainty in the former group stage and the current incomplete round-robin league phase of the UEFA Champions League, under both accurate and inaccurate seedings. We also break down the impact of the 2024/25 reform into various components. The new format is found to decrease draw uncertainty, but the reduction is mainly attributable to the inaccurate seeding system used by UEFA. Consequently, the primary benefit of an incomplete round-robin tournament compared to the standard group stage lies in the robustness of its draw uncertainty to the seeding of the teams.

stat.AP↗

arXiv2025

Ranking matters: Does the new format select the best teams for the knockout phase in the UEFA Champions League?

Starting in the 2024/25 season, the Union of European Football Associations (UEFA) has fundamentally changed the format of its club competitions: the group stage has been replaced by a league phase played by 36 teams in an incomplete round robin format. This makes ranking the teams based on their results challenging because teams play against different sets of opponents, whose strengths vary. In this research note, we apply several well-known ranking methods for incomplete round robin tournaments to the 2024/25 UEFA Champions League league phase in order to check the robustness of the official ranking, as well as to call the attention of organizers to the non-trivial issue of ranking in these competitions. Our results show that it is doubtful whether the currently used point-based system provides the best ranking of the teams.

physics.soc-ph↗

arXiv2023

Scheduling jobs that change over time

We consider a 1-machine scheduling problem where the temperature of a job rises during processing, and cools down when not being processed according to given linear heating and cooling rates. No job's temperature is allowed to rise above a given threshold, and no job's temperature can cool below 0. Another crucial property of our problem is that jobs can be preempted an arbitrary number of times, and even more, we allow that a job is processed for an infinitely small amount of time. We consider two objectives: minimize the makespan, and minimize the sum of completion times. Our results are as follows. We show how to compactly represent a solution. Further, we prove that the problem of minimizing the sum of completion times can be solved in polynomial time by formulating it as a linear program, and deriving a structural property. This result can be extended to hold for any number of machines. Further, we show that a minimum makespan can be found in $O(n)$ time, even when heating and cooling rates are job-dependent.

math.OC↗

arXiv2022

How to Design a Stable Serial Knockout Competition

We investigate a new tournament format that consists of a series of individual knockout tournaments; we call this new format a Serial Knockout Competition (SKC). This format has recently been adopted by the Professional Darts Corporation. Depending on the seedings of the players used for each of the knockout tournaments, players can meet in the various rounds (eg first round, second round, ..., semi-final, final) of the knockout tournaments. Following a fairness principle of treating all players equal, we identify an attractive property of an SKC: each pair of players should potentially meet equally often in each of the rounds of the SKC. If the seedings are such that this property is indeed present, we call the resulting SKC stable. In this note we formalize this notion, and we address the question: do there exist seedings for each of the knockout tournaments such that the resulting SKC is stable? We show, using a connection to the Fano plane, that the answer is yes for 8 players. We show how to generalize this to any number of players that is a power of 2, and we provide stable schedules for competitions on 16 and 32 players

math.CO↗

arXiv2022

Integer Programming Models for Round Robin Tournaments

Round robin tournaments are omnipresent in sport competitions and beyond. We propose two new integer programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. We analytically compare their linear relaxations with the linear relaxation of a well-known traditional formulation. We find that the matching formulation is stronger than the other formulations, while its LP relaxation is still being solvable in polynomial time. In addition, we provide an exponentially sized class of valid inequalities for the matching formulation. Complementing our theoretical assessment of the strength of the different formulations, we also experimentally show that the matching formulation is superior on a broad set of instances. Finally, we describe a branch-and-price algorithm for finding round robin tournaments that is based on the matching formulation.

math.OC↗

arXiv2021

Minimum Scan Cover and Variants -- Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.

cs.CG↗