Joachim Gudmundsson

Joachim Gudmundsson, Yuan Sha, Sampson Wong

In the celebrated paper of Henzinger, Klein, Rao and Subramanian (1997), it was shown that planar graphs admit a linear time single-source shortest path algorithm. Their algorithm unfortunately does not extend to Euclidean graph classes. We give criteria and prove that any Euclidean graph class satisfying the criteria admits a linear time single-source shortest path algorithm. As a main ingredient, we show that the contracted graphs of these Euclidean graph classes admit sublinear separators.

Sarita de Berg, Joachim Gudmundsson, Peter Kramer et al.

We investigate multiple fundamental variants of the classic coordinated motion planning (CMP) problem for unit square robots in the plane under the $L_1$ metric. In coordinated motion planning, we are given two arrangements of $k$ robots and are tasked with finding a movement schedule that minimizes a certain objective function. The two most prominent objective functions are the sum of distances traveled (Min-Sum) and the latest time of arrival (Min-Makespan). Both objectives have previously been studied extensively. We introduce a new objective function for CMP in the plane. The proposed Min-Exposure objective function defines a set of polygonal regions in the plane that provide cover and asks for a schedule with minimal elapsed time during which at least one robot is partially or fully outside of these regions. We give an $\mathcal{O}(n^4\log n)$ time algorithm that computes exposure-minimal schedules for $k=2$ robots, and an XP algorithm for arbitrary $k$. As a result of independent interest, we leverage new insights to prove that both the Min-Makespan and Min-Sum objectives are fixed-parameter tractable (FPT) parameterized by the number of robots. Our parameterized complexity results generalize known FPT results for rectangular grid graphs [Eiben, Ganian, and Kanj, SoCG'23].

Lintao Wang, Shiwen Xu, Michael Horton et al.

Predicting soccer match outcomes is a challenging task due to the inherently unpredictable nature of the game and the numerous dynamic factors influencing results. While it conventionally relies on meticulous feature engineering, deep learning techniques have recently shown a great promise in learning effective player and team representations directly for soccer outcome prediction. However, existing methods often overlook the heterogeneous nature of interactions among players and teams, which is crucial for accurately modeling match dynamics. To address this gap, we propose HIGFormer (Heterogeneous Interaction Graph Transformer), a novel graph-augmented transformer-based deep learning model for soccer outcome prediction. HIGFormer introduces a multi-level interaction framework that captures both fine-grained player dynamics and high-level team interactions. Specifically, it comprises (1) a Player Interaction Network, which encodes player performance through heterogeneous interaction graphs, combining local graph convolutions with a global graph-augmented transformer; (2) a Team Interaction Network, which constructs interaction graphs from a team-to-team perspective to model historical match relationships; and (3) a Match Comparison Transformer, which jointly analyzes both team and player-level information to predict match outcomes. Extensive experiments on the WyScout Open Access Dataset, a large-scale real-world soccer dataset, demonstrate that HIGFormer significantly outperforms existing methods in prediction accuracy. Furthermore, we provide valuable insights into leveraging our model for player performance evaluation, offering a new perspective on talent scouting and team strategy analysis.

Joachim Gudmundsson, Martin P. Seybold, John Pfeifer

Recent advances in tracking sensors and pose estimation software enable smart systems to use trajectories of skeleton joint locations for supervised learning. We study the problem of accurately recognizing sign language words, which is key to narrowing the communication gap between hard and non-hard of hearing people. Our method explores a geometric feature space that we call `sub-skeleton' aspects of movement. We assess similarity of feature space trajectories using natural, speed invariant distance measures, which enables clear and insightful nearest neighbor classification. The simplicity and scalability of our basic method allows for immediate application in different data domains with little to no parameter tuning. We demonstrate the effectiveness of our basic method, and a boosted variation, with experiments on data from different application domains and tracking technologies. Surprisingly, our simple methods improve sign recognition over recent, state-of-the-art approaches.

Joachim Gudmundsson, Michael Horton, John Pfeifer et al.

We present a scalable approach for range and $k$ nearest neighbor queries under computationally expensive metrics, like the continuous Fréchet distance on trajectory data. Based on clustering for metric indexes, we obtain a dynamic tree structure whose size is linear in the number of trajectories, regardless of the trajectory's individual sizes or the spatial dimension, which allows one to exploit low `intrinsic dimensionality' of data sets for effective search space pruning. Since the distance computation is expensive, generic metric indexing methods are rendered impractical. We present strategies that (i) improve on known upper and lower bound computations, (ii) build cluster trees without any or very few distance calls, and (iii) search using bounds for metric pruning, interval orderings for reduction, and randomized pivoting for reporting the final results. We analyze the efficiency and effectiveness of our methods with extensive experiments on diverse synthetic and real-world data sets. The results show improvement over state-of-the-art methods for exact queries, and even further speed-ups are achieved for queries that may return approximate results. Surprisingly, the majority of exact nearest-neighbor queries on real data sets are answered without any distance computations.

Kevin Buchin, Anne Driemel, Joachim Gudmundsson et al.

The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $\mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $\mathcal{C}$ of $k$ centers (not necessarily part of $\mathcal{G}$) such that the maximum distance between a point in $\mathcal{G}$ and its nearest neighbor in $\mathcal{C}$ is minimized. In this paper we study the corresponding $(k,\ell)$-center problem for polygonal curves under the Fréchet distance, that is, given a set $\mathcal{G}$ of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity $m$, determine a set $\mathcal{C}$ of $k$ polygonal curves in $\mathbb{R}^d$, each of complexity $\ell$, such that the maximum Fréchet distance of a curve in $\mathcal{G}$ to its closest curve in $\mathcal{C}$ is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension $2$ and higher. We show that, if $\ell$ is part of the input, then there is no polynomial-time approximation scheme unless $\mathsf{P}=\mathsf{NP}$. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to $2.598$. This result also holds when $k=1$, and the $\mathsf{NP}$-hardness extends to the case that $\ell=\infty$, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a $3$-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.

Younjin Chung, Joachim Gudmundsson, Masahiro Takatsuka

Estimating output changes by input changes is the main task in causal analysis. In previous work, input and output Self-Organizing Maps (SOMs) were associated for causal analysis of multivariate and nonlinear data. Based on the association, a weight distribution of the output conditional on a given input was obtained over the output map space. Such a weighted SOM pattern of the output changes when the input changes. In order to analyze the change, it is important to measure the difference of the patterns. Many methods have been proposed for the dissimilarity measure of patterns. However, it remains a major challenge when attempting to measure how the patterns change. In this paper, we propose a visualization approach that simplifies the comparison of the difference in terms of the pattern property. Using this approach, the change can be analyzed by integrating colors and star glyph shapes representing the property dissimilarity. Ecological data is used to demonstrate the usefulness of our approach and the experimental results show that our approach provides the change information effectively.

Michael Horton, Joachim Gudmundsson, Sanjay Chawla et al.

A knowledgeable observer of a game of football (soccer) can make a subjective evaluation of the quality of passes made between players during the game. We investigate the problem of producing an automated system to make the same evaluation of passes. We present a model that constructs numerical predictor variables from spatiotemporal match data using feature functions based on methods from computational geometry, and then learns a classification function from labelled examples of the predictor variables. Furthermore, the learned classifiers are analysed to determine if there is a relationship between the complexity of the algorithm that computed the predictor variable and the importance of the variable to the classifier. Experimental results show that we are able to produce a classifier with 85.8% accuracy on classifying passes as Good, OK or Bad, and that the predictor variables computed using complex methods from computational geometry are of moderate importance to the learned classifiers. Finally, we show that the inter-rater agreement on pass classification between the machine classifier and a human observer is of similar magnitude to the agreement between two observers.

Haris Aziz, Serge Gaspers, Joachim Gudmundsson et al.

We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show that computing the winner for proportional approval voting is NP-hard, closing a long standing open problem. As none of the rules are strategyproof, even for dichotomous preferences, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents. In many settings, we show that it is NP-hard for an agent or agents to compute how best to vote given a fixed set of approval ballots from the other agents.