arXiv subjects

Philip Mayer

Publications and source records attributed to Philip Mayer.

4 recordsLinked to original sources

arXiv2026

A Separator-based Algorithm for the Graph Edit Distance Problem

The Graph Edit Distance (GED) is a widely used graph similarity measure asking for the minimum cost of a sequence of edits transforming one (labeled) graph into another. The considered edit operations are deletion, insertion, and relabeling of nodes and edges. Special cases include the Graph Isomorphism problem, as well as many other graph problems that ask for the existence or minimum cost of a certain substructure, like the Traveling Salesman or Maximum Clique problem. We present a novel exponential time algorithm to compute the exact GED and a corresponding edit sequence in $O^*(4 + \varepsilon)^n$ time and polynomial space, provided one of the two graphs admits strictly sublinear balanced separators. In particular, the claimed runtime holds if one of the graphs is $K_h$-minor free (e.g., planar), or has bounded treewidth, which is the case for many real-world applications (e.g., all instances in GEDLIB). This substantially improves the best known worst-case running time bounds of $O^*(n!)$ for these graph classes.

cs.DS↗

arXiv2025

Bicriteria Polygon Aggregation with Arbitrary Shapes

We study the problem of aggregating a set of polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Equivalently, this can be seen as a fence enclosure problem, where the goal is to surround the polygons with a set of closed curves. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions, which naturally extends other fencing problems and has applications in geographical information systems. Previously, this objective was only studied in a restricted variant, in which the boundary curves of the regions must be selected from a fixed subdivision of the plane. It is natural to ask whether the problem is still tractable if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of an algorithm with runtime $\mathcal{\tilde{O}}(n^4)$, where $n$ is the number of input vertices. To achieve this, we fully characterize the optimal solutions by showing that their boundaries are composed of input edges and circular arcs of constant radius. Additionally, we consider the parametric problem, where for every weighting factor we seek a solution that is optimal for that trade-off of area and perimeter. We show that $\mathcal{O}(n^2)$ combinatorial solutions suffice to describe all optimal solutions across all weighting factors, and provide both an exact algorithm and an approximation scheme. To make the algorithms scalable in practice, we develop engineering techniques that exploit structural properties of the solutions. Our experimental evaluation on real-world data shows linear runtime in practice, even for the parametric variant.

cs.CG↗

arXiv2024

A Simpler Approach for Monotone Parametric Minimum Cut: Finding the Breakpoints in Order

We present parametric breadth-first search (PBFS), a new algorithm for solving the parametric minimum cut problem in a network with source-sink-monotone capacities. The objective is to find the set of breakpoints, i.e., the points at which the minimum cut changes. It is well known that this problem can be solved in the same asymptotic runtime as the static minimum cut problem. However, existing algorithms that achieve this runtime bound involve fairly complicated steps that are inefficient in practice. PBFS uses a simpler approach that discovers the breakpoints in ascending order, which allows it to achieve the desired runtime bound while still performing well in practice. We evaluate our algorithm on benchmark instances from polygon aggregation and computer vision. Polygon aggregation was recently proposed as an application for parametric minimum cut, but the monotonicity property has not been exploited fully. PBFS outperforms the state of the art on most benchmark instances, usually by a factor of 2-3. It is particularly strong on instances with many breakpoints, which is the case for polygon aggregation. Compared to the existing min-cut-based approach for polygon aggregation, PBFS scales much better with the instance size. On large instances with millions of vertices, it is able to compute all breakpoints in a matter of seconds.

cs.DS↗

arXiv2022

Minimum-Error Triangulations for Sea Surface Reconstruction

We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al.~(Computers \& Geosciences, 157:104920, 2021) who have suggested to learn a triangulation $D$ of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by $D$ to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay ($k$-OD) triangulations and used an integer linear program in order to solve the resulting optimization problem. In geometric terms, the input to our problem consists of two sets of points in $\mathbb{R}^2$ with elevations: a set $\mathcal{S}$ that is to be triangulated, and a set $\mathcal{R}$ of reference points. Intuitively, we define the error of a triangulation as the average vertical distance of a point in $\mathcal{R}$ to the triangulated surface that is obtained by interpolating elevations of $\mathcal{S}$ linearly in each triangle. Our goal is to find the triangulation of $\mathcal{S}$ that has minimum error with respect to $\mathcal{R}$. In our work, we prove that the minimum-error triangulation problem is NP-hard and cannot be approximated within any multiplicative factor in polynomial time unless $P=NP$. At the same time we show that the problem instances that occur in our application (considering sea level data from several hundreds of tide gauge stations worldwide) can be solved relatively fast using dynamic programming when restricted to $k$-OD triangulations for $k\le 7$. In particular, instances for which the number of connected components of the so-called $k$-OD fixed-edge graph is small can be solved within few seconds.

cs.CG↗