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Oliver Schaudt

Publications and source records attributed to Oliver Schaudt.

At least 19 recordsLinked to original sources
arXiv2026

Minimal obstructions to $C_5$-coloring in hereditary graph classes

For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Note that if $H$ is the triangle, then $H$-colorings are equivalent to $3$-colorings. In this paper we are interested in the case that $H$ is the five-vertex cycle $C_5$. A minimal obstruction to $C_5$-coloring is a graph that does not have a $C_5$-coloring, but every proper induced subgraph thereof has a $C_5$-coloring. In this paper we are interested in minimal obstructions to $C_5$-coloring in $F$-free graphs, i.e., graphs that exclude some fixed graph $F$ as an induced subgraph. Let $P_t$ denote the path on $t$ vertices, and let $S_{a,b,c}$ denote the graph obtained from paths $P_{a+1},P_{b+1},P_{c+1}$ by identifying one of their endvertices. We show that there is only a finite number of minimal obstructions to $C_5$-coloring among $F$-free graphs, where $F \in \{ P_8, S_{2,2,1}, S_{3,1,1}\}$ and explicitly determine all such obstructions. This extends the results of Kamiński and Pstrucha [Discr. Appl. Math. 261, 2019] who proved that there is only a finite number of $P_7$-free minimal obstructions to $C_5$-coloring, and of Dębski et al. [ISAAC 2022 Proc.] who showed that the triangle is the unique $S_{2,1,1}$-free minimal obstruction to $C_5$-coloring. We complement our results with a construction of an infinite family of minimal obstructions to $C_5$-coloring, which are simultaneously $P_{13}$-free and $S_{2,2,2}$-free. We also discuss infinite families of $F$-free minimal obstructions to $H$-coloring for other graphs $H$.

math.CO↗
arXiv2025

Efficient Quantum Convolutional Neural Networks for Image Classification: Overcoming Hardware Constraints

While classical convolutional neural networks (CNNs) have revolutionized image classification, the emergence of quantum computing presents new opportunities for enhancing neural network architectures. Quantum CNNs (QCNNs) leverage quantum mechanical properties and hold potential to outperform classical approaches. However, their implementation on current noisy intermediate-scale quantum (NISQ) devices remains challenging due to hardware limitations. In our research, we address this challenge by introducing an encoding scheme that significantly reduces the input dimensionality. We demonstrate that a primitive QCNN architecture with 49 qubits is sufficient to directly process $28\times 28$ pixel MNIST images, eliminating the need for classical dimensionality reduction pre-processing. Additionally, we propose an automated framework based on expressibility, entanglement, and complexity characteristics to identify the building blocks of QCNNs, parameterized quantum circuits (PQCs). Our approach demonstrates advantages in accuracy and convergence speed with a similar parameter count compared to both hybrid QCNNs and classical CNNs. We validated our experiments on IBM's Heron r2 quantum processor, achieving $96.08\%$ classification accuracy, surpassing the $71.74\%$ benchmark of traditional approaches under identical training conditions. These results represent one of the first implementations of image classifications on real quantum hardware and validate the potential of quantum computing in this area.

quant-ph↗
arXiv2020

Fault-Tolerant Edge-Disjoint Paths -- Beyond Uniform Faults

The overwhelming majority of survivable (fault-tolerant) network design models assume a uniform fault model. Such a model assumes that every subset of the network resources (edges or vertices) of a given cardinality $k$ may fail. While this approach yields problems with clean combinatorial structure and good algorithms, it often fails to capture the true nature of the scenario set coming from applications. One natural refinement of the uniform model is obtained by partitioning the set of resources into vulnerable and safe resources. The scenario set contains every subset of at most $k$ faulty resources. This work studies the Fault-Tolerant Path (FTP) problem, the counterpart of the Shortest Path problem in this fault model and the Fault-Tolerant Flow problem (FTF), the counterpart of the $\ell$-disjoint Shortest $s$-$t$ Path problem. We present complexity results alongside exact and approximation algorithms for both models. We emphasize the vast increase in the complexity of the problem with respect to the uniform analogue, the Edge-Disjoint Paths problem.

cs.DS↗
arXiv2020

Discrete Optimization Methods for Group Model Selection in Compressed Sensing

In this article we study the problem of signal recovery for group models. More precisely for a given set of groups, each containing a small subset of indices, and for given linear sketches of the true signal vector which is known to be group-sparse in the sense that its support is contained in the union of a small number of these groups, we study algorithms which successfully recover the true signal just by the knowledge of its linear sketches. We derive model projection complexity results and algorithms for more general group models than the state-of-the-art. We consider two versions of the classical Iterative Hard Thresholding algorithm (IHT). The classical version iteratively calculates the exact projection of a vector onto the group model, while the approximate version (AM-IHT) uses a head- and a tail-approximation iteratively. We apply both variants to group models and analyse the two cases where the sensing matrix is a Gaussian matrix and a model expander matrix. To solve the exact projection problem on the group model, which is known to be equivalent to the maximum weight coverage problem, we use discrete optimization methods based on dynamic programming and Benders' Decomposition. The head- and tail-approximations are derived by a classical greedy-method and LP-rounding, respectively.

math.OC↗
arXiv2020

Better 3-coloring algorithms: excluding a triangle and a seven vertex path

We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a special case of the problem solved in [Combinatorica 38(4):779--801, 2018], that does not require the absence of triangles, the algorithm here is both faster and conceptually simpler. The complexity of the algorithm is $O(|V(G)|^5(|V(G)|+|E(G)|))$, and if $G$ is bipartite, it improves to $O(|V(G)|^2(|V(G)|+|E(G)|))$. Moreover, we prove that there are finitely many minimal obstructions to list 3-coloring $\{P_t,C_3\}$-free graphs if and only if $t \leq 7$. This implies the existence of a polynomial time certifying algorithm for list 3-coloring in $\{P_7,C_3\}$-free graphs. We furthermore determine other cases of $t, \ell$, and $k$ such that the family of minimal obstructions to list $k$-coloring in $\{P_t,C_{\ell}\}$-free graphs is finite.

math.CO↗
arXiv2019

Erdős-Pósa property for labelled minors: 2-connected minors

In the 1960s, Erdős and Pósa proved that there is a packing-covering duality for cycles in graphs. As part of the graph minor project, Robertson and Seymour greatly extended this: there is such a duality for $H$-expansions in graphs if and only if $H$ is a planar graph (this includes the previous result for $H=K_3$). We consider vertex labelled graphs and minors and provide such a characterisation for $2$-connected labelled graphs $H$.

math.CO↗
arXiv2019

Improved approximation algorithms for hitting 3-vertex paths

We study the problem of deleting a minimum cost set of vertices from a given vertex-weighted graph in such a way that the resulting graph has no induced path on three vertices. This problem is often called cluster vertex deletion in the literature and admits a straightforward 3-approximation algorithm since it is a special case of the vertex cover problem on a 3-uniform hypergraph. Recently, You, Wang, and Cao described an efficient 5/2-approximation algorithm for the unweighted version of the problem. Our main result is a 9/4-approximation algorithm for arbitrary weights, using the local ratio technique. We further conjecture that the problem admits a 2-approximation algorithm and give some support for the conjecture. This is in sharp contrast with the fact that the similar problem of deleting vertices to eliminate all triangles in a graph is known to be UGC-hard to approximate to within a ratio better than 3, as proved by Guruswami and Lee.

cs.DS↗
arXiv2018

How to Secure Matchings Against Edge Failures

Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We provide efficient exact and approximation algorithms. In particular, for the unit-cost problem, we provide a $\log_2 n$-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we settle the approximability of the problem and show a close relation to the Directed Steiner Forest Problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. Our methods rely on a close relationship to the classical strong connectivity augmentation problem and directed Steiner problems.

cs.DS↗
arXiv2018

Obstructions for three-coloring and list three-coloring $H$-free graphs

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the case when $H$ is connected. This solves a problem posed by Golovach et al. As a second result, we characterize all graphs $H$ for which there are only finitely many $H$-free minimal obstructions for list 3-colorability.

math.CO↗
arXiv2017

Fast Algorithms for Delta-Separated Sparsity Projection

We describe a fast approximation algorithm for the $Δ$-separated sparsity projection problem. The $Δ$-separated sparsity model was introduced by Hegde, Duarte and Cevher (2009) to capture the firing process of a single Poisson neuron with absolute refractoriness. The running time of our projection algorithm is linear for an arbitrary (but fixed) precision and it is both a head and a tail approximation. This solves a problem of Hegde, Indyk and Schmidt (2015). We also describe how our algorithm fits into the approximate model iterative hard tresholding framework of Hegde, Indyk and Schmidt (2014) that allows to recover $Δ$-separated sparse signals from noisy random linear measurements. The resulting recovery algorithm is substantially faster than the existing one, at least for large data sets.

cs.DS↗
arXiv2017

The Parameterized Complexity of the Equidomination Problem

A graph $G=(V,E)$ is called equidominating if there exists a value $t \in \mathbb{N}$ and a weight function $ω: V \rightarrow \mathbb{N}$ such that the total weight of a subset $D\subseteq V$ is equal to $t$ if and only if $D$ is a minimal dominating set. To decide whether or not a given graph is equidominating is referred to as the Equidomination problem. In this paper we show that two parameterized versions of the Equidomination problem are fixed-parameter tractable: the first parameterization considers the target value $t$ leading to the Target-$t$ Equidomination problem. The second parameterization allows only weights up to a value $k$, which yields the $k$-Equidomination problem. In addition, we characterize the graphs whose every induced subgraph is equidominating. We give a finite forbidden induced subgraph characterization and derive a fast recognition algorithm.

cs.CC↗
arXiv2016

On bounding the difference between the maximum degree and the chromatic number by a constant

We provide a finite forbidden induced subgraph characterization for the graph class $\varUpsilon_k$, for all $k \in \mathbb{N}_0$, which is defined as follows. A graph is in $\varUpsilon_k$ if for any induced subgraph, $Δ\leq χ-1 + k$ holds, where $Δ$ is the maximum degree and $χ$ is the chromatic number of the subgraph. We compare these results with those given in [O. Schaudt, V. Weil, On bounding the difference between the maximum degree and the clique number, Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI: 10.1007/s00373-014-1468-3], where we studied the graph class $\varOmega_k$, for $k \in \mathbb{N}_0$, whose graphs are such that for any induced subgraph, $Δ\leq ω-1 + k$ holds, where $ω$ denotes the clique number of a graph. In particular, we give a characterization in terms of $\varOmega_k$ and $\varUpsilon_k$ of those graphs where the neighborhood of every vertex is perfect.

cs.DM↗
arXiv2016

Approximately coloring graphs without long induced paths

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.

math.CO↗
arXiv2015

Exhaustive generation of $k$-critical $\mathcal H$-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.

math.CO↗
arXiv2015

Long cycles through prescribed vertices have the Erdős-Pósa property

We prove that for every graph, any vertex subset $S$, and given integers $k,\ell$: there are $k$ disjoint cycles of length at least $\ell$ that each contain at least one vertex from $S$, or a vertex set of size $O(\ell \cdot k \log k)$ that meets all such cycles. This generalises previous results of Fiorini and Hendrickx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in $O(k \log k \cdot s^2 \cdot (f(\ell) \cdot n+m))$ time, where $s$ denotes the cardinality of $S$.

math.CO↗