Graph partitioning in restricted graph classes

2024 [ to top ]

"Horev, Yinon"; "Shay, Shiraz"; "Cohen, Sarel"; "Friedrich, Tobias"; "Issac, Davis"; "Kamma, Lior"; "Niklanovits, Aikaterini"; "Simonov, Kirill" A Contraction Tree SAT Encoding for Computing Twin-Width 2024
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Twin-width is a structural width parameter and matrix invariant introduced by Bonnet et al. [FOCS 2020], that has been gaining attention due to its various fields of applications. In this paper, inspired by the SAT approach of Schidler and Szeider [ALENEX 2022], we provide a new SAT encoding for computing twin-width. The encoding aims to encode the contraction sequence as a binary tree. The asymptotic size of the formula under our encoding is smaller than in the state-of-the-art relative encoding of Schidler and Szeider. We also conduct an experimental study, comparing the performance of the new encoding and the relative encoding.
@inproceedings{horev2024contraction, abstract = {Twin-width is a structural width parameter and matrix invariant introduced by Bonnet et al. [FOCS 2020], that has been gaining attention due to its various fields of applications. In this paper, inspired by the SAT approach of Schidler and Szeider [ALENEX 2022], we provide a new SAT encoding for computing twin-width. The encoding aims to encode the contraction sequence as a binary tree. The asymptotic size of the formula under our encoding is smaller than in the state-of-the-art relative encoding of Schidler and Szeider. We also conduct an experimental study, comparing the performance of the new encoding and the relative encoding.}, author = {"Horev, Yinon" and "Shay, Shiraz" and "Cohen, Sarel" and "Friedrich, Tobias" and "Issac, Davis" and "Kamma, Lior" and "Niklanovits, Aikaterini" and "Simonov, Kirill"}, keywords = {sarelcohen yinonhorev davisissac kirillsimonov aikaterininiklanovits tobiasfriedrich year2024 shirazshay liorkamma pakdd}, title = {A Contraction Tree SAT Encoding for Computing Twin-Width}, year = 2024 }

2023 [ to top ]

Casel, Katrin; Friedrich, Tobias; Issac, Davis; Niklanovits, Aikaterini; Zeif, Ziena Efficient Constructions for the Gyori-Lovasz Theorem on Almost Chordal GraphsWorkshop Graph-Theoretic Concepts in Computer Science (WG) 2023: 143–156
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In the 1970s, GyHo}ri and Lovász showed that for a \(k\)-connected \(n\)-vertex graph, a given set of terminal vertices \(t_1, \dots, t_k\) and natural numbers \(n_1, \dots, n_k\) satisfying \(\sum_{i=1^k n_i = n\), a connected vertex partition \(S_1, \dots, S_k\) satisfying \(t_i \in S_i\) and \(|S_i| = n_i\) exists. However, polynomial algorithms to actually compute such partitions are known so far only for \(k \leq 4\). This motivates us to take a new approach and constrain this problem to particular graph classes instead of restricting the values of \(k\). More precisely, we consider \(k\)-connected chordal graphs and a broader class of graphs related to them. For the first class, we give an algorithm with \(\mathcal{O(n^2)\) running time that solves the problem exactly, and for the second, an algorithm with \(\mathcal{O(n^4)\) running time that deviates on at most one vertex from the required vertex partition sizes.
@inproceedings{casel2023efficient, abstract = {In the 1970s, Gy\H{o}ri and Lov{á}sz showed that for a \(k\)-connected \(n\)-vertex graph, a given set of terminal vertices \(t_1, \dots, t_k\) and natural numbers \(n_1, \dots, n_k\) satisfying \(\sum_{i=1}^{k} n_i = n\), a connected vertex partition \(S_1, \dots, S_k\) satisfying \(t_i \in S_i\) and \(|S_i| = n_i\) exists. However, polynomial algorithms to actually compute such partitions are known so far only for \(k \leq 4\). This motivates us to take a new approach and constrain this problem to particular graph classes instead of restricting the values of \(k\). More precisely, we consider \(k\)-connected chordal graphs and a broader class of graphs related to them. For the first class, we give an algorithm with \(\mathcal{O}(n^2)\) running time that solves the problem exactly, and for the second, an algorithm with \(\mathcal{O}(n^4)\) running time that deviates on at most one vertex from the required vertex partition sizes.}, author = {Casel, Katrin and Friedrich, Tobias and Issac, Davis and Niklanovits, Aikaterini and Zeif, Ziena}, booktitle = {Workshop Graph-Theoretic Concepts in Computer Science (WG)}, keywords = {sys:relevantfor:ae zienazeif katrincasel davisissac aikaterininiklanovits tobiasfriedrich year2023 wg}, pages = {143–156}, title = {Efficient Constructions for the Gyori-Lovasz Theorem on Almost Chordal Graphs}, year = 2023 }

2021 [ to top ]

Borndörfer, Ralf; Casel, Katrin; Issac, Davis; Niklanovits, Aikaterini; Schwartz, Stephan; Zeif, Ziena Connected k-Partition of k-Connected Graphs and c-Claw-Free GraphsApproximation Algorithms for Combinatorial Optimization Problems (APPROX) 2021: 27:1–27:14
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A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. In these settings, it is often desirable to partition into a fixed number of parts of roughly of the same size or weight. The resulting computational problem is called Balanced Connected Partition (BCP). The two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on \(c\)-claw-free graphs, the class of graphs that do not have \(K_{1,c}\) as an induced subgraph, and present efficient \((c-1)\)-approximation algorithms for both objectives. In particular, for \(3\)-claw-free graphs, also simply known as claw-free graphs, we obtain a \(2\)-approximation. Due to the claw-freeness of line graphs, this also implies a \(2\)-approximation for the edge-partition version of BCP in general graphs. A harder connected partition problem arises from demanding a connected partition into k parts that have (possibly) heterogeneous target weights \(w_1, \dots, w_k\). In the 1970s Győri and Lovász showed that if \(G\) is \(k\)-connected and the target weights sum to the total size of \(G\), such a partition exists. However, to this day no polynomial algorithm to compute such partitions exists for \(k > 4\). Towards finding such a partition \(T_1, \dots, T_k\) in \(k\)-connected graphs for general \(k\), we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each \(w_i\) is greater than the weight of the heaviest vertex. In particular, we give a \(3\)-approximation for both the lower and the upper bounded version i.e. we guarantee that each \(T_i\) has weight at least \(\frac{w_i}{3}\) or that each \(T_i\) has weight most \(3 w_i\), respectively. Also, we present a both-side bounded version that produces a connected partition where each \(T_i\) has size at least \(\frac{w_i}{3}\) and at most \(\max(\{r, 3\}) w_i \), where \(r \geq 1\) is the ratio between the largest and smallest value in \(w_1, \dots, w_k\). In particular for the balanced version, i.e. \(w_1 = w_2 =, \dots , = w k\), this gives a partition with \(\frac{w_i3 leq w(T_i) \leq 3 w_i\).
@inproceedings{borndorfer2021connected, abstract = {A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. In these settings, it is often desirable to partition into a fixed number of parts of roughly of the same size or weight. The resulting computational problem is called Balanced Connected Partition (BCP). The two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on \(c\)-claw-free graphs, the class of graphs that do not have \(K_{1,c}\) as an induced subgraph, and present efficient \((c-1)\)-approximation algorithms for both objectives. In particular, for \(3\)-claw-free graphs, also simply known as claw-free graphs, we obtain a \(2\)-approximation. Due to the claw-freeness of line graphs, this also implies a \(2\)-approximation for the edge-partition version of BCP in general graphs. A harder connected partition problem arises from demanding a connected partition into k parts that have (possibly) heterogeneous target weights \(w_1, \dots, w_k\). In the 1970s Győri and Lovász showed that if \(G\) is \(k\)-connected and the target weights sum to the total size of \(G\), such a partition exists. However, to this day no polynomial algorithm to compute such partitions exists for \(k > 4\). Towards finding such a partition \(T_1, \dots, T_k\) in \(k\)-connected graphs for general \(k\), we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each \(w_i\) is greater than the weight of the heaviest vertex. In particular, we give a \(3\)-approximation for both the lower and the upper bounded version i.e. we guarantee that each \(T_i\) has weight at least \(\frac{w_i}{3}\) or that each \(T_i\) has weight most \(3 w_i\), respectively. Also, we present a both-side bounded version that produces a connected partition where each \(T_i\) has size at least \(\frac{w_i}{3}\) and at most \(\max(\{r, 3\}) w_i \), where \(r \geq 1\) is the ratio between the largest and smallest value in \(w_1, \dots, w_k\). In particular for the balanced version, i.e. \(w_1 = w_2 =, \dots , = w k\), this gives a partition with \(\frac{w_i}{3} \leq w(T_i) \leq 3 w_i\).}, author = {Borndörfer, Ralf and Casel, Katrin and Issac, Davis and Niklanovits, Aikaterini and Schwartz, Stephan and Zeif, Ziena}, booktitle = {Approximation Algorithms for Combinatorial Optimization Problems (APPROX)}, keywords = {sys:relevantfor:ae zienazeif katrincasel davisissac aikaterininiklanovits approx year2021 ralfborndörfer stephanschwartz}, pages = {27:1-27:14}, title = {Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs}, year = 2021 }

Casel, Katrin; Friedrich, Tobias; Issac, Davis; Niklanovits, Aikaterini; Zeif, Ziena Balanced Crown Decomposition for Connectivity ConstraintsEuropean Symposium on Algorithms (ESA) 2021: 26:1–26:15
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We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a \(3\)-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.
@inproceedings{casel2021balanced, abstract = {We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a \(3\)-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.}, author = {Casel, Katrin and Friedrich, Tobias and Issac, Davis and Niklanovits, Aikaterini and Zeif, Ziena}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {sys:relevantfor:ae zienazeif esa katrincasel davisissac aikaterininiklanovits tobiasfriedrich year2021}, pages = {26:1–26:15}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {Balanced Crown Decomposition for Connectivity Constraints}, volume = 204, year = 2021 }