arXiv subjects

Aikaterini Niklanovits

Publications and source records attributed to Aikaterini Niklanovits.

7 recordsLinked to original sources

arXiv2026

Connected (Dense) Partition for Tree-Like Graphs

We focus on two variants of graph partitioning problems, connected partition and dense partition. Formally, given a graph $G=(V,E)$ and a partition of its vertices $\mathcal P=\{P_1,\ldots, P_k\}$ we say that $\mathcal P$ is a connected partition of $G$ if each $P_i$ induces a connected graph in $G$. Many classical variants of this problem impose additional restrictions both on the number of parts as well as on the size of each part. Moreover, given a partition $\mathcal P=\{P_1,\ldots, P_k\}$ we define its density by $d(\mathcal P):=\sum_{i=1}^k |E(P_i)|/|V(P_i)|$. The problem Maximum Dense Graph Partition asks to construct a partition of maximum density. We study this problem both with and without fixed number of sets $k$. We prove the following results: 1. A polynomial time algorithm for Maximum Dense Graph Partition of thick forests, a subclass of chordal graphs, generalizing the previously known polynomial time algorithm on block graphs. 2. A generic dynamic programming algorithm to construct (if possible) a connected partition into $k$ sets of prescribed sizes on graphs with bounded treewidth. This yields algorithms for both variants of Dense Graph Partition and an efficient construction for the Győri-Lovász theorem. 3. The $\mathsf{NP}$-hardness of Maximum Dense Graph Partition to $k$ parts restricted to split graphs, indicating that thick trees are the boundary for the polynomial computability of this problem.

cs.DS↗

arXiv2025

Connected Partitions via Connected Dominating Sets

The classical theorem due to Győri and Lovász states that any $k$-connected graph $G$ admits a partition into $k$ connected subgraphs, where each subgraph has a prescribed size and contains a prescribed vertex, as long as the total size of target subgraphs is equal to the size of $G$. However, this result is notoriously evasive in terms of efficient constructions, and it is still unknown whether such a partition can be computed in polynomial time, even for $k = 5$. We make progress towards an efficient constructive version of the Győri--Lovász theorem by considering a natural strengthening of the $k$-connectivity requirement. Specifically, we show that the desired connected partition can be found in polynomial time, if $G$ contains $k$ disjoint connected dominating sets. As a consequence of this result, we give several efficient approximate and exact constructive versions of the original Győri--Lovász theorem: 1. On general graphs, a Győri--Lovász partition with $k$ parts can be computed in polynomial time when the input graph has connectivity $Ω(k \cdot \log^2 n)$; 2. On convex bipartite graphs, connectivity of $4k$ is sufficient; 3. On biconvex graphs and interval graphs, connectivity of $k$ is sufficient, meaning that our algorithm gives a ``true'' constructive version of the theorem on these graph classes.

math.CO↗

arXiv2024

Combining Crown Structures for Vulnerability Measures

Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number of vertices required to decompose a graph into fragments, but also take into account the size of the largest remaining component. The main focus of our paper is on kernelization algorithms tailored to both measures. We capitalize on the structural attributes inherent in different crown decompositions, strategically combining them to introduce novel kernelization algorithms that advance the current state of the field. In particular, we extend the scope of the balanced crown decomposition provided by Casel et al.~[7] and expand the applicability of crown decomposition techniques. In summary, we improve the vertex kernel of VI from $p^3$ to $p^2$, and of wVI from $p^3$ to $3(p^2 + p^{1.5} p_{\ell})$, where $p_{\ell} < p$ represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from $\mathcal{O}(k^2W + kW^2)$ to $3μ(k + \sqrtμW)$, where $μ= \max(k,W)$. We also give a combinatorial algorithm that provides a $2kW$ vertex kernel in FPT-runtime when parameterized by $r$, where $r \leq k$ is the size of a maximum $(W+1)$-packing. We further show that the algorithm computing the $2kW$ vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when $W=1$, which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a $2k$ vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.

cs.DS↗

arXiv2024

A New Approach for Approximating Directed Rooted Networks

We consider the k-outconnected directed Steiner tree problem (k-DST). Given a directed edge-weighted graph $G=(V,E,w)$, where $V=\{r\}\cup S \cup T$, and an integer $k$, the goal is to find a minimum cost subgraph of $G$ in which there are $k$ edge-disjoint $rt$-paths for every terminal $t\in T$. The problem is know to be NP-hard. Furthermore, the question on whether a polynomial time, subpolynomial approximation algorithm exists for $k$-DST was answered negatively by Grandoni et al. (2018), by proving an approximation hardness of $Ω(|T|/\log |T|)$ under $NP\neq ZPP$. Inspired by modern day applications, we focus on developing efficient algorithms for $k$-DST in graphs where terminals have out-degree $0$, and furthermore constitute the vast majority in the graph. We provide the first approximation algorithm for $k$-DST on such graphs, in which the approximation ratio depends (primarily) on the size of $S$. We present a randomized algorithm that finds a solution of weight at most $\mathcal O(k|S|\log |T|)$ times the optimal weight, and with high probability runs in polynomial time.

cs.DS↗

arXiv2023

Efficient Constructions for the Győri-Lovász Theorem on Almost Chordal Graphs

In the 1970s, Győri 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, we give an algorithm with $O(n^2)$ running time that solves the problem exactly, and for the second, an algorithm with $O(n^4)$ running time that deviates on at most one vertex from the given required vertex partition sizes.

cs.DS↗

arXiv2021

Connected $k$-partition of $k$-connected graphs and $c$-claw-free graphs

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. We consider Balanced Connected Partitions (BCP), where 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, due to the (3-)claw-freeness of line graphs, this also implies a 2-approximations for the edge-partition version of BCP in general graphs. In the 1970s Győri and Lovász showed for natural numbers $w_1,\dots,w_k$ where $\sum_i w_i$ is the vertex size, that if $G$ is k-connected, then there exist a connected k-partition with part sizes $w_1,\dots,w_k$. However, to this day no polynomial algorithm to compute such partitions exists for k>4. Towards finding such a partition $T_1,\dots, T_k$, 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 $3w_i$, 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 $\frac{1}{3}w_i \leq w(T_i) \leq 3w_i$.

math.CO↗

arXiv2021

Balanced Crown Decomposition for Connectivity Constraints

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 kernelization and approximation algorithms 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.

cs.DS↗