arXiv subjects

Dorota Osula

Publications and source records attributed to Dorota Osula.

6 recordsLinked to original sources

arXiv2021

The Complexity of Bicriteria Tree-Depth

The tree-depth problem can be seen as finding an elimination tree of minimum height for a given input graph $G$. We introduce a bicriteria generalization in which additionally the width of the elimination tree needs to be bounded by some input integer $b$. We are interested in the case when $G$ is the line graph of a tree, proving that the problem is NP-hard and obtaining a polynomial-time additive $2b$-approximation algorithm. This particular class of graphs received significant attention in the past, mainly due to a number of potential applications, e.g. in parallel assembly of modular products, or parallel query processing in relational databases, as well as purely combinatorial applications, including searching in tree-like partial orders (which in turn generalizes binary search on sorted data).

cs.DS↗

arXiv2019

On the connected and weakly convex domination numbers

In this paper we study relations between connected and weakly convex domination numbers. We show that in general the difference between these numbers can be arbitrarily large and we focus on the graphs for which a weakly convex domination number equals a connected domination number. We also study the influence of the edge removing on the weakly convex domination number, in particular we prove that the weakly convex domination number is an interpolating function.

math.CO↗

arXiv2019

Twin domination number of Tournaments

Let $D=(V,A)$ be a digraph. A subset $S$ of $V$ is called a twin dominating set of $D$ if for every vertex $v\in V-S$, there exists vertices $u_1,u_2 \in S$ such that $(v,u_1)$ and $(u_2,v)$ are arcs in $D$. The minimum cardinality of a twin dominating set in $D$ is called the twin domination number of $D$ and is denoted by $γ^{*}(D)$. The upper orientable twin domination number of a graph $G$ is $DOM^{*}(G)=\max\{ γ^{*}(D)|D \ \text{is an orientation of G} \}.$ It has been conjectured that for the complete graph $K_n$ with $n\geq 8$, $DOM^{*}(K_n)=\left\lceil \frac{n+1}{2}\right\rceil$. In this work we prove $DOM^{*}(K_8)= DOM^{*}(K_9)= 4$ and establish new upper bounds for $DOM^{*}(K_n)$, disproving the same above conjecture for all $n \geq 8$.

math.CO↗

arXiv2019

Minimizing the Cost of Team Exploration

A group of mobile agents is given a task to explore an edge-weighted graph $G$, i.e., every vertex of $G$ has to be visited by at least one agent. There is no centralized unit to coordinate their actions, but they can freely communicate with each other. The goal is to construct a deterministic strategy which allows agents to complete their task optimally. In this paper we are interested in a cost-optimal strategy, where the cost is understood as the total distance traversed by agents coupled with the cost of invoking them. Two graph classes are analyzed, rings and trees, in the off-line and on-line setting, i.e., when a structure of a graph is known and not known to agents in advance. We present algorithms that compute the optimal solutions for a given ring and tree of order $n$, in $O(n)$ time units. For rings in the on-line setting, we give the $2$-competitive algorithm and prove the lower bound of $3/2$ for the competitive ratio for any on-line strategy. For every strategy for trees in the on-line setting, we prove the competitive ratio to be no less than $2$, which can be achieved by the $DFS$ algorithm.

cs.DM↗

arXiv2018

Finding small-width connected path decompositions in polynomial time

A connected path decomposition of a simple graph $G$ is a path decomposition $(X_1,\ldots,X_l)$ such that the subgraph of $G$ induced by $X_1\cup\cdots\cup X_i$ is connected for each $i\in\{1,\ldots,l\}$. The connected pathwidth of $G$ is then the minimum width over all connected path decompositions of $G$. We prove that for each fixed $k$, the connected pathwidth of any input graph can be computed in polynomial-time. This answers an open question raised by Fedor V. Fomin during the GRASTA 2017 workshop, since connected pathwidth is equivalent to the connected (monotone) node search game.

cs.DS↗

arXiv2017

The Snow Team Problem (Clearing Directed Subgraphs by Mobile Agents)

We study several problems of clearing subgraphs by mobile agents in digraphs. The agents can move only along directed walks of a digraph and, depending on the variant, their initial positions may be pre-specified. In general, for a given subset~$\mathcal{S}$ of vertices of a digraph $D$ and a positive integer $k$, the objective is to determine whether there is a subgraph $H=(\mathcal{V}_H,\mathcal{A}_H)$ of $D$ such that (a) $\mathcal{S} \subseteq \mathcal{V}_H$, (b) $H$ is the union of $k$ directed walks in $D$, and (c) the underlying graph of $H$ includes a Steiner tree for $\mathcal{S}$ in $D$. We provide several results on the polynomial time tractability, hardness, and parameterized complexity of the problem.

cs.DM↗