arXiv subjects

Piotr Borowiecki

Publications and source records attributed to Piotr Borowiecki.

4 recordsLinked to original sources

arXiv2026

A framework for distributed discrete evacuation strategies

In this paper, we study discrete evacuation in networks, where agents know the network topology and designated exit nodes but do not know the number and initial positions of other agents. Each agent initially occupies a distinct node and must reach any exit node. Operating in a synchronous distributed model with local communication, the agents aim to minimize the time when the last agent reaches an exit. We introduce a general algorithmic framework for constructing evacuation strategies on arbitrary graphs. As a key application, we demonstrate that the framework yields asymptotically optimal evacuation strategies -- achieving a constant competitive ratio -- for grid networks, with natural extensions to triangular and hexagonal grids.

cs.DM↗

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↗

arXiv2015

Independence in Uniform Linear Triangle-free Hypergraphs

The independence number $α(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free graphs (J. Comb. Theory, Ser. B 53 (1991) 300-307), and considerably improving recent results of Li and Zang (SIAM J. Discrete Math. 20 (2006) 96-104) and Chishti et al. (Acta Univ. Sapientiae, Informatica 6 (2014) 132-158), we show that $$α(H)\geq \sum_{u\in V(H)}f_r(d_H(u))$$ for an $r$-uniform linear triangle-free hypergraph $H$ with $r\geq 2$, where \begin{eqnarray*} f_r(0)&=&1\mbox{, and }\\ f_r(d)&=&\frac{1+\Big((r-1)d^2-d\Big)f_r(d-1)}{1+(r-1)d^2}\mbox{ for $d\geq 1$.} \end{eqnarray*}

math.CO↗

arXiv2013

Minimum order of graphs with given coloring parameters

A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $φ:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number $χ(G)$ and achromatic number $ψ(G)$, respectively), and the Grundy number $Γ(G)$ defined as the largest $k$ admitting a complete coloring $φ$ with exactly $k$ colors such that every vertex $v\in V$ of color $φ(v)$ has a neighbor of color $i$ for all $1\le i<φ(v)$. The inequality chain $χ(G)\le Γ(G)\le ψ(G)$ obviously holds for all graphs $G$. A triple $(f,g,h)$ of positive integers at least 2 is called realizable if there exists a connected graph $G$ with $χ(G)=f$, $Γ(G)=g$, and $ψ(G)=h$. Chartrand et al. (A note on graphs with prescribed complete coloring numbers, J. Combin. Math. Combin. Comput. LXXIII (2010) 77-84) found the list of realizable triples. In this paper we determine the minimum number of vertices in a connected graph with chromatic number $f$, Grundy number $g$, and achromatic number $h$, for all realizable triples $(f,g,h)$ of integers. Furthermore, for $f=g=3$ we describe the (two) extremal graphs for each $h \geq 6$. For $h=4$ and $5$, there are more extremal graphs, their description is contained as well.

cs.DM↗