arXiv subjects

Bartłomiej Bosek

Publications and source records attributed to Bartłomiej Bosek.

29 records · Page 2Linked to original sources

arXiv2020

Majority choosability of countable graphs

In any vertex coloring of a graph some edges have differently colored ends (\emph{good} edges) and some are monochromatic (\emph{bad} edges). In a proper coloring all edges are good. In a \emph{majority coloring} it is enough that for every vertex $v$, the number of bad edges incident to $v$ does not exceed the number of good edges incident to $v$. A well known result of Lovász \cite{Lovasz} asserts that every finite graph has a majority $2$-coloring. A similar statement for countably infinite graphs is a challenging open problem, known as the \emph{Unfriendly Partition Conjecture}. We consider a natural list variant of majority coloring. A graph is \emph{majority $k$-choosable} if it has a majority coloring from any lists of size $k$ assigned arbitrarily to the vertices. We prove that every countable graph is majority $4$-choosable. We also consider a natural analog of majority coloring for directed graphs. We prove that every countable digraph is also majority $4$-choosable. We pose list and directed analogs of the Unfriendly Partition Conjecture, stating that every countable graph is majority $2$-choosable and every countable digraph is majority $3$-choosable.

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arXiv2019

Planar Posets that are Accessible from Below Have Dimension at Most 6

Planar posets can have arbitrarily large dimension. However, a planar poset of height $h$ has dimension at most $192h+96$, while a planar poset with $t$ minimal elements has dimension at most $2t+1$. In particular, a planar poset with a unique minimal element has dimension at most $3$. In this paper, we extend this result by showing that a planar poset has dimension at most $6$ if it has a plane diagram in which every minimal element is accessible from below.

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arXiv2019

Hat chromatic number of graphs

We study the hat chromatic number of a graph defined in the following way: there is one player at each vertex of a loopless graph $G$, an adversary places a hat of one of $K$ colors on the head of each player, two players can see each other's hats if and only if they are at adjacent vertices. All players simultaneously try to guess the color of their hat. The players cannot communicate but collectively determine a strategy before the hats are placed. The hat chromatic number, $μ(G)$, is the largest number $K$ of colors such that the players are able to fix a strategy that will ensure that for every possible placement of hats at least one of the guesses correctly. We compute $μ(G)$ for several classes of graphs, for others we establish some bounds. We establish connections between the hat chromatic number, the chromatic number and the coloring number. We also introduce several variants of the game: with multiple guesses, restrictions on allowed strategies or restrictions on colorings. We show examples how the modified games can be used to obtain interesting results for the original game.

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arXiv2018

Centroidal localization game

One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r \in V(G)$. In a more realistic setting, instead of getting the exact distance information, given devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the entity's location $r$ and the devices (for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set $\{v_1,\cdots,v_k\}$ of vertices is probed and then the relative distances between the vertices $v_i$ and the current location $r$ of the entity are given. If not located, the moving entity may move along one edge. Let $ζ^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We prove that $ζ^* (T)\leq 2$ for every tree $T$ and give an upper bound on $ζ^*(G\square H)$ in cartesian product of graphs $G$ and $H$. Our main result is that $ζ^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $ζ^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $ζ^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn.

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arXiv2018

An easy subexponential bound for online chain partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.

cs.DS↗

arXiv2018

On-line Chain Partitioning Approach to Scheduling

An on-line chain partitioning algorithm receives the points of the poset from some externally determined list. Being presented with a new point the algorithm learns the comparability status of this new point to all previously presented ones. As each point is received, the algorithm assigns this new point to a chain in an irrevocable manner and this assignment is made without knowledge of future points. Kierstead presented an algorithm using $(5^w-1)/4$ chains to cover each poset of width $w$. Felsner proved that width $2$ posets can be partitioned on-line into $5$ chains. We present an algorithm using $16$ chains on posets of width $3$. This result significantly narrows down the previous bound of $31$. Moreover, we address the on-line chain partitioning problem for interval orders. Kierstead and Trotter presented an algorithm using $3w-2$ chains. We deal with an up-growing version of an on-line chain partition of interval orders, i.e. we restrict possible inputs by the rule that each new point is maximal at the moment of its arrival. We present an algorithm using $2w-1$ chains and show that there is no better one. These problems come from a need for better algorithms that can be applied to scheduling. Each on-line chain partitioning algorithm schedules tasks in a multiprocessor environment, and therefore can be applied in order to minimize number of processors.

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arXiv2017

A Tight Bound for Shortest Augmenting Paths on Trees

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree $T=(W \uplus B, E)$ is being revealed online, i.e., in each round one vertex from $B$ with its incident edges arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis, R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with augmentations. In INFOCOM 2009] that the total length of all shortest augmenting paths found is $O(n \log n)$. In this paper, we prove a tight $O(n \log n)$ upper bound for the total length of shortest augmenting paths for trees improving over $O(n \log^2 n)$ bound [B. Bosek, D. Leniowski, P. Sankowski, and A. Zych. Shortest augmenting paths for online matchings on trees. In WAOA 2015].

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arXiv2017

Localization game on geometric and planar graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph $G$ we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $ζ(G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth $2$ and unbounded $ζ(G)$. On a positive side, we prove that $ζ(G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $ζ(G)$ is NP-hard in graphs with diameter at most $2$. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane.

math.CO↗

arXiv2016

Majority choosability of digraphs

A \emph{majority coloring} of a digraph is a coloring of its vertices such that for each vertex $v$, at most half of the out-neighbors of $v$ has the same color as $v$. A digraph $D$ is \emph{majority $k$-choosable} if for any assignment of lists of colors of size $k$ to the vertices there is a majority coloring of $D$ from these lists. We prove that every digraph is majority $4$-choosable. This gives a positive answer to a question posed recently by Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain this result as a consequence of a more general theorem, in which majority condition is profitably extended. For instance, the theorem implies also that every digraph has a coloring from arbitrary lists of size three, in which at most $2/3$ of the out-neighbors of any vertex share its color. This solves another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture stating that every digraph is majority $3$-colorable.

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arXiv2014

On the Duality of Semiantichains and Unichain Coverings

We study a min-max relation conjectured by Saks and West: For any two posets $P$ and $Q$ the size of a maximum semiantichain and the size of a minimum unichain covering in the product $P\times Q$ are equal. For positive we state conditions on $P$ and $Q$ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.

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arXiv2012

Additive colorings of planar graphs

An \emph{additive coloring} of a graph $G$ is an assignment of positive integers $\{1,2,...,k\}$ to the vertices of $G$ such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number $k$ for which there exists an additive coloring of $G$ is denoted by $η(G)$. We prove that $η(G)\leqslant 468$ for every planar graph $G$. This improves a previous bound $η(G)\leqslant 5544$ due to Norin. The proof uses Combinatorial Nullstellensatz and coloring number of planar hypergrahs. We also demonstrate that $η(G)\leqslant 36$ for 3-colorable planar graphs, and $η(G)\leqslant 4$ for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each $r\geqslant 2$ there is an $r$-chromatic graph $G_{r}$ with no additive coloring by elements of any Abelian group of order $r$.

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