arXiv subjects

Bartłomiej Bosek

Publications and source records attributed to Bartłomiej Bosek.

At least 19 recordsLinked to original sources

arXiv2026

A rainbow partition theorem for trees and connected maximin share allocations of chores

Xiao, Qiu, and Huang (AAMAS 2023) and independently Lonc (personal communication) asked whether indivisible chores located at the vertices of a tree can always be allocated to $n$ agents in connected bundles so that the cost of every agent is at most its connected maximin share; for goods, this is a theorem of Bouveret, Cechlárová, Elkind, Igarashi, and Peters. We answer the question affirmatively, even for monotone costs. The answer follows from a combinatorial theorem: if $\mathcal P_1,\ldots,\mathcal P_k$ are partitions of the vertex set of a finite tree, each into at most $k$ connected parts, then the vertex set can be split into disjoint sets $B_1,\ldots,B_k$, some possibly empty, such that each nonempty $B_i$ is connected and contained in a part of $\mathcal P_i$. Equivalently, if each of $k$ colours occurs on at most $k-1$ edges of a tree, then the vertices can be partitioned into connected sets labelled by distinct colours, none containing an edge of its own colour; in particular, one can choose for every colour a component of the forest obtained by deleting that colour so that the chosen components cover the tree. The bound is already best possible for paths, and for additive costs, the theorem is equivalent to the fair-division statement. The proof reduces the problem to inward partitions of oriented trees, which we obtain from the colourful KKM theorem on a simplex of edge weights, using a leaf-elimination labelling that remains compatible when weights vanish. We also give an algorithm running in time $k^{O(k)}$ plus polynomial time.

math.CO↗

arXiv2026

Online and Incremental Fractional Vertex Cover on Trees

In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.

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arXiv2026

A tight lower bound for malicious online bipartite matching with limited recourse budget

We study one-sided online bipartite matching with recourse. In this setting, one side of a bipartite graph is known in advance, while vertices on the other side arrive online together with their incident edges. After each arrival, the algorithm must maintain a maximum-cardinality matching while minimizing the total number of reallocations, also known as the recourse budget. Despite extensive work, the exact recourse complexity of the problem remains unsettled: the best lower bound is $Ω(n \log n)$, whereas the best upper bound is $\mathcal{O}(n \log^2 n)$, where $n$ denotes the number of online vertices. Tight upper bounds of $\mathcal{O}(n \log n)$ are known only for restricted graph classes, such as forests. The best known upper bounds are attained by a very simple and natural algorithm SAP, which after each arrival applies a shortest augmenting path, and it is conjectured to be optimal. All known upper bound analyses of this algorithm do not depend on the particular maximum matching maintained by the algorithm. Consequently, they also apply to a more difficult problem, which we call the malicious matching setting: after each arrival, the maintained matching is replaced by a worst-case maximum matching for the next step. This led to the conjecture that the malicious setting still admits an $\mathcal{O}(n \log n)$ recourse bound, in line with the conjectured optimal complexity of the original model. Our main result is an $Ω(n \log^2 n)$ lower bound for the malicious matching setting, thus disproving the conjecture. Together with the previous upper bound, this settles the asymptotic recourse complexity of the malicious variant of the problem. We complement our lower bound with an upper bound of $\mathcal{O}(n \log n)$ for expander graphs.

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arXiv2026

Alon-Tarsi for hypergraphs

Given a hypergraph $H=(V,E)$, define for every edge $e\in E$ a linear expression with arguments corresponding to the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal is to find a relationship between the Alon-Tarsi number of $p_H$ and the edge density of $H$. We prove that $AT(p_H)=\lceil \mathrm{ed}(H)\rceil+1$ if all the coefficients in $p_H$ are equal to $1$ and the base field has characteristic zero. Our main result is that, over an arbitrary field, if on every edge the coefficients are not all equal, then they can be permuted within the edges so that for the resulting polynomial $p_H^\prime$, $AT(p_H^\prime)\leq 2\lceil \mathrm{ed}(H)\rceil+1$ holds. We conjecture that this bound holds for every hypergraph polynomial without permuting its coefficients. If this were true, then in particular a significant generalization of the famous 1-2-3 Conjecture would follow.

math.CO↗

arXiv2026

Arboricity Nearly Bounds Degeneracy

Arboricity and degeneracy are two fundamental and closely related graph parameters that measure the sparsity of a graph. Every $k$-degenerate graph is $k$-arboric, but some $k$-arboric graphs are only $(2k-1)$-degenerate. However, every maximal $k$-arboric multigraph with $n$ vertices and every maximal $k$-degenerate multigraph with $n$ vertices has exactly $k(n-1)$ edges. These basic observations lead to a natural structural question: How far are $k$-arboric graphs from being $k$-degenerate? We answer this question by showing that: By at most a $(k-1)$-bounded-degree graph apart. More specifically, we prove that a $k$-arboric multigraph admits a $(k,k-1)$-decomposition, that is, its edges can be partitioned into two multisets such that one spans a $k$-degenerate multigraph and the other spans a multigraph with every vertex having degree at most $k-1$. Moreover, we provide a complete characterisation of all possible such decomposition types. Namely, for any integers $k \ge 1$ and $d,h \ge 0$ we show that every $k$-arboric multigraph admits a $(d,h)$-decomposition if and only if $d\geq k$ and $d+h\geq 2k-1$. Our proofs are constructive and we present a polynomial time algorithm that produces such decompositions. By contrast, we show that related decision problems for general graphs (without constraints on the arboricity) are NP-complete.

math.CO↗

arXiv2026

Dynamic domination and independence in sparse graphs

Let $\mathscr{C}$ be a class of graphs of bounded expansion and $r,k\in \mathbb{N}$ be fixed. We give a dynamic data structure that for a given dynamic graph $G$, updated by edge insertions and deletions subject to the promise that $G\in \mathscr{C}$ at all times, maintains the answer to the following two queries: (a) Does $G$ contain a distance-$r$ dominating set of size $k$? (b) Does $G$ contain a distance-$r$ independent set of size $k$? The data structure is randomized with error probability bounded by $\varepsilon$, for a parameter $\varepsilon>0$ fixed upon the initialization. The amortized update time is $\log^c n\cdot \log \frac{1}{\varepsilon}$, where $n$ is the vertex count of $G$ and $c$ is a constant that depends only on $r$, $k$, and $\mathscr{C}$. In the case of the first query, the data structure can also output a distance-$r$ dominating set of size $k$, if existent. We also prove that when $r=1$, our data structure for the dominating set query can be implemented even if we only assume that the maintained graph $G$ has degeneracy bounded by a constant $d$, yielding a simpler data structure with an improved amortized update time of $2^{k^{{\cal O}(d)}}\cdot \log^3 n\cdot \log \frac{1}{\varepsilon}$. Finally, we prove that in graphs of degeneracy at most $d$, one can maintain an ${\cal O}(d^2)$-approximation of the minimum size of a (distance-$1$) dominating set with amortized expected update time $d^{{\cal O}(1)}\cdot \log n$.

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arXiv2026

Epistemic fair division of independence structures

We study the problem of fair division of indivisible goods with constraints imposed by a prescribed independence structure, that is, a family of subsets of goods closed under taking subsets. As a motivating example, imagine that the goods to be divided are the available connections in a logistic, financial, or social network. The admissible bundle of goods for each agent must correspond to an acyclic set of edges, corresponding to a basic feasible solution to a linear network problem to be solved. Suppose that all agents assign the same value to each good (in the example, the network connections are equally important for every agent) and evaluate each bundle by summing the values of its goods. Is there a fair partition of the goods into such acyclic bundles? Surprisingly, the answer is yes, provided that the number of agents is at least the arboricity of $G$, and the fairness requirement is envy-freeness up to one good (EF1). The situation becomes more mysterious when agents have arbitrary additive valuations. Our main result guarantees that, in this case, epistemic EF1 partitions always exist, which means that each agent receives an acyclic bundle for which there exists a feasible partition of the remaining goods into acyclic bundles that they do not envy up to one good. We derive this conclusion from a general result for abstract independence structures defined on the sets of goods. We also discuss connections with several conjectures concerning matroids. In particular, we prove that any Hamiltonian matroid partitionable into two independent sets admits an EF1 bipartition with respect to a common monotone valuation. We complement our results with a constructive perspective: we present explicitly two algorithms for computing the fair allocations described above. Finally, we provide illustrative examples to demonstrate these algorithms on specific instances.

math.OC↗

arXiv2026

Dynamic data structures for twin-ordered matrices

We present a dynamic data structure for representing binary $n\times n$ matrices that are $d$-twin-ordered, for a~fixed parameter $d$. Our structure supports cell queries and single-cell updates both in $\Oh(\log \log n)$ expected worst case time, while using $\Oh_d(n)$ memory; here, the $\Oh_d(\cdot)$ notation

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arXiv2025

Mrs. Correct and Majority Colorings

A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for paintability - an on-line analog of the list coloring.

math.CO↗

arXiv2024

First-Fit Coloring of Forests in Random Arrival Model

We consider a graph coloring algorithm that processes vertices in order taken uniformly at random and assigns colors to them using First-Fit strategy. We show that this algorithm uses, in expectation, at most $(1 + o(1))\cdot \ln n \,/\, \ln\ln n$ different colors to color any forest with $n$ vertices. We also construct a family of forests that shows that this bound is best possible.

cs.DM↗

arXiv2024

A Note About Majority Colorings of Countable DAGs

A majority coloring of an undirected graph is a vertex coloring in which for each vertex there are at least as many bi-chromatic edges containing that vertex as monochromatic ones. It is known that for every countable graph a majority 3-coloring always exists. The Unfriendly Partition Conjecture states that every countable graph admits a majority 2-coloring. Since the 3-coloring result extends to countable DAGs, a variant of the conjecture states that 2 colors are enough to majority color every countable DAG. We show that this is false by presenting a DAG for which 3 colors are necessary. Presented construction is strongly based on a StackExchange conversation regarding labellings of infinite graphs that is linked in the references.

cs.DM↗

arXiv2022

Recoloring Unit Interval Graphs with Logarithmic Recourse Budget

In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals is allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains $k$-colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of $O(k^7 \log n)$ while maintaining a proper coloring with $k$ colors. This is an exponential improvement over the result in [Bosek et al., Recoloring Interval Graphs with Limited Recourse Budget. SWAT 2020] in terms of both $k$ and $n$. We complement this result by showing the lower bound of $Ω(n)$ on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented. As a byproduct of independent interest we include a new result on coloring proper circular arc graphs. Let $L$ be the maximum number of arcs intersecting in one point for some set of unit circular arcs $\mathcal{A}$. We show that if there is a set $\mathcal{A}'$ of non-intersecting unit arcs of size $L^2-1$ such that $\mathcal{A} \cup \mathcal{A}'$ does not contain $L+1$ arcs intersecting in one point, then it is possible to color $\mathcal{A}$ with $L$ colors. This complements the work on unit circular arc coloring, which specifies sufficient conditions needed to color $\mathcal{A}$ with $L+1$ colors or more.

cs.DM↗

arXiv2021

Graph Polynomials and Group Coloring of Graphs

Let $Γ$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $Γ$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow Γ$, there exists a vertex coloring $c$ by the elements of $Γ$ such that $c(y)-c(x)\neq \ell(e)$, for every edge $e=xy$ (oriented from $x$ to $y$). Langhede and Thomassen proved recently that every planar graph on $n$ vertices has at least $2^{n/9}$ different $\mathbb{Z}_5$-colorings. By using a different approach based on graph polynomials, we extend this result to $K_5$-minor-free graphs in the more general setting of field coloring. More specifically, we prove that every such graph on $n$ vertices is $\mathbb{F}$-$5$-choosable, whenever $\mathbb{F}$ is an arbitrary field with at least $5$ elements. Moreover, the number of colorings (for every list assignment) is at least $5^{n/4}$.

math.CO↗

arXiv2021

On a Problem of Steinhaus

Let $N$ be a positive integer. A sequence $X=(x_1,x_2,\ldots,x_N)$ of points in the unit interval $[0,1)$ is piercing if $\{x_1,x_2,\ldots,x_n\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. In 1958 Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most $74$ elements. This was later improved to the best possible value of $17$ by Warmus, and independently by Berlekamp and Graham. In this paper we study a more general variant of piercing sequences. Let $f(n)\geq n$ be an infinite nondecreasing sequence of positive integers. A sequence $X=(x_1,x_2,\ldots,x_{f(N)})$ is $f$-piercing if $\{x_1,x_2,\ldots,x_{f(n)}\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. A special case of $f(n)=n+d$, with $d$ a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each $d\geq 0$, the maximum length of any $(n+d)$-piercing sequence is finite. Expressing this maximum length as $s(d)+d$, they obtained an exponential upper bound on the function $s(d)$, which was later improved to $s(d)=O(d^3)$ by Graham and Levy. Recently, Konyagin proved that $2d\leqslant s(d)< 200d$ holds for all sufficiently big $d$. Using a different technique based on the Farey fractions and stick-breaking games, we prove here that the function $s(d)$ satisfies $\left\lfloor{}c_1d\right\rfloor{}\leqslant s(d)\leqslant c_2d+o(d)$, where $c_1=\frac{\ln 2}{1-\ln 2}\approx2.25$ and $c_2=\frac{1+\ln2}{1-\ln2}\approx5.52$. We also prove that there exists an infinite $f$-piercing sequence with $f(n)= γn+o(n)$ if and only if $γ\geq\frac{1}{\ln 2}\approx 1.44$.

math.NT↗

arXiv2020

Local Dimension is Unbounded for Planar Posets

In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width, and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height.

math.CO↗

arXiv2020

Reflections on the Erd\H {o}s Discrepancy Problem

We consider some coloring issues related to the famous Erd\H {o}s Discrepancy Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in \mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove that for every fixed $k$ there exists a $2$-coloring of $\mathbb N$ such that every set $A_{s,k}$ is \emph{perfectly balanced} (the numbers of red and blue elements in the set $A_{s,k}$ differ by at most one). This prompts reflection on various restricted versions of Erd\H {o}s' problem, obtained by imposing diverse confinements on parameters $s,k$. In a slightly different direction, we discuss a \emph{majority} variant of the problem, in which each set $A_{s,k}$ should have an excess of elements colored differently than the first element in the set. This problem leads, unexpectedly, to some deep questions concerning completely multiplicative functions with values in $\{+1,-1\}$. In particular, whether there is such a function with partial sums bounded from above.

math.CO↗

arXiv2020

Dilworth's Theorem for Borel Posets

A famous theorem of Dilworth asserts that any finite poset of width $k$ can be decomposed into $k$ chains. We study the following problem: given a Borel poset $P$ of finite width $k$, is it true that it can be decomposed into $k$ Borel chains? We give a positive answer in a special case of Borel posets embeddable into the real line. We also prove a dual theorem for posets whose comparability graphs are locally countable.

math.CO↗

arXiv2020

On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.

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