arXiv subjects

Konstanty Junosza-Szaniawski

Publications and source records attributed to Konstanty Junosza-Szaniawski.

15 recordsLinked to original sources

arXiv2026

How Close is a Tree to a Euclidean Minimum Spanning Tree?

Let $Γ$ be a straight-line crossing-free drawing of a tree $T$. A \emph{bad pair} in $Γ$ is a pair of non-adjacent vertices of $T$ whose Euclidean distance in $Γ$ is smaller than the length of the longest edge in the path connecting them in~$Γ$. When $Γ$ has no bad pairs, $Γ$ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be \NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For $n$-vertex trees with maximum vertex degree $Δ$, we prove the $Δ^2n\log n$ upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.

cs.CG↗

arXiv2026

Modelling Network Resilience: The Complexity of Some Graph Division Games

Motivated by the controller placement problems in software-defined networks and the fair division principles of classical "cake cutting", we investigate the following two-player zero-sum game. In our model, a defender places a limited number of controllers on graph vertices, while an attacker deletes a limited number of vertices. The defender score is the total number of surviving vertices reachable from any remaining controller. We formalize the computational problems associated with various game dynamics (defender plays first; attacker plays first; players play simultaneously; pure or mixed strategies). We show that these natural problems are $\mathsf{NP}$-complete or $Σ^\mathsf{P}_2$-complete, depending on the specific variant. These hardness results provide limitations for optimal controller placement algorithms under different notions of quality of a solution. Finally, we present structural insights that yield efficient algorithms for restricted graph classes (namely interval graphs and graphs of bounded treewidth).

cs.CC↗

arXiv2026

The Parameterized Complexity of Coloring Mixed Graphs

A mixed graph contains (undirected) edges as well as (directed) arcs, thus generalizing undirected and directed graphs. A proper coloring $c$ of a mixed graph $G$ assigns a positive integer to each vertex such that $c(u)\neq c(v)$ for every edge $\{u,v\}$ and $c(u)<c(v)$ for every arc $(u,v)$ of $G$. As in classical coloring, the objective is to minimize the number of colors. Thus, mixed (graph) coloring generalizes classical coloring of undirected graphs and allows for more general applications, such as scheduling with precedence constraints, modeling metabolic pathways, and process management in operating systems; see a survey by Sotskov [Mathematics, 2020]. We initiate the systematic study of the parameterized complexity of mixed coloring. We focus on structural graph parameters that lie between cliquewidth and vertex cover, primarily with respect to the underlying undirected graph. Unlike classical coloring, which is fixed-parameter tractable (FPT) parameterized by treewidth or neighborhood diversity, we show that mixed coloring is W[1]-hard for treewidth and even paraNP-hard for neighborhood diversity. To utilize the directedness of arcs, we introduce and analyze natural generalizations of neighborhood diversity and cliquewidth to mixed graphs, and show that mixed coloring becomes FPT when parameterized by mixed neighborhood diversity. Further, we investigate how these parameters are affected if we add transitive arcs, which do not affect colorings. Finally, we provide tight bounds on the chromatic number of mixed graphs, generalizing known bounds on mixed interval graphs.

cs.CC↗

arXiv2023

Coloring and Recognizing Directed Interval Graphs

A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph $G$, an interval $u$ receives a lower (different) color than an interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a $\min \{ω(G), λ(G)+1 \}$-approximation algorithm, where $ω(G)$ is the size of a largest clique and $λ(G)$ is the length of a longest directed path in $G$. For the subclass of \emph{bidirectional interval graphs} (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call \emph{containment interval graphs}. In such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$, and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.

cs.DM↗

arXiv2022

Exact and approximation algorithms for sensor placement against DDoS attacks

In a DDoS attack (Distributed Denial of Service), an attacker gains control of many network users through a virus. Then the controlled users send many requests to a victim, leading to its resources being depleted. DDoS attacks are hard to defend because of their distributed nature, large scale and various attack techniques. One possible mode of defense is to place sensors in a network that can detect and stop an unwanted request. However, such sensors are expensive so there is a natural question as to the minimum number of sensors and the optimal placement required to get the necessary level of safety. Presented below are two mixed integer models for optimal sensor placement against DDoS attacks. Both models lead to a trade-off between the number of deployed sensors and the volume of uncontrolled flow. Since the above placement problems are NP-hard, two efficient heuristics are designed, implemented and compared experimentally with exact mixed integer linear programming solvers.

cs.DS↗

arXiv2022

Online coloring of disk graphs

In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of discs with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of whichis an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online L(2, 1)-labeling.

math.CO↗

arXiv2022

Coloring distance graphs on the plane

We consider the coloring of certain distance graphs on the Euclidean plane. Namely, we ask for the minimal number of colors needed to color all points of the plane in such a way that pairs of points at distance in the interval $[1,b]$ get different colors. The classic Hadwiger-Nelson problem is a special case of this question -- obtained by taking $b=1$. The main results of the paper are improved lower and upper bounds on the number of colors for some values of $b$. In particular, we determine the minimal number of colors for two ranges of values of $b$ - one of which is enlarging an interval presented by Exoo and the second is completely new. Up to our knowledge, these are the only known families of distance graphs on the plane with a determined nontrivial chromatic number. Moreover, we present the first $8$-coloring for $b$ larger than values of $b$ for the known $7$-colorings. As a byproduct, we give some bounds and exact values for bounded parts of the plane, specifically by coloring certain annuli.

math.CO↗

arXiv2020

Online Coloring of Short Intervals

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in $[1,σ]$. For $σ=1$ it is the class of unit interval graphs, and for $σ=\infty$ the class of all interval graphs. Our focus is on intermediary classes. We present a $(1+σ)$-competitive algorithm, which beats the state of the art for $1 < σ< 2$, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than $5/3$-competitive for any $σ>1$, nor better than $7/4$-competitive for any $σ>2$, and that no algorithm beats the $5/2$ asymptotic competitive ratio for all, arbitrarily large, values of $σ$. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than $2$.

math.CO↗

arXiv2017

Fixing improper colorings of graphs

In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper $r$-coloring $φ$ of a graph $G$. We investigate the problem of finding a proper $r$-coloring of $G$, which is "the most similar" to $φ$, i.e. the number $k$ of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any $r \geq 3$, even for bipartite planar graphs. On the other hand, the problem is fixed-parameter tractable, when parameterized by the number of allowed transformations $k$. We provide an $2^n \cdot n^{\mathcal{O}(1)}$ algorithm for the problem (for any fixed $r$) and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the {\em fixing number} of a graph $G$. It is the maximum possible distance (in the number of transformations) between some non-proper coloring of $G$ and a proper one.

cs.DM↗

arXiv2016

Homothetic Polygons and Beyond: Intersection Graphs, Recognition, and Maximum Clique

We study the {\sc Clique} problem in classes of intersection graphs of convex sets in the plane. The problem is known to be NP-complete in convex-set intersection graphs and straight-line-segment intersection graphs, but solvable in polynomial time in intersection graphs of homothetic triangles. We extend the latter result by showing that for every convex polygon $P$ with sides parallel to $k$ directions, every $n$-vertex graph which is an intersection graph of homothetic copies of $P$ contains at most $n^{k}$ inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so called $k_{\text{DIR}}-\text{CONV}$, which are intersection graphs of convex polygons whose sides are parallel to some fixed $k$ directions. Moreover, we provide some lower bounds on the numbers of maximal cliques, discuss the complexity of recognizing these classes of graphs and present a relationship with other classes of convex-set intersection graphs. Finally, we generalize the upper bound on the number of maximal cliques to intersection graphs of higher-dimensional convex polytopes in Euclidean space.

cs.DM↗

arXiv2015

Fractional and $j$-fold colouring of the plane

We present results referring to the Hadwiger-Nelson problem which asks for the minimum number of colours needed to colour the plane with no two points at distance $1$ having the same colour. Exoo considered a more general problem concerning graphs $G_{[a,b]}$ with $\mathbb{R}^2$ as the vertex set and two vertices adjacent if their distance is in the interval $[a,b]$. Exoo conjectured $χ(G_{[a,b]}) = 7$ for sufficiently small but positive difference between $a$ and $b$. We partially answer this conjecture by proving that $χ(G_{[a,b]}) \geq 5$ for $b > a$. A $j$-fold colouring of graph $G = (V,E)$ is an assignment of $j$-elemental sets of colours to the vertices of $G$, in such a way that the sets assigned to any two adjacent vertices are disjoint. The fractional chromatic number $χ_f(G)$ is the infimum of fractions $k/j$ for $j$-fold colouring of $G$ using $k$ colours. We generalize a method by Hochberg and O'Donnel (who proved that $G_{[1,1]} \leq 4.36$) for fractional colouring of graphs $G_{[a,b]}$, obtaining a bound dependant on $\frac{a}{b}$. We also present few specific and two general methods for $j$-fold coloring of $G_{[a,b]}$ for small $j$, in particular for $G_{[1,1]}$ and $G_{[1,2]}$. The $j$-fold colouring for small $j$ has strong practical motivation especially in scheduling theory, while graph $G_{[1,2]}$ is often used to model hidden conflicts in radio networks.

math.CO↗

arXiv2015

First non-trivial upper bound on the circular chromatic number of the plane

We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of the plane with $r<7$ in the case when we require disjoint arcs for points at distance belonging to the internal $[\frac{10}{11}, \frac{12}{11}]$.

math.CO↗

arXiv2015

Counting independent sets via Divide Measure and Conquer method

In this paper we give an algorithm for counting the number of all independent sets in a given graph which works in time $O^*(1.1394^n)$ for subcubic graphs and in time $O^*(1.2369^n)$ for general graphs, where $n$ is the number of vertices in the instance graph, and polynomial space. The result comes from combining two well known methods "Divide and Conquer" and "Measure and Conquer". We introduce this new concept of Divide, Measure and Conquer method and expect it will find applications in other problems. The algorithm of Björklund, Husfeldt and Koivisto for graph colouring with our algorithm used as a subroutine has complexity $O^*(2.2369^n)$ and is currently the fastest graph colouring algorithm in polynomial space.

cs.DM↗

arXiv2013

An Exact Algorithm for the Generalized List $T$-Coloring Problem

The generalized list $T$-coloring is a common generalization of many graph coloring models, including classical coloring, $L(p,q)$-labeling, channel assignment and $T$-coloring. Every vertex from the input graph has a list of permitted labels. Moreover, every edge has a set of forbidden differences. We ask for such a labeling of vertices of the input graph with natural numbers, in which every vertex gets a label from its list of permitted labels and the difference of labels of the endpoints of each edge does not belong to the set of forbidden differences of this edge. In this paper we present an exact algorithm solving this problem, running in time $\mathcal{O}^*((τ+2)^n)$, where $τ$ is the maximum forbidden difference over all edges of the input graph and $n$ is the number of its vertices. Moreover, we show how to improve this bound if the input graph has some special structure, e.g. a bounded maximum degree, no big induced stars or a perfect matching.

cs.DM↗

arXiv2011

Determining L(2,1)-Span in Polynomial Space

A $k$-L(2,1)-labeling of a graph is a function from its vertex set into the set $\{0,...,k\}$, such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. It is known that finding the smallest $k$ admitting the existence of a $k$-L(2,1)-labeling of any given graph is NP-Complete. In this paper we present an algorithm for this problem, which works in time $O(\complexity ^n)$ and polynomial memory, where $\eps$ is an arbitrarily small positive constant. This is the first exact algorithm for L(2,1)-labeling problem with time complexity $O(c^n)$ for some constant $c$ and polynomial space complexity.

cs.DM↗