arXiv subjects

Thomas Kalinowski

Publications and source records attributed to Thomas Kalinowski.

At least 19 recordsLinked to original sources

arXiv2023

Independent domination in the graph defined by two consecutive levels of the $n$-cube

Fix a positive integer $n$ and consider the bipartite graph whose vertices are the $3$-element subsets and the $2$-element subsets of $[n]=\{1,2,\dots,n\}$, and there is an edge between $A$ and $B$ if $A\subset B$. We prove that the domination number of this graph is $\binom{n}{2}-\lfloor\frac{(n+1)^2}{8}\rfloor$, we characterize the dominating sets of minimum size, and we observe that the minimum size dominating set can be chosen as an independent set. This is an exact version of an asymptotic result by Balogh, Katona, Linz and Tuza (2021). For the corresponding bipartite graph between the $(k+1)$-element subsets and the $k$-elements subsets of $[n]$ ($k\geq 3$), we provide a new construction for small independent dominating sets. This improves on a construction by Gerbner, Kezegh, Lemons, Palmer, P\'alv\"olgyi and Patk\'os (2012), who studied these independent dominating sets under the name saturating flat antichains.

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arXiv2023

Sizes of flat maximal antichains of subsets

This is the second of two papers investigating for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only $l$-sets and $(l+1)$-sets for some $l$.

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arXiv2022

Maximal antichains of subsets II: Constructions

This is the second in a sequence of three papers investigating the question for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the previous paper we characterized those $m$ between $\binom{n}{\lceil n/2\rceil}-\lceil n/2\rceil^2$ and the maximum size $\binom{n}{\lceil n/2 \rceil}$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.

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arXiv2022

The saturation spectrum for antichains of subsets

Extending a classical theorem of Sperner, we characterize the integers $m$ such that there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$, that is, the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers $t$ and $k$, we ask which integers $s$ have the property that there exists a family $\mathcal F$ of $k$-sets with $\lvert\mathcal F\rvert=t$ such that the shadow of $\mathcal F$ has size $s$, where the shadow of $\mathcal F$ is the collection of $(k-1)$-sets that are contained in at least one member of $\mathcal F$. We provide a complete answer for $t\leqslant k+1$. Moreover, we prove that the largest integer which is not the shadow size of any family of $k$-sets is $\sqrt 2k^{3/2}+\sqrt[4]{8}k^{5/4}+O(k)$.

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arXiv2021

Minimum Weight Flat Antichains of Subsets

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in $\mathcal{A}$. Given reals $α,β>0$, we say the weight of $\mathcal F$ is $α\cdot|\mathcal{A}|+β\cdot|\mathcal{B}|$. We characterize the minimum weight antichains $\mathcal F$ for any given $n,k,α,β$, and we do the same when in addition $\mathcal F$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function.

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arXiv2020

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed.

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arXiv2020

Convex Hulls for Graphs of Quadratic Functions With Unit Coefficients: Even Wheels and Complete Split Graphs

We study the convex hull of the graph of a quadratic function $f(\mathbf{x})=\sum_{ij\in E}x_ix_j$, where the sum is over the edge set of a graph $G$ with vertex set $\{1,\dots,n\}$. Using an approach proposed by Gupte et al. (Discrete Optimization $\textbf{36}$, 2020, 100569), we investigate minimal extended formulations using additional variables $y_{ij}$, $1\leq i<j\leq n$, representing the products $x_ix_j$. The basic idea is to identify a set of facets of the Boolean Quadric Polytope which is sufficient for characterizing the convex hull for the given graph. Our main results are extended formulations for the cases that the underlying graph $G$ is either an even wheel or a complete split graph.

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arXiv2019

Maintenance scheduling in a railway corricdor

We investigate a novel scheduling problem which is motivated by an application in the Australian railway industry. Given a set of maintenance jobs and a set of train paths over a railway corridor with bidirectional traffic, we seek a schedule of jobs such that a minimum number of train paths are cancelled due to conflict with the job schedule. We show that the problem is NP-complete in general. In a special case of the problem when every job under any schedule just affects one train path, and the speed of trains is bounded from above and below, we show that the problem can be solved in polynomial time. Moreover, in another special case of the problem where the traffic is unidirectional, we show that the problem can be solved in time $O(n^4)$.

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arXiv2019

Feasible bases for a polytope related to the Hamilton cycle problem

We study a certain polytope depending on a graph $G$ and a parameter $β\in(0,1)$ which arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Eshragh \emph{et al.} conjectured a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress towards a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter $β$ when the parameter is close to 1, (2) the polytope can be interpreted as a generalized network flow polytope and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.

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arXiv2019

Extended formulations for convex hulls of graphs of bilinear functions

We consider the problem of characterizing the convex hull of the graph of a bilinear function $f$ on the $n$-dimensional unit cube $[0,1]^n$. Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a systematic study of properties of $f$ that guarantee that certain classes of BQP facets are sufficient for an extended formulation. We use a modification of Zuckerberg's geometric method for proving convex hull characterizations [Geometric proofs for convex hull defining formulations, Operations Research Letters \textbf{44} (2016), 625--629] to prove some initial results in this direction. In particular, we provide small-sized extended formulations for bilinear functions whose corresponding graph is either a cycle with arbitrary edge weights or a clique or an almost clique with unit edge weights.

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arXiv2019

Hamiltonian cycles and subsets of discounted occupational measures

We study a certain polytope arising from embedding the Hamiltonian cycle problem in a discounted Markov decision process. The Hamiltonian cycle problem can be reduced to finding particular extreme points of a certain polytope associated with the input graph. This polytope is a subset of the space of discounted occupational measures. We characterize the feasible bases of the polytope for a general input graph $G$, and determine the expected numbers of different types of feasible bases when the underlying graph is random. We utilize these results to demonstrate that augmenting certain additional constraints to reduce the polyhedral domain can eliminate a large number of feasible bases that do not correspond to Hamiltonian cycles. Finally, we develop a random walk algorithm on the feasible bases of the reduced polytope and present some numerical results. We conclude with a conjecture on the feasible bases of the reduced polytope.

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arXiv2018

Zero forcing in iterated line digraphs

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications.

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arXiv2018

Tight MIP formulations for bounded length cyclic sequences

We study cyclic binary strings with bounds on the lengths of the intervals of consecutive ones and zeros. This is motivated by scheduling problems where such binary strings can be used to represent the state (on/off) of a machine. In this context the bounds correspond to minimum and maximum lengths of on- or off-intervals, and cyclic strings can be used to model periodic schedules. Extending results for non-cyclic strings is not straight forward. We present a non-trivial tight compact extended network flow formulation, as well as valid inequalities in the space of the state and start-up variables some of which are shown to be facet-defining. Applying a result from disjunctive programming, we also convert the extended network flow formulation into an extended formulation over the space of the state and start-up variables.

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arXiv2018

Minimum rank and zero forcing number for butterfly networks

The minimum rank of a simple graph $G$ is the smallest possible rank over all symmetric real matrices $A$ whose nonzero off-diagonal entries correspond to the edges of $G$. Using the zero forcing number, we prove that the minimum rank of the butterfly network is $\frac19\left[(3r+1)2^{r+1}-2(-1)^r\right]$ and that this is equal to the rank of its adjacency matrix.

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arXiv2018

A lower bound on the zero forcing number

In this note, we study a dynamic vertex coloring for a graph $G$. In particular, one starts with a certain set of vertices black, and all other vertices white. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a \emph{zero forcing set} if by iterating this process, all of the vertices in $G$ become black. The \emph{zero forcing number} of $G$ is the minimum cardinality of a zero forcing set in $G$, and is denoted by $Z(G)$. Davila and Kenter have conjectured in 2015 that $Z(G)\geq (g-3)(δ-2)+δ$ where $g$ and $δ$ denote the girth and the minimum degree of $G$, respectively. This conjecture has been proven for graphs with girth $g \leq 10$. In this note, we present a proof for $g \geq 5$, $δ\geq 2$, thereby settling the conjecture.

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arXiv2017

$H$-supermagic labelings for firecrackers, banana trees and flowers

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ is contained in a subgraph $H'=(V',E')$ of $G$ which is isomorphic to $H$. In this case we say that $G$ is $H$-supermagic if there is a bijection $f:V\cup E\to\{1,\ldots\lvert V\rvert+\lvert E\rvert\}$ such that $f(V)=\{1,\ldots,\lvert V\rvert\}$ and $\sum_{v\in V(H')}f(v)+\sum_{e\in E(H')}f(e)$ is constant over all subgraphs $H'$ of $G$ which are isomorphic to $H$. In this paper, we show that for odd $n$ and arbitrary $k$, the firecracker $F_{k,n}$ is $F_{2,n}$-supermagic, the banana tree $B_{k,n}$ is $B_{1,n}$-supermagic and the flower $F_n$ is $C_3$-supermagic.

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arXiv2017

The metric dimension of the circulant graph $C(n,\pm\{1,2,3,4\})$

Let $G=(V,E)$ be a connected graph and let $d(u,v)$ denote the distance between vertices $u,v \in V$. A metric basis for $G$ is a set $B\subseteq V$ of minimum cardinality such that no two vertices of $G$ have the same distances to all points of $B$. The cardinality of a metric basis of $G$ is called the metric dimension of $G$, denoted by $\dim(G)$. In this paper we determine the metric dimension of the circulant graphs $C(n,\pm\{1,2,3,4\})$ for all values of $n$.

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arXiv2017

Zero forcing number of graphs

A subset $S$ of initially infected vertices of a graph $G$ is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects this neighbour. The forcing number of $G$ is the minimum cardinality of a forcing set in $G$. In the present paper, we study the forcing number of various classes of graphs, including graphs of large girth, $H$-free graphs for a fixed bipartite graph $H$, random and pseudorandom graphs.

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