arXiv subjects

Geir Agnarsson

Publications and source records attributed to Geir Agnarsson.

At least 19 recordsLinked to original sources

arXiv2026

Elements represented as intersections of sets

For a natural number $n$ let $[n] = \{1,\ldots,n\}$. We say that a family ${\cal{S}}\subseteq 2^{[n]}$ is \emph{representing} if every singleton set of $[n]$ is an intersection of some sets from ${\cal{S}}$. We show that the smallest possible cardinality of a representing set for $[n]$ is the discrete inverse $s(n)$ of the Sperner's function $n\mapsto \binom{n}{\lfloor n/2\rfloor}$, which by Sperner's Theorem is the maximum number of elements in an antichain in $2^{[n]}$ when viewed as subset (or boolean) lattice. Specifically, $s(n)$ is then the smallest positive integer such that $2^{[n]}$ contains an $n$-element antichain. Some generalization, further applications and asymptotics in terms of the second real branch of the Lambert $W$ function are presented.

math.CO↗

arXiv2026

Tur\'{a}n results for posets and their alternating cycles

For a partially ordered set ${\mathbb{P}} = (X,\leq)$ there exist hypergraphs where the vertices are the set of ordered tuples of either all incomparable elements of ${\mathbb{P}}$ or all the critical pairs of ${\mathbb{P}}$, and the edges are formed by the duals of either all the alternating cycles of ${\mathbb{P}}$ or all the strict alternating cycles of ${\mathbb{P}}$. The weak chromatic numbers of these hypergraphs are all equal to the order dimension of ${\mathbb{P}}$. Here are established upper bounds on the number of strict alternating cycles a poset ${\mathbb{P}}=(X,\leq)$ can have in terms of $n = |X|$, the cardinality of the groundset of ${\mathbb{P}}$, and the width $w$ of ${\mathbb{P}}$. These bounds also apply to the number of hyperedges of the associated hypergraph ${\mathcal{H}}^s(\mathbb{P})$, with incomparable pairs as vertices and strict alternating cycles dual to its hyperedges.

math.CO↗

arXiv2024

Minkowski ideals and rings

\emph{Minkowski rings} are certain rings of simple functions on the Euclidean space $W = {\mathbb{R}}^d$ with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set ${\cal{P}}$ of indicator functions of $n$ polytopes then the ring can be presented as ${\mathbb{C}}[x_1,\ldots,x_n]/I$ when viewed as a ${\mathbb{C}}$-algebra, where $I$ is the ideal describing all the relations implied by identities among Minkowski sums of elements of ${\cal{P}}$. We discuss in detail the $1$-dimensional case, the $d$-dimensional box case and the affine Coxeter arrangement in ${\mathbb{R}}^2$ where the convex sets are formed by closed half-planes with bounding lines making the regular triangular grid in ${\mathbb{R}}^2$. We also consider, for a given polytope $P$, the Minkowski ring $M^\pm_F(P)$ of the collection ${\cal{F}}(P)$ of the nonempty faces of $P$ and their multiplicative inverses. Finally we prove some general properties of identities in the Minkowski ring of ${\cal{F}}(P)$; in particular, we show that Minkowski rings behave well under Cartesian product, namely that $M^\pm_F(P\times Q) \cong M^{\pm}_F(P)\otimes M^{\pm}_F(Q)$ as ${\mathbb{C}}$-algebras where $P$ and $Q$ are polytopes.

math.CO↗

arXiv2023

On locally finite ordered rooted trees and their rooted subtrees

In this article we compare the known dynamical polynomial time algorithm for the game-over attack strategy, to that of the brute force approach; of checking all the ordered rooted subtrees of a given tree that represents a given computer network. Our approach is purely enumerative and combinatorial in nature. We first revisit known results about a doubly exponential sequence and generalize them. We then consider both finite and locally finite ordered rooted trees (LFOR-trees), and the class of their finite ordered rooted subtrees of bounded height, describing completely the LFOR-trees with no leaves where the number of ordered rooted subtrees of height at most $h$ are bounded by a polynomial in $h$. We finally consider general LFOR-trees where each level can have leaves and determine conditions for the number of ordered rooted subtrees of height at most $h$ to be bounded by a polynomial in $h$.

math.CO↗

arXiv2023

On posets, monomial ideals, Gorenstein ideals and their combinatorics

In this article we first compare the set of elements in the socle of an ideal of a polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ that are not in the ideal itself and Macaulay's inverse systems of such polynomial algebras in a purely combinatorial way for monomial ideals, and then develop some closure operational properties for the related poset ${{\nats}_0^d}$. We then derive some algebraic propositions of $Γ$-graded rings that then have some combinatorial consequences. Interestingly, some of the results from this part that uniformly hold for polynomial rings are always false when the ring is local. We finally delve into some direct computations, w.r.t.~a given term order of the monomials, for general zero-dimensional Gorenstein ideals and deduce a few explicit observations and results for the inverse systems from some recent results about socles.

math.AC↗

arXiv2023

Power-closed ideals of polynomial and Laurent polynomial rings

We investigate the structure of power-closed ideals of the complex polynomial ring $R = \mathbb{C}[x_1,\ldots,x_d]$ and the Laurent polynomial ring $R^{\pm} = \mathbb{C}[x_1,\ldots,x_d]^{\pm} = M^{-1}\mathbb{C}[x_1,\ldots,x_d]$, where $M$ is the multiplicative sub-monoid $M = [x_1,\ldots,x_d]$ of $R$. Here, an ideal $I$ is {\em power-closed} if $f(x_1,\ldots,x_d)\in I$ implies $f(x_1^i,\ldots,x_d^i)\in I$ for each natural $i$. In particular, we investigate related closure and interior operators on the set of ideals of $R$ and $R^{\pm}$. Finally, we give a complete description of principal power-closed ideals and of the radicals of general power-closed ideals of $R$ and $R^{\pm}$.

math.AC↗

arXiv2019

On a special presentation of matrix algebras

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}\}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)\times(m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropriate powers $a^i$ and $a^j$ of a single element $a$ respectively. In general very little is known about the structure of the ring $S$. In this article we study in depth the case $m=n=1$ when $R\cong M_2(S)$. More specifically we study the universal algebra over a commutative ring $A$ with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We describe completely the structure of these $A$-algebras and their underlying rings when $\gcd(i,j)=1$. Finally we obtain results that fully determine when there are surjections onto $M_2({\mathbb F})$ when ${\mathbb F}$ is a base field ${\mathbb Q}$ or ${\mathbb Z}_p$ for a prime number $p$.

math.RA↗

arXiv2018

On monomial ideals and their socles

For a finite subset $M\subset [x_1,\ldots,x_d]$ of monomials, we describe how to constructively obtain a monomial ideal $I\subseteq R = K[x_1,\ldots,x_d]$ such that the set of monomials in $\text{Soc}(I)\setminus I$ is precisely $M$, or such that $\overline{M}\subseteq R/I$ is a $K$-basis for the the socle of $R/I$. For a given $M$ we obtain a natural class of monomials $I$ with this property. This is done by using solely the lattice structure of the monoid $[x_1,\ldots,x_d]$. We then present some duality results by using anti-isomorphisms between upsets and downsets of $(\mathbb Z^d,\preceq)$. Finally, we define and analyze zero-dimensional monomial ideals of $R$ of type $k$, where type $1$ are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in $\mathbb Z^d$.

math.AC↗

arXiv2016

On a special class of general permutahedra

Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices $Δ_{k-1}$ in ${\mathbb{R}}^n$ where $k$ and $n$ are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their {\em exponential generating flag function}. These polytopes are simple, include both the simplex $Δ_{n-1}$ and the permutahedron $Π_{n-1}$, and form a Minkowski basis for more general permutahedra.

math.CO↗

arXiv2016

The structure and topology of rooted weighted trees modeling layered cyber-security systems

In this paper we consider a layered-security model in which the containers and their nestings are given in the form of a rooted tree $T$. A {\em cyber-security model\/} is an ordered three-tuple $M = (T, C, P)$ where $C$ and $P$ are multisets of {\em penetration costs\/} for the containers and {\em target-acquisition values\/} for the prizes that are located within the containers, respectively, both of the same cardinality as the set of the non-root vertices of $T$. The problem that we study is to assign the penetration costs to the edges and the target-acquisition values to the vertices of the tree $T$ in such a way that minimizes the total prize that an attacker can acquire given a limited {\em budget}. For a given assignment of costs and target values we obtain a {\em security system}, and we discuss three types of them: {\em improved}, {\em good}, and {\em optimal}. We show that in general it is not possible to develop an optimal security system for a given cyber-security model $M$. We define P- and C-models where the penetration costs and prizes, respectively, all have unit value. We show that if $T$ is a rooted tree such that any P- or C-model $M = (T,C,P)$ has an optimal security system, then $T$ is one of the following types: (i) a rooted path, (ii) a rooted star, (iii) a rooted 3-caterpillar, or (iv) a rooted 4-spider. Conversely, if $T$ is one of these four types of trees, then we show that any P- or C-model $M = (T,C,P)$ does have an optimal security system\@. Finally, we study a duality between P- and C-models that allows us to translate results for P-models into corresponding results for C-models and vice versa. The results obtained give us some mathematical insights into how layered-security defenses should be organized.

cs.DM↗

arXiv2015

The complexity of cyber attacks in a new layered-security model and the maximum-weight, rooted-subtree problem

In our cyber security model we define the concept of {\em penetration cost}, which is the cost that must be paid in order to break into the next layer of security. Given a tree $T$ rooted at a vertex $r$, a {\em penetrating cost} edge function $c$ on $T$, a {\em target-acquisition} vertex function $p$ on $T$, the attacker's {\em budget} and the {\em game-over threshold} $B,G \in {\mathbb{Q}}^{+}$ respectively, we consider the problem of determining the existence of a rooted subtree $T'$ of $T$ within the attacker's budget (that is, the sum of the costs of the edges in $T'$ is less than or equal to $B$) with total acquisition value more than the game-over threshold (that is, the sum of the target values of the nodes in $T'$ is greater than or equal to $G$). We prove that the general version of this problem is intractable, but does admit a polynomial time approximation scheme. We also analyze the complexity of three restricted versions of the problems, where the penetration cost is the constant function, integer-valued, and rational-valued among a given fixed number of distinct values.

cs.DS↗

arXiv2013

Extremal subgraphs of the $d$-dimensional grid graph

For each natural number $n$ we determine, both asymptotically and exactly, the maximum number of edges an induced subgraph of order $n$ of the $d$-dimension a grid graph ${\ints}^d$ can have. The asymptotic bound is obtained by using a theorem Bollobás and Thomason, and the exact bound is obtained by induction. This generalizes some earlier results for the case $d=2$ on one hand, and for $n\leq 2^d$ on the other.

math.CO↗

arXiv2011

Induced subgraphs of hypercubes

Let $Q_k$ denote the $k$-dimensional hypercube on $2^k$ vertices. A vertex in a subgraph of $Q_k$ is {\em full} if its degree is $k$. We apply the Kruskal-Katona Theorem to compute the maximum number of full vertices an induced subgraph on $n\leq 2^k$ vertices of $Q_k$ can have, as a function of $k$ and $n$. This is then used to determine $\min(\max(|V(H_1)|, |V(H_2)|))$ where (i) $H_1$ and $H_2$ are induced subgraphs of $Q_k$, and (ii) together they cover all the edges of $Q_k$, that is $E(H_1)\cup E(H_2) = E(Q_k)$.

math.CO↗

arXiv2011

On the number of hypercubic bipartitions of an integer

We revisit a well-known divide-and-conquer maximin recurrence $f(n) = \max(\min(n_1,n_2) + f(n_1) + f(n_2))$ where the maximum is taken over all proper bipartitions $n = n_1+n_2$, and we present a new characterization of the pairs $(n_1,n_2)$ summing to $n$ that yield the maximum $f(n) = \min(n_1,n_2) + f(n_1) + f(n_2)$. This new characterization allows us, for a given $n\in\nats$, to determine the number $h(n)$ of these bipartitions that yield the said maximum $f(n)$. We present recursive formulae for $h(n)$, a generating function $h(x)$, and an explicit formula for $h(n)$ in terms of a special representation of $n$.

math.CO↗

arXiv2010

The flag polynomial of the Minkowski sum of simplices

For a polytope we define the {\em flag polynomial}, a polynomial in commuting variables related to the well-known flag vector and describe how to express the the flag polynomial of the Minkowski sum of $k$ standard simplices in a direct and canonical way in terms of the {\em $k$-th master polytope} $P(k)$ where $k\in\nats$. The flag polynomial facilitates many direct computations. To demonstrate this we provide two examples; we first derive a formula for the $f$-polynomial and the maximum number of $d$-dimensional faces of the Minkowski sum of two simplices. We then compute the maximum discrepancy between the number of $(0,d)$-chains of faces of a Minkowski sum of two simplices and the number of such chains of faces of a simple polytope of the same dimension and on the same number of vertices.

math.CO↗

arXiv2010

On integer radii coin representations of the wheel graph

A {\em flower} is a coin graph representation of the wheel graph. A {\em petal} of the wheel graph is an edge to the center vertex. In this paper we investigate flowers whose coins have integer radii. For an $n$-petaled flower we show there is a unique irreducible polynomial $P_n$ in $n$ variables over the integers $\ints$, the affine variety of which contains the cosines of the internal angles formed by the petals of the flower. We also establish a recursion that these irreducible polynomials satisfy. Using the polynomials $P_n$, we develop a parameterization for all the integer radii of the coins of the 3-petal flower.

math.AC↗

arXiv2009

On the maximum number of edges of non-flowerable coin graphs

For $n\in\nats$ and $3\leq k\leq n$ we compute the exact value of $E_k(n)$, the maximum number of edges of a simple planar graph on $n$ vertices where each vertex bounds an $\ell$-gon where $\ell\geq k$. The lower bound of $E_k(n)$ is obtained by explicit construction, and the matching upper bound is obtained by using Integer Programming (IP.) We then use this result to conjecture the maximum number of edges of a non-flowerable coin graph on $n$ vertices. A {\em flower} is a coin graph representation of the wheel graph. A collection of coins or discs in the Euclidean plane is {\em non-flowerable} if no flower can be formed by coins from the collection.

math.CO↗

arXiv2008

The Complexity of the Evolution of Graph Labelings

We study the {\sc Graph Relabeling Problem}--given an undirected, connected, simple graph $G = (V,E)$, two labelings $L$ and $L'$ of $G$, and label {\em flip} or {\em mutation} functions determine the complexity of transforming or evolving the labeling $L$ into $L'$\@. The transformation of $L$ into $L'$ can be viewed as an evolutionary process governed by the types of flips or mutations allowed. The number of applications of the function is the duration of the evolutionary period. The labels may reside on the vertices or the edges. We prove that vertex and edge relabelings have closely related computational complexities. Upper and lower bounds on the number of mutations required to evolve one labeling into another in a general graph are given. Exact bounds for the number of mutations required to evolve paths and stars are given. This corresponds to computing the exact distance between two vertices in the corresponding {\em Cayley graph}. We finally explore both vertex and edge relabeling with {\em privileged labels}, and resolve some open problems by providing precise characterizations of when these problems are solvable. Many of our results include algorithms for solving the problems, and in all cases the algorithms are polynomial-time. The problems studied have applications in areas such as bioinformatics, networks, and VLSI.

math.CO↗