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Alan Frieze

Publications and source records attributed to Alan Frieze.

At least 55 records · Page 3Linked to original sources
arXiv2023

Rainbow Hamilton Cycles in Random Geometric Graphs

Let $X_1,X_2,\ldots,X_n$ be chosen independently and uniformly at random from the unit $d$-dimensional cube $[0,1]^d$. Let $r$ be given and let $\cal X=\{X_1,X_2,\ldots,X_n\}$. The random geometric graph $G=G_{\cal X,r}$ has vertex set $\cal X$ and an edge $X_iX_j$ whenever $\|X_i-X_j\|\leq r$. We show that if each edge of $G$ is colored independently from one of $n+o(n)$ colors and $r$ has the smallest value such that $G$ has minimum degree at least two, then $G$ contains a rainbow Hamilton cycle a.a.s.

math.CO↗
arXiv2023

Giant Rainbow Trees in Sparse Random Graphs

For any small constant $ε>0$, the Erdős-Rényi random graph $G(n,\frac{1+ε}{n})$ with high probability has a unique largest component which contains $(1\pm O(ε))2εn$ vertices. Let $G_c(n,p)$ be obtained by assigning each edge in $G(n,p)$ a color in $[c]$ independently and uniformly. Cooley, Do, Erde, and Missethan proved that for any fixed $α>0$, $G_{αn}(n,\frac{1+ε}{n})$ with high probability contains a rainbow tree (a tree that does not repeat colors) which covers $(1\pm O(ε))\fracα{α+1}εn$ vertices, and conjectured that there is one which covers $(1\pm O(ε))2εn$. In this paper, we achieve the correct leading constant and prove their conjecture correct up to a logarithmic factor in the error term, as we show that with high probability $G_{αn}(n,\frac{1+ε}{n})$ contains a rainbow tree which covers $(1\pm O(ε\log(1/ε)))2εn$ vertices.

math.CO↗
arXiv2023

Rainbow Greedy Matching Algorithms

We consider the problem of finding a large rainbow matching in a random graph with randomly colored edges. In particular we analyze the performance of two greedy algorithms for this problem. The algorithms we study are colored versions of algorithms that were previously used to find large matchings in random graphs (i.e. the color-free version of our present problem).

math.CO↗
arXiv2022

Spanners in randomly weighted graphs: Euclidean case

Given a connected graph $G=(V,E)$ and a length function $\ell:E\to {\mathbb R}$ we let $d_{v,w}$ denote the shortest distance between vertex $v$ and vertex $w$. A $t$-spanner is a subset $E'\subseteq E$ such that if $d'_{v,w}$ denotes shortest distances in the subgraph $G'=(V,E')$ then $d'_{v,w}\leq t d_{v,w}$ for all $v,w\in V$. We study the size of spanners in the following scenario: we consider a random embedding of $G_{n,p}$ into the unit square with Euclidean edge lengths. For $ε>0$ constant, we prove the existence w.h.p. of $(1+ε)$-spanners for ${\mathcal X}_p$ that have $O_ε(n)$ edges. These spanners can be constructed in $O_ε(n^2\log n)$ time. (We will use $O_ε$ to indicate that the hidden constant depends on $ε$.) There are constraints on $p$ preventing it going to zero too quickly.

math.CO↗
arXiv2022

Colorful Hamilton cycles in random graphs

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C=\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)\in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,\ldots,P_r$, where $P_i$ contains $m_i$ edges of color $c_i$. We study $hcp(G_{n,p})$ when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when $hcp(G_{n,p})=\{(m_1,m_2,\ldots,m_r)\in [0,n]^r: m_1+m_2+\cdots+m_r=n\}$.

math.CO↗
arXiv2022

Hamilton cycles in a semi-random graph model

We show that with high probability we can build a Hamilton cycle after at most $1.85 n$ rounds in a particular semi-random model. In this model, in one round, we are given a {uniform random} $v\in[n]$ and then we can add an {arbitrary} edge $\{v,w\}$. Our result improves on $2.016 n$ in a recent paper of Gao, MacRury, and Pralat.

math.CO↗
arXiv2022

Rank of the vertex-edge incidence matrix of $r$-out hypergraphs

We consider a space of sparse Boolean matrices of size $n \times n$, which have finite co-rank over $GF(2)$ with high probability. In particular, the probability such a matrix has full rank, and is thus invertible, is a positive constant with value about $0.2574$ for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs The result that the null space is finite, can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices $n$, \cite{ACO}, \cite{CFP}. For fields of higher order, the co-rank is typically Poisson distributed.

math.CO↗
arXiv2022

Subexponential mixing for partition chains on grid-like graphs

We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph. For the case where we want to have partitions with linearly many pieces of bounded size, we obtain approximate sampling algorithms based on Glauber dynamics which are fixed-parameter tractable with respect to the bandwidth of $G$, with simple-exponential dependence on the bandwidth. For example, for rectangles of constant or logarithmic width this gives polynomial-time sampling algorithms. More generally, this gives sub-exponential algorithms for bounded-degree graphs without large expander subgraphs (for example, we obtain $O(2^{\sqrt n})$ time algorithms for square grids). In the case where we instead want partitions with a small number of pieces of linear size, we show that Glauber dynamics can have exponential mixing time, even just for the case of 2 pieces, and even for 2-connected subgraphs of the grid with bounded bandwidth.

math.PR↗
arXiv2022

On the cover time of the emerging giant

Let $p=\frac{1+\varepsilon}{n}$. It is known that if $N=\varepsilon^3n\to\infty$ then w.h.p. $G_{n,p}$ has a unique giant largest component. We show that if in addition, $\varepsilon=\varepsilon(n)\to 0$ then w.h.p. the cover time of $G_{n,p}$ is asymptotic to $n\log^2N$; previously Barlow, Ding, Nachmias and Peres had shown this up to constant multiplicative factors.

math.CO↗
arXiv2021

Maker Breaker on Digraphs

We study two biassed Maker-Breaker games played on the complete digraph $\vec{K}_n$. In the strong connectivity game, Maker wants to build a strongly connected subgraph. We determine the asymptotic optimal bias for this game viz. $\frac{n}{\log n}$. In the Hamiltonian game, Maker wants to build a Hamiltonian subgraph. We determine the asymptotic optimal bias for this game up to a constant factor.

math.CO↗
arXiv2021

Spanners in randomly weighted graphs: independent edge lengths

Given a connected graph $G=(V,E)$ and a length function $\ell:E\to {\mathbb R}$ we let $d_{v,w}$ denote the shortest distance between vertex $v$ and vertex $w$. A $t$-spanner is a subset $E'\subseteq E$ such that if $d'_{v,w}$ denotes shortest distances in the subgraph $G'=(V,E')$ then $d'_{v,w}\leq t d_{v,w}$ for all $v,w\in V$. We show that for a large class of graphs with suitable degree and expansion properties with independent exponential mean one edge lengths, there is w.h.p.~a 1-spanner that uses $\approx \frac12n\log n$ edges and that this is best possible. In particular, our result applies to the random graphs $G_{n,p}$ for $np\gg \log n$.

math.CO↗
arXiv2021

Multitrees in random graphs

Let $N=\binom{n}{2}$ and $s\geq 2$. Let $e_{i,j},\,i=1,2,\ldots,N,\,j=1,2,\ldots,s$ be $s$ independent permutations of the edges $E(K_n)$ of the complete graph $K_n$. A {\em MultiTree} is a set $I\subseteq [N]$ such that the edge sets $E_{I,j}$ induce spanning trees for $j=1,2,\ldots,s$. In this paper we study the following question: what is the smallest $m=m(n)$ such that w.h.p. $[m]$ contains a MultiTree. We prove a hitting time result for $s=2$ and an $O(n\log n)$ bound for $s\geq 3$.

math.CO↗
arXiv2021

Karp's patching algorithm on dense digraphs

We consider the following question. We are given a dense digraph $D$ with minimum in- and out-degree at least $αn$, where $α>1/2$ is a constant. The edges of $D$ are given edge costs $C(e),e\in E(D)$, where $C(e)$ is an independent copy of the uniform $[0,1]$ random variable $U$. Let $C(i,j),i,j\in[n]$ be the associated $n\times n$ cost matrix where $C(i,j)=\infty$ if $(i,j)\notin E(D)$. We show that w.h.p. the patching algorithm of Karp finds a tour for the asymmetric traveling salesperson problem that is asymptotically equal to that of the associated assignment problem. Karp's algorithm runs in polynomial time.

cs.DS↗
arXiv2021

Isomorphism for Random $k$-Uniform Hypergraphs

We study the isomorphism problem for random hypergraphs. We show that it is solvable in polynomial time for the binomial random $k$-uniform hypergraph $H_{n,p;k}$, for a wide range of $p$. We also show that it is solvable w.h.p. for random $r$-regular, $k$-uniform hypergraphs $H_{n,r;k},r=O(1)$.

math.CO↗
arXiv2020

Giant descendant trees, matchings and independent sets in the age-biased attachment graphs

We study two models of an age-biased graph process: the $δ$-version of the preferential attachment graph model (PAM) and the uniform attachment graph model (UAM), with $m$ attachments for each of incoming vertices. We show that almost surely the scaled size of a breadth-first (descendant) tree rooted at a fixed vertex converges, for $m=1$, to a limit whose distribution is a mixture of two beta-distributions and a single beta-distribution respectively, and that for $m>1$ the limit is $1$. We also analyze the likely performance of two greedy (online) algorithms, for a large matching set and a large independent set, and determine--for each model and each greedy algorithm--both a limiting fraction of vertices involved and an almost sure convergence rate.

math.CO↗
arXiv2020

On the existence of Hamilton cycles with a periodic pattern in a random digraph

We consider Hamilton cycles in the random digraph $D_{n,m}$ where the orientation of edges follows a pattern other than the trivial orientation in which the edges are oriented in the same direction as we traverse the cycle. We show that if the orientation forms a periodic pattern, other than the trivial pattern, then approximately half the usual $n\log n$ edges are needed to guarantee the existence of such Hamilton cycles a.a.s.

math.CO↗
arXiv2020

Random volumes in d-dimensional polytopes

Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Füredi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.

math.PR↗