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Alan Frieze

Publications and source records attributed to Alan Frieze.

At least 19 recordsLinked to original sources
arXiv2026

Expected cost in Combinatorial Optimization under color constraints

We present an average case model of classical problems in combinatorial optimization where there are color constraints. In all cases we seek some (spanning) sub-structure of a complete graph of minimum cost. The edges are randomly colored either red or blue. We bias against the red edges by placing a bound on the number of them that are allowed in our structure. This bound will be lower w.h.p. than what would occur without discrimination. We examine the effect of this bias on the minimum cost of a desired structure. We consider minimum cost spanning trees, shortest paths, minimum cost perfect matchings and the asymmetric traveling salesperson problem.

math.CO↗
arXiv2026

Karp's patching algorithm on random perturbations of dense digraphs

We consider the following question. We are given a dense digraph $D_0$ with minimum in- and out-degree at least $αn$, where $α>0$ is a constant. We then add random edges $R$ to $D_0$ to create a digraph $D$. Here an edge $e$ is placed independently into $R$ with probability $n^{-ε}$ where $ε>0$ is a small positive constant. The edges $E(D)$ of $D$ are given independent edge costs $C=C(e),e\in E(D)$, where $C$ has a density $f(x)=a+bx+o(x)$ as $x\to 0$. Here $a>0,b$ are constants. The prime examples will be the uniform $[0,1]$ distribution ($a=1,b=0$) and the exponential mean 1 distribution $EXP(1)$ ($a=1,b=-1$). 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 whose cost is asymptotically equal to the cost of the associated assignment problem. Karp's algorithm runs in polynomial time.

math.CO↗
arXiv2026

Karp's patching algorithm on dense digraph

We consider the following question. We are given a dense digraph $D$ with $n$ vertices and minimum in- and out-degree at least $αn$, where $α>1/2$ is a constant. The edges $E(D)$ of $D$ are given independent edge costs $C(e),e\in E(D)$, such that (i) $C$ has a density $f$ that satisfies $f(x)=a+bx+O(x^2)$, for constants $a>0,b$ as $x\to 0$ and such that in general either (ii) $\Pr(C\geq x)\leq \a e^{-\b x}$ for constants $\a,\b>0$, or $f(x)=0$ for $x>\n$ for some constant $\n>0$. Let $C(i,j),i,j\in[n]$ be the associated $n\times n$ cost matrix where $C(i,j)=\infty$ if $(i,j)\notin E$. We show that w.h.p. (a small modification to) 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. The algorithm runs in polynomial time.

math.CO↗
arXiv2026

Solving a Random Asymmetric TSP Exactly in Quasi-Polynomial Time w.h.p

Let the costs $C(i,j)$ for an instance of the Asymmetric Traveling Salesperson Problem (ATSP) be independent copies of a non-negative random variable $C$ from a class of distributions that include the uniform $[0,1]$ distribution and the exponential mean 1 distribution with mean 1. We describe an algorithm that solves ATSP exactly in time $e^{\log^{2+o(1)}n}$, w.h.p.

cs.DS↗
arXiv2026

Binomial Random Matroids

Let $\mathcal B=\mathcal B_{k,n,p}$ be a random collection of $k$-subsets of $[n]$ where each possible set is present independently with probability $p$. Let $\cal E_{\mathcal B}$ be the event that $\mathcal B$ defines the set of bases of a matroid. We prove that If $p= 1-\frac{c_n}{(k(n-k)\binom nk)^{1/2}}$ where $0\leq c_n\leq \infty$, then \[ \lim_{n\to\infty}\Pr[\cal E_{\cal B}\mid |\cal B|\geq2]=\begin{cases}1&c_n\to0.\\e^{-c^2/2}&c_n\to c.\\0&c_n\to \infty.\end{cases}\] In addition, we identify a condition preventing the occurence of $\cal E_{\cal B}$ and prove a hitting time version for the occurence of $\cal B$. We also prove that when $\cal E_{\mathcal B}$ occurs, $\mathcal B$ defines a sparse paving matroid w.h.p. In addition, study a greedy algorithm that produces a random matroid defined by a collection of hyperplanes. We use this to improve the estimates in \cite{HPV} on $\log m(n,k),\log p(n,k), \log s(n,k)$ where $ m(n, k), p(n, k), s(n, k)$ denote the number of matroids, paving matroids, and sparse paving matroids (respectively) of rank $k$ on $[n]$. Our improvement lies in that we can deal with $k$ growing slowly with $n$ as opposed to $k=O(1)$ in \cite{HPV}. More generally, we obtain estimates for the number of matchings in nearly-regular hypergraphs with small codegree, which may be of independent interest.

math.CO↗
arXiv2026

Coloring powers of random graphs

Given a graph $G$ and an integer $r\ge 1$, the $r$th power $G^r$ of $G$ is the graph obtained from $G$ by adding edges for all pairs of distinct vertices at distance at most $r$ from each other. We focus on two basic structural properties of the $r$th power of the binomial random graph $G_{n,p}$, namely, the maximum degree $Δ(G_{n,p}^r)$ and the chromatic number $χ(G_{n,p}^r)$, and give with high probability (w.h.p.) bounds. In the sparse case that $p=d/n$ for some fixed constant $d>0$, we prove the following. We prove that w.h.p.~$Δ(G_{n,p}^r) \sim \frac{\log n}{\log_{(r+1)}n}$ (where $\log_{(1)}n=\log n$ and $\log_{(r+1)}n=\log\log_{(r)}n$) and that w.h.p.~$Δ(G_{n,p}^{\lfloor{r/2}\rfloor})+1 \le χ(G_{n,p}^r) \le Δ(G_{n,p}^{r-1})+1$. For $r=2$, we show the upper bound holds with equality. For denser cases, for $d$ satisfying $d=ω(\log n)$ and $d\le n^{1/r-Ω(1)}$ as $n\to\infty$, we have $χ(G_{n,p}^r) = Θ(d^r/\log d)$ w.h.p.

math.CO↗
arXiv2026

Edge disjoint Hamilton cycles in random digraphs of constant minimum degree

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(δ\geq k+1)}$. Let $m=cn$ and $c\ge c_k$, where $c_k$ is a sufficiently large constant. We prove that w.h.p. $D_{n,m}^{(δ\geq k+1)}$ contains $k$ edge disjoint Hamilton cycles.

math.CO↗
arXiv2026

The effect of adding randomly weighted edges

We consider the following question. We have a dense regular graph $G$ with degree $αn$, where $α>0$ is a constant. We add $m=o(n^2)$ random edges. The edges of the augmented graph $G(m)$ are given independent edge weights $X(e)$, $e\in E(G(m))$. We estimate the minimum weight of some specified combinatorial structures. We show that in certain cases, we can obtain the same estimate as is known for the complete graph, but scaled by a factor $α^{-1}$. We consider spanning trees, shortest paths, perfect matchings in (pseudo-random) bipartite graphs.

math.CO↗
arXiv2026

Loose paths in random ordered hypergraphs

We consider the length of {\em ordered loose paths} in the random $r$-uniform hypergraph $H=H^{(r)}(n, p)$. A ordered loose path is a sequence of edges $E_1,E_2,\ldots,E_\ell$ where $\max\{j\in E_i\}=\min\{j\in E_{i+1}\}$ for $1\leq i<\ell$. We establish fairly tight bounds on the length of the longest ordered loose path in $H$ that hold with high probability.

math.CO↗
arXiv2025

O(1) Insertion for Random Walk d-ary Cuckoo Hashing up to the Load Threshold

The random walk $d$-ary cuckoo hashing algorithm was defined by Fotakis, Pagh, Sanders, and Spirakis to generalize and improve upon the standard cuckoo hashing algorithm of Pagh and Rodler. Random walk $d$-ary cuckoo hashing has low space overhead, guaranteed fast access, and fast in practice insertion time. In this paper, we give a theoretical insertion time bound for this algorithm. More precisely, for every $d\ge 3$ random hashes, let $c_d^*$ be the sharp threshold for the load factor at which a valid assignment of $cm$ objects to a hash table of size $m$ exists with high probability. We show that for any $d\ge 3$ hashes and load factor $c<c_d^*$, the expectation of the random walk insertion time is $O(1)$, that is, a constant depending only on $d$ and $c$ but not $m$.

cs.DS↗
arXiv2025

Some Maker-Breaker games on hypergraphs

We consider some biased Maker-Breaker games. Starting with the complete $k$-uniform hypergraph on $n$ vertices, at each turn Maker claims one edge, and then Breaker claims $b$ edges. Maker's goal is to obtain a set of edges having some increasing property. We consider the case where Maker's desired property is minimum degree $m$, or having a full-rank incidence matrix, or having a Berge Hamilton cycle or having a perfect matching or a Hamilton $\ell$-cycle.

math.CO↗
arXiv2025

Random walks on edge colored random graphs

We consider random walks on edge coloured random graphs, where the colour of an edge reflects the cost of using it. In the simplest instance, the edges are coloured red or blue. Blue edges are free to use, whereas red edges incur a unit cost every time they are traversed.

math.CO↗
arXiv2025

Rainbow copies of spanning subgraphs

Let $G_{n,p}^{[κ]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set $[κ]$. We consider the threshold for the existence of rainbow colored copies of a spanning subgraph $H$. We provide lower bounds on $p$ and $κ$ sufficient to prove the existence of such copies w.h.p.

math.CO↗
arXiv2025

Hamilton cycles in random digraphs with minimum degree at least one

We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{(δ\geq1)}$. We prove that if $m=\tfrac n2(\log n+2\log\log n+c_n)$ then \[ \lim_{n\to\infty}\Pr(D_{n,m}^{(δ\geq1)}\text{ is Hamiltonian})=\begin{cases}0&c_n\to-\infty.\\e^{-e^{-c}/4}&c_n\to c.\\1&c_n\to\infty.\end{cases} \]

math.CO↗
arXiv2025

Cover time of random subgraphs of the hypercube

$Q_{n,p}$, the random subgraph of the $n$-vertex hypercube $Q_n$, is obtained by independently retaining each edge of $Q_n$ with probability $p$. We give precise values for the cover time of $Q_{n,p}$ above the connectivity threshold.

math.CO↗
arXiv2025

The threshold for loose Hamilton cycles in random hypergraph

We show that w.h.p.\ the random $r$-uniform hypergraph $H_{n,m}$ contains a loose Hamilton cycle, provided $r\geq 3$ and $m\geq \frac{(1+ε)n\log n}{r}$, where $ε$ is an arbitrary positive constant. This is asymptotically best possible, as if $m\leq \frac{(1-ε)n\log n}{r}$ then w.h.p.\ $H_{n,m}$ contains isolated vertices.

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