arXiv subjects
Publications and source records attributed to Alan Frieze.
Let F be a uniformly distributed random k-SAT formula with n variables and m clauses. We prove that the Walksat algorithm from Papadimitriou (FOCS 1991)/Schoning (FOCS 1999) finds a satisfying assignment of F in polynomial time w.h.p. if m/n<ρ2^k/k for a certain constant ρ>0. This is an improvement by a factor of $Θ(k)$ over the best previous analysis of Walksat from Coja-Oghlan, Feige, Frieze, Krivelevich, Vilenchik (SODA 2009).
We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in all. When the minimum degree $δ(G)\geq (\frac12+ε)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$; Hamiltonian for $k$ sufficiently large. When $δ(G) \geq m$, then $G_k$ has a cycle of length $(1-ε)m$ for $k\geq k_ε$. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function $ϕ(n)$ (or $ϕ(m)$) where $\lim_{n\to\infty}ϕ(n)=0$.
We introduce and analyze the Walker-Breaker game, a variant of Maker-Breaker games where Maker is constrained to choose edges of a walk or path in a given graph G, with the goal of visiting as many vertices of the underlying graph as possible.
We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n,\,1<c\leq 4$. We show that for any positive integer $\ell$, there exists $ε=ε(\ell)$ such that if $c=1+ε$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $χ(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$.
For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on t vertices and ε> 0, the random graph G_{n,p}, with n a multiple of t, with high probability contains a family of edge-disjoint T-factors covering all but an ε-fraction of its edges, as long as ε^4 n p >> (log n)^2. Assuming stronger divisibility conditions, the edge probability can be taken down to p > (C log n)/n. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.
We consider the length $L(n)$ of the longest path in a randomly generated Apollonian Network (ApN) ${\cal A}_n$. We show that w.h.p. $L(n)\leq ne^{-\log^cn}$ for any constant $c<2/3$.
Let $HP_{n,m,k}$ be drawn uniformly from all $k$-uniform, $k$-partite hypergraphs where each part of the partition is a disjoint copy of $[n]$. We let $HP^{(\k)}_{n,m,k}$ be an edge colored version, where we color each edge randomly from one of $\k$ colors. We show that if $\k=n$ and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if $n$ is even and $m=Kn\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in $G^{(n)}_{n,m}$. Here $G^{(n)}_{n,m}$ denotes a random edge coloring of $G_{n,m}$ with $n$ colors. When $n$ is odd, our proof requires $m=\om(n\log n)$ for there to be a rainbow Hamilton cycle.
We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors.
Beautiful formulas are known for the expected cost of random two-dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural "Axial" and "Planar" versions, both of which are NP-hard. For 3-dimensional Axial random assignment instances of size $n$, the cost scales as $Ω(1/n)$, and a main result of the present paper is a linear-time algorithm that, with high probability, finds a solution of cost $O(n^{-1+o(1)})$. For 3-dimensional Planar assignment, the lower bound is $Ω(n)$, and we give a new efficient matching-based algorithm that with high probability returns a solution with cost $O(n \log n)$.
In a Maker-Breaker game on a graph $G$, Breaker and Maker alternately claim edges of $G$. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker-Breaker games played on random geometric graphs. For each of our four games we show that if we add edges between $n$ points chosen uniformly at random in the unit square by order of increasing edge-length then, with probability tending to one as $n\to\infty$, the graph becomes Maker-win the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the $H$-game as soon as there is a subgraph from a finite list of "minimal graphs". These results also allow us to give precise expressions for the limiting probability that $G(n,r)$ is Maker-win in each case, where $G(n,r)$ is the graph on $n$ points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most $r$.
In this paper we asymptotically count $d$-regular $k$-uniform hypergraphs on $n$ vertices, provided $k$ is fixed and $d=d(n)=o(n^{1/2})$. In doing so, we extend to hypergraphs a switching technique of McKay and Wormald.
Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $ε>0$, if $p=\frac{1+ε}{r}$, then $G_p$ is non-planar with probability approaching 1 as $r$ grows. This generalizes classical results on planarity of binomial random graphs.
We establish a relation between two uniform models of random $k$-graphs (for constant $k \ge 3$) on $n$ labeled vertices: $H(n,m)$, the random $k$-graph with exactly $m$ edges, and $H(n,d)$, the random $d$-regular $k$-graph. By extending to $k$-graphs the switching technique of McKay and Wormald, we show that, for some range of $d = d(n)$ and a constant $c > 0$, if $m \sim cnd$, then one can couple $H(n,m)$ and $H(n,d)$ so that the latter contains the former with probability tending to one as $n \to \infty$. In view of known results on the existence of a loose Hamilton cycle in $H(n,m)$, we conclude that $H(n,d)$ contains a loose Hamilton cycle when $\log n = o(d)$ (or just $d \ge C log n$, if $k = 3$) and $d = o(n^{1/2})$.
We study the expected value of the length $L_n$ of the minimum spanning tree of the complete graph $K_n$ when each edge $e$ is given an independent uniform $[0,1]$ edge weight. We sharpen the result of Frieze \cite{F1} that $\lim_{n\to\infty}\E(L_n)=\z(3)$ and show that $\E(L_n)=\z(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}$ where $c_1,c_2$ are explicitly defined constants.
The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. Begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is chosen uniformly at random out of all remaining triangles. This stochastic process terminates once it arrives at a triangle-free graph, and a longstanding open problem is to estimate the final number of edges, or equivalently the time it takes the process to conclude. The intuition that the edge distribution is roughly uniform at all times led to a folklore conjecture that the final number of edges is $n^{3/2+o(1)}$ with high probability, whereas the best known upper bound is $n^{7/4+o(1)}$. It is no coincidence that various methods break precisely at the exponent 7/4 as it corresponds to the inherent barrier where co-degrees become comparable to the variations in their values that arose earlier in the process. In this work we significantly improve upon the previous bounds by establishing that w.h.p. the number of edges in the final graph is at most $ n^{5/3+o(1)} $. Our approach relies on a system of martingales used to control key graph parameters, where the crucial new idea is to harness the self-correcting nature of the process in order to control these parameters well beyond the point where their early variation matches the order of their expectation.
We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph $G=\gc$. In this model $G$ is drawn uniformly from graphs with vertex set $[n]$, $m$ edges and minimum degree at least three. We focus on the case where $m=cn$ for constant $c$. If $c$ is sufficiently large then our algorithm runs in $O(n^{1+o(1)})$ time and succeeds w.h.p.
An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold $p=\frac{\log n+\om}{n}$ where $\om=\om(n)\to\infty$ and ${\om}=o(\log{n})$ and of random $r$-regular graphs where $r \geq 3$ is a fixed integer. Specifically, we prove that the rainbow connectivity $rc(G)$ of $G=G(n,p)$ satisfies $rc(G) \sim \max\set{Z_1,diameter(G)}$ with high probability (\whp). Here $Z_1$ is the number of vertices in $G$ whose degree equals 1 and the diameter of $G$ is asymptotically equal to $\diam$ \whp. Finally, we prove that the rainbow connectivity $rc(G)$ of the random $r$-regular graph $G=G(n,r)$ satisfies $rc(G) =O(\log^2{n})$ \whp.
A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\binom{[k]}{2}$ such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph $G$, denoted $τ_2(G)$ is the smallest $k$ such that $G$ admits a proper 2-tone $k$ coloring. In this paper, we prove that w.h.p. for $p\ge Cn^{-1/4}\ln^{9/4}n$, $τ_2(G_{n,p})=(2+o(1))χ(G_{n,p})$ where $χ$ represents the ordinary chromatic number. For sparse random graphs with $p=c/n$, $c$ constant, we prove that $τ_2(G_{n,p}) = \lceil{{\sqrt{8Δ+1} +5}/{2}}\rceil$ where $Δ$ represents the maximum degree. For the more general concept of $t$-tone coloring, we achieve similar results.