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Pu Gao

Publications and source records attributed to Pu Gao.

At least 37 records · Page 2Linked to original sources
arXiv2022

The satisfiability threshold for random linear equations

Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ be a random vector chosen independently of $A$. We identify the threshold $m/n$ up to which the linear system $A x=y$ has a solution with high probability and analyse the geometry of the set of solutions. In the special case $q=2$, known as the random $k$-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof technique was subsequently extended to the cases $q=3,4$ [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to $q>3$. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.

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arXiv2022

The number of perfect matchings, and the nesting properties, of random regular graphs

We prove that the number of perfect matchings in ${\mathcal G}(n,d)$ is asymptotically normal when $n$ is even, $d\to\infty$ as $n\to\infty$, and $d=O(n^{1/7}/\log^2 n)$. This is the first distributional result of spanning subgraphs of ${\mathcal G}(n,d)$ when $d\to\infty$. Moreover, we prove that ${\mathcal G}(n,d-1)$ and ${\mathcal G}(n,d)$ can be coupled so that ${\mathcal G}(n,d-1)$ is a subgraph of ${\mathcal G}(n,d)$ with high probability when $d\to\infty$ and $d=o(n^{1/3})$. Further, if $d=Ω(\log^7 n)$, $d=O(n^{1/7}/\log^2n)$, and $d\le d'\le n-1$ then ${\mathcal G}(n,d)$ and ${\mathcal G}(n,d')$ can be coupled so that asymptotically almost surely ${\mathcal G}(n,d)$ is a subgraph of ${\mathcal G}(n,d')$.

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arXiv2022

Perfect Matchings in the Semi-random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a perfect matching in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves a perfect matching in $βn$ rounds, where the value of $β< 1.206$ is derived from a solution to some system of differential equations. This improves upon the previously best known upper bound of $(1+2/e+o(1)) \, n < 1.736 \, n$ rounds. We also improve the previously best lower bound of $(\ln 2 + o(1)) \, n > 0.693 \, n$ and show that the player cannot achieve the desired property in less than $αn$ rounds, where the value of $α> 0.932$ is derived from a solution to another system of differential equations. As a result, the gap between the upper and lower bounds is decreased roughly four times.

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arXiv2022

The full rank condition for sparse random matrices

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.

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arXiv2021

Linear-time uniform generation of random sparse contingency tables with specified marginals

We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics. When $Δ^4< M/5$, where $Δ$ is the maximum of the row and column sums and $M$ is the sum of all entries of the matrix, our algorithm runs in time linear in $M$ in expectation. Most previously published algorithms for this problem are approximate samplers based on Markov chain Monte Carlo, whose provable bounds on the mixing time are typically polynomials with rather large degrees.

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arXiv2021

Fast uniform generation of random graphs with given degree sequences

In this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that $Δ^4=O(m)$, where $Δ$ is the maximal degree and $m$ is the number of edges,the algorithm runs in expected time $O(m)$. Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time $O(m^2Δ^2)$ for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for power-law degree sequences in cases where the previous best was $O(n^{4.081})$, and (ii) $O(nd+d^4)$ for $d$-regular graphs when $d=o(\sqrt n)$, where the previous best was $O(nd^3)$.

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arXiv2021

Uniform generation of random regular graphs

We develop a new approach for uniform generation of combinatorial objects, and apply it to derive a uniform sampler REG for d-regular graphs. REG can be implemented such that each graph is generated in expected time O(nd^3), provided that d=o(n^{1/2}). Our result significantly improves the previously best uniform sampler, which works efficiently only when d=O(n^{1/3}), with essentially the same running time for the same d. We also give a linear-time approximate sampler REG*, which generates a random d-regular graph whose distribution differs from the uniform by o(1) in total variation distance, when d=o(n^{1/2}).

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arXiv2020

Mixing time of the switch Markov chain and stable degree sequences

The switch chain is a well-studied Markov chain which can be used to sample approximately uniformly from the set $Ω(\boldsymbol{d})$ of all graphs with a given degree sequence $\boldsymbol{d}$. Polynomial mixing time (rapid mixing) has been established for the switch chain under various conditions on the degree sequences. Amanatidis and Kleer introduced the notion of strongly stable families of degree sequences, and proved that the switch chain is rapidly mixing for any degree sequence from a strongly stable family. Using a different approach, Erdős et al. recently extended this result to the (possibly larger) class of P-stable degree sequences, introduced by Jerrum and Sinclair in 1990. We define a new notion of stability for a given degree sequence, namely $k$-\emph{stability}, and prove that if a degree sequence $\boldsymbol{d}$ is 8-stable then the switch chain on $Ω(\boldsymbol{d})$ is rapidly mixing. We also provide sufficient conditions for P-stability, strong stability and 8-stability. Using these sufficient conditions, we give the first proof of P-stability for various families of heavy-tailed degree sequences, including power-law degree sequences, and show that the switch chain is rapidly mixing for these families. We further extend these notions and results to directed degree sequences.

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arXiv2020

Hamilton Cycles in the Semi-random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a Hamilton cycle in as few rounds as possible. In particular, we present a novel strategy for the player which achieves a Hamiltonian cycle in $(2+4e^{-2}+0.07+o(1)) \, n < 2.61135 \, n$ rounds, assuming that a specific non-convex optimization problem has a negative solution (a premise we numerically support). Assuming that this technical condition holds, this improves upon the previously best known upper bound of $3 \, n$ rounds. We also show that the previously best lower bound of $(\ln 2 + \ln (1+\ln 2) + o(1)) \, n$ is not tight.

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arXiv2020

The threshold of symmetry in random graphs with specified degree sequences

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random graphs. This phase transition coincides with that of the graph connectivity. However, when the maximum degree is a growing function as the number of vertices tends to infinity, our results suggest that these two thresholds do not coincide any more

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arXiv2019

Hamiltonicity of random graphs in the stochastic block model

We study the Hamiltonicity of the following model of a random graph. Suppose that we partition [n] into V_1,V_2,...,V_k and add edge {x,y} to our graph with probability p if there exists i such that x,y\in V_i. Otherwise, we add the edge with probbability q. We denote this model by G(n, p,q) and give tight results for Hamiltonicity, including a critical window analysis, under various conditions.

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arXiv2019

Uniform generation of spanning regular subgraphs of a dense graph

Let $H_n$ be a graph on $n$ vertices and let $\ber{H_n}$ denote the complement of $H_n$. Suppose that $Δ= Δ(n)$ is the maximum degree of $\ber{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of $H_n$. This is equivalent to uniformly sampling $d$-regular graphs which avoid a set $E(\ber{H_n})$ of forbidden edges. Here $d=d(n)$ is a positive integer which may depend on $n$. Two of these algorithms produce a uniformly random $d$-factor of $H_n$ in expected runtime which is linear in $n$ and low-degree polynomial in $d$ and $Δ$. The first algorithm applies when $(d+Δ)dΔ= o(n)$. This improves on an earlier algorithm by the first author, which required constant $d$ and at most a linear number of edges in $\ber{H_n}$. The second algorithm applies when $H_n$ is regular and $d^2+Δ^2 = o(n)$, adapting an approach developed by the first author together with Wormald. The third algorithm is a simplification of the second, and produces an approximately uniform $d$-factor of $H_n$ in time $O(dn)$. Here the output distribution differs from uniform by $o(1)$ in total variation distance, provided that $d^2+Δ^2 = o(n)$.

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arXiv2018

Counting triangles in power-law uniform random graphs

We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $τ\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two random neighbors of a vertex of degree $k$ are connected. We find that the number of triangles, as well as the local clustering coefficient, scale similarly as in the erased configuration model, where all self-loops and multiple edges of the configuration model are removed. Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence. The number of triangles in uniform random graphs is closely related to that in a version of the rank-1 inhomogeneous random graph, where all vertices are equipped with weights, and the edge probabilities are moderated by asymptotically linear functions of the products of these vertex weights.

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arXiv2018

The rank of random matrices over finite fields

We determine the rank of a random matrix A over a finite field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula verifies a conjecture of Lelarge [Proc. IEEE Information Theory Workshop 2013]. The proofs are based on coupling arguments and the interpolation method from mathematical physics.

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arXiv2018

Full rainbow matchings in graphs and hypergraphs

Let $G$ be a simple graph that is properly edge coloured with $m$ colours and let $\M=\{M_1,\ldots, M_m\}$ be the set of $m$ matchings induced by the colours in $G$. Suppose that $m\le n-n^{c}$, where $c>9/10$, and every matching in $\M$ has size $n$. Then $G$ contains a full rainbow matching, i.e.\ a matching that contains exactly one edge from $M_i$ for each $1\le i\le m$. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalisation of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs. Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

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arXiv2017

Uniform generation of random graphs with power-law degree sequences

We give a linear-time algorithm that approximately uniformly generates a random simple graph with a power-law degree sequence whose exponent is at least 2.8811. While sampling graphs with power-law degree sequence of exponent at least 3 is fairly easy, and many samplers work efficiently in this case, the problem becomes dramatically more difficult when the exponent drops below 3; ours is the first provably practicable sampler for this case. We also show that with an appropriate rejection scheme, our algorithm can be tuned into an exact uniform sampler. The running time of the exact sampler is O(n^{2.107}) with high probability, and O(n^{4.081}) in expectation.

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arXiv2017

The stripping process can be slow: part I

Given an integer k, we consider the parallel k-stripping process applied to a hypergraph H: removing all vertices with degree less than k in each iteration until reaching the k-core of H. Take H as H_r(n,m): a random r-uniform hypergraph on n vertices and m hyperedges with the uniform distribution. Fixing k,r\ge 2 with (k,r)\neq (2,2), it has previously been proved that there is a constant c_{r,k} such that for all m=cn with constant c\neq c_{r,k}, with high probability, the parallel k-stripping process takes O(\log n) iterations. In this paper we investigate the critical case when c=c_{r,k}+o(1). We show that the number of iterations that the process takes can go up to some power of n, as long as c approaches c_{r,k} sufficiently fast. A second result we show involves the depth of a non-k-core vertex v: the minimum number of steps required to delete v from H_r(n,m) where in each step one vertex with degree less than k is removed. We will prove lower and upper bounds on the maximum depth over all non-k-core vertices.

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