arXiv subjects

Wesley Pegden

Publications and source records attributed to Wesley Pegden.

At least 55 records · Page 3Linked to original sources

arXiv2019

Separating effect from significance in Markov chain tests

We give qualitative and quantitative improvements to theorems which enable significance testing in Markov Chains, with a particular eye toward the goal of enabling strong, interpretable, and statistically rigorous claims of political gerrymandering. Our results can be used to demonstrate at a desired significance level that a given Markov Chain state (e.g., a districting) is extremely unusual (rather than just atypical) with respect to the fragility of its characteristics in the chain. We also provide theorems specialized to leverage quantitative improvements when there is a product structure in the underlying probability space, as can occur due to geographical constraints on districtings.

math.PR↗

arXiv2019

Scalefree hardness of average-case Euclidean TSP approximation

We show that if P$\neq$NP, then a wide class of TSP heuristics fail to approximate the length of the TSP to asymptotic optimality, even for random Euclidean instances. Previously, this result was not even known for any heuristics (greedy, etc) used in practice. As an application, we show that when using a heuristic from this class, a natural class of branch-and-bound algorithms takes exponential time to find an optimal tour (again, even on a random point-set), regardless of the particular branching strategy or lower-bound algorithm used.

cs.CC↗

arXiv2019

On the cover time of dense graphs

We consider arbitrary graphs $G$ with $n$ vertices and minimum degree at least $δn$ where $δ>0$ is constant. If the conductance of $G$ is sufficiently large then we obtain an asymptotic expression for the cover time $C_G$ of $G$ as the solution to an explicit transcendental equation. Failing this, if the mixing time of a random walk on $G$ is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic estimate via a decomposition into a bounded number of dense sub-graphs with high conductance. Failing this we give a deterministic asymptotic (2+o(1))-approximation of $C_G$.

math.CO↗

arXiv2019

A note on the localization number of random graphs: diameter two case

We study the localization game on dense random graphs. In this game, a {\em cop} $x$ tries to locate a {\em robber} $y$ by asking for the graph distance of $y$ from every vertex in a sequence of sets $W_1,W_2,\ldots,W_\ell$. We prove high probability upper and lower bounds for the minimum size of each $W_i$ that will guarantee that $x$ will be able to locate $y$.

math.CO↗

arXiv2019

Extremal Collections of $k$-Uniform Vectors

We show any matrix of rank $r$ over $\mathbb{F}_q$ can have $\leq \binom{r}{k}(q-1)^k$ distinct columns of weight $k$ if $ k \leq O_q(\sqrt{\log r})$ (up to divisibility issues), and $\leq \binom{r}{k}(q-1)^{r-k}$ distinct columns of co-weight $k$ if $k \leq O_q(r^{2/3})$. This shows the natural examples consisting of only $r$ rows are optimal for both, and the proofs will recover some form of uniqueness of these examples in all cases.

math.CO↗

arXiv2019

Minors of a random binary matroid

Let ${\bf A}={\bf A}_{n,m,k}$ be a random $n\times m$ matrix over $\mathbf{GF}_2$ wher each column consists of $k$ randomly chosen ones. Let $M$ be an arbirary fixed binary matroid. We show that if $m/n$ and $k$ are sufficiently large then as $n\to\infty$ the binary matroid induced by {\bf A} contains $M$ as a minor.

math.CO↗

arXiv2019

On random multi-dimensional assignment problems

We study random multidimensional assignment problems where the costs decompose into the sum of independent random variables. In particular, in three dimensions, we assume that the costs $W_{i,j,k}$ satisfy $W_{i,j,k}=a_{i,j}+b_{i,k}+c_{j,k}$ where the $a_{i,j},b_{i,k},c_{j,k}$ are independent exponential rate 1 random variables. Our objective is to minimize the total cost and we show that w.h.p. a simple greedy algorithm is a $(3+o(1))$-approximation. This is in contrast to the case where the $W_{i,j,k}$ are independent exponential rate 1 random variables. Here all that is known is an $n^{o(1)}$-approximation, due to Frieze and Sorkin.

math.CO↗

arXiv2018

Constraining the clustering transition for colorings of sparse random graphs

Let $Ω_q$ denote the set of proper $q$-colorings of the random graph $G_{n,m}, m=dn/2$ and let $H_q$ be the graph with vertex set $Ω_q$ and an edge $\{σ,τ\}$ where $σ,τ$ are mappings $[n]\to[q]$ iff $h(σ,τ)=1$. Here $h(σ,τ)$ is the Hamming distance $|\{v\in [n]:σ(v)\neqτ(v)\}|$. We show that w.h.p. $H_q$ contains a single giant component containing almost all colorings in $Ω_q$ if $d$ is sufficiently large and $q\geq \frac{cd}{\log d}$ for a constant $c>3/2$.

math.CO↗

arXiv2017

Diffusion limited aggregation in the Boolean lattice

In the Diffusion Limited Aggregation (DLA) process on on $\mathbb{Z}^2$, or more generally $\mathbb{Z}^d$, particles aggregate to an initially occupied origin by arrivals on a random walk. The scaling limit of the result, empirically, is a fractal with dimension strictly less than $d$. Very little has been shown rigorously about the process, however. We study an analogous process on the Boolean lattice $\{0,1\}^n$, in which particles take random decreasing walks from $(1,\dots,1)$, and stick at the last vertex before they encounter an occupied site for the first time; the vertex $(0,\dots,0)$ is initially occupied. In this model, we can rigorously prove that lower levels of the lattice become full, and that the process ends by producing an isolated path of unbounded length reaching $(1,\dots,1)$.

math.PR↗

arXiv2017

A partisan districting protocol with provably nonpartisan outcomes

We design and analyze a protocol for dividing a state into districts, where parties take turns proposing a division, and freezing a district from the other party's proposed division. We show that our protocol has predictable and provable guarantees for both the number of districts in which each party has a majority of supporters, and the extent to which either party has the power to pack a specific population into a single district.

cs.GT↗

arXiv2017

An analysis of the Act 43 Wisconsin Assembly district map using the $\sqrt{\varepsilon}$ test

In previous work [arXiv:1608.02014], we developed a rigorous statistical test for outlier status in a reversible Markov Chain, and demonstrated its utilization with an application to detecting gerrymandering in Pennsylvania's Congressional districting. In this note, we apply our test to the current (Act 43) assembly districting of the state of Wisconsin, and find that the districting is indeed an outlier among the the landscape of valid districtings of Wisconsin. Outlier status is significant at betwee p=.0002 and p=.0008, depending on assumptions.

stat.AP↗

arXiv2017

The Apollonian structure of integer superharmonic matrices

We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely characterizes the PDE which determines the continuum scaling limit of the Abelian sandpile on the lattice $\mathbb{Z}^2$.

math.AP↗

arXiv2017

The distribution of minimum-weight cliques and other subgraphs in graphs with random edge weights

We determine, asymptotically in $n$, the distribution and mean of the weight of a minimum-weight $k$-clique (or any strictly balanced graph $H$) in a complete graph $K_n$ whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions. For the clique, we also provide explicit (non-asymptotic) bounds on the distribution's CDF in a form obtained directly from the Stein-Chen method, and in a looser but simpler form. The direct form extends to other subgraphs and other edge-weight distributions. We illustrate the clique results for various values of $k$ and $n$. The results may be applied to evaluate whether an observed minimum-weight copy of a graph $H$ in a network provides statistical evidence that the network's edge weights are not independently distributed but have some structure.

math.PR↗

arXiv2017

Assessing significance in a Markov chain without mixing

We present a new statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to demonstrate rigorously that the presented state is an outlier with respect to the values, by establishing a $p$-value for observations we make about the state under the null hypothesis that it was chosen uniformly at random. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain, and compare these to the rank of the presented state; if the presented state is a $0.1\%$-outlier compared to the sampled ranks (i.e., its rank is in the bottom $0.1\%$ of sampled ranks) then this should correspond to a $p$-value of $0.001$. This test is not rigorous, however, without good bounds on the mixing time of the Markov chain, as one must argue that the observed states on the trajectory approximate the stationary distribution. Our test is the following: given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an $\varepsilon$-outlier on the walk is significant at $p=\sqrt {2\varepsilon}$, under the null hypothesis that the state was chosen from a stationary distribution. Our result assumes nothing about the structure of the Markov chain beyond reversibility, and we construct examples to show that significance at $p\approx\sqrt \varepsilon$ is essentially best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districtings.

math.PR↗

arXiv2017

Online purchasing under uncertainty

Suppose there is a collection $x_1,x_2,\dots,x_N$ of independent uniform $[0,1]$ random variables, and a hypergraph $\cF$ of \emph{target structures} on the vertex set $\{1,\dots,N\}$. We would like to buy a target structure at small cost, but we do not know all the costs $x_i$ ahead of time. Instead, we inspect the random variables $x_i$ one at a time, and after each inspection, choose to either keep the vertex $i$ at cost $x_i$, or reject vertex $i$ forever. In the present paper, we consider the case where $\{1,\dots,N\}$ is the edge-set of some graph, and the target structures are the spanning trees of a graph, spanning arborescences of a digraph, the paths between a fixed pair of vertices, perfect matchings, Hamilton cycles or the cliques of some fixed size.

cs.DS↗

arXiv2016

Looking for vertex number one

Given an instance of the preferential attachment graph $G_n=([n],E_n)$, we would like to find vertex 1, using only 'local' information about the graph; that is, by exploring the neighborhoods of small sets of vertices. Borgs et. al gave an an algorithm which runs in time $O(\log^4 n)$, which is local in the sense that at each step, it needs only to search the neighborhood of a set of vertices of size $O(\log^4 n)$. We give an algorithm to find vertex 1, which w.h.p. runs in time $O(ω\log n)$ and which is local in the strongest sense of operating only on neighborhoods of single vertices. Here $ω=ω(n)$ is any function that goes to infinity with $n$.

math.CO↗

arXiv2015

Separating subadditive Euclidean functionals

If we are given $n$ random points in the hypercube $[0,1]^d$, then the minimum length of a Traveling Salesperson Tour through the points, the minimum length of a spanning tree, and the minimum length of a matching, etc., are known to be asymptotically $βn^{\frac{d-1}{d}}$ a.s., where $β$ is an absolute constant in each case. We prove separation results for these constants. In particular, concerning the constants $β_{\mathrm{TSP}}^d$, $β_{\mathrm{MST}}^d$, $β_{\mathrm{MM}}^d$, and $β_{\mathrm{TF}}^d$ from the asymptotic formulas for the minimum length TSP, spanning tree, matching, and 2-factor, respectively, we prove that $β_{\mathrm{MST}}^d<β_{\mathrm{TSP}}^d$, $2β_{\mathrm{MM}}^d<β_{\mathrm{TSP}}^d$, and $β_{\mathrm{TF}}^d<β_{\mathrm{TSP}}^d$ for all $d\geq 2$. We also asymptotically separate the TSP from its linear programming relaxation in this setting. Our results have some computational relevance, showing that a certain natural class of simple algorithms cannot solve the random Euclidean TSP efficiently.

math.PR↗

arXiv2014

Traveling in randomly embedded random graphs

We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the Beardwood-Halton-Hammersley theorem to this setting.

math.PR↗