arXiv subjects

Wesley Pegden

Publications and source records attributed to Wesley Pegden.

At least 37 records · Page 2Linked to original sources

arXiv2023

Direct sampling of short paths for contiguous partitioning

In this paper, we provide a family of dynamic programming based algorithms to sample nearly-shortest self avoiding walks between two points of the integer lattice $\mathbb{Z}^2$. We show that if the shortest path of between two points has length $n$, then we can sample paths (self-avoiding-walks) of length $n+O(n^{1-δ})$ in polynomial time. As an example of an application, we will show that the Glauber dynamics Markov chain for partitions of the Aztec Diamonds in $\mathbb{Z}^2$ into two contiguous regions with nearly tight perimeter constraints has exponential mixing time, while the algorithm provided in this paper can be used be used to uniformly (and exactly) sample such partitions efficiently.

math.PR↗

arXiv2023

Comb inequalities for typical Euclidean TSP instances

We prove that even in average case, the Euclidean Traveling Salesman Problem exhibits an integrality gap of $(1+ε)$ for $ε>0$ when the Held-Karp Linear Programming relaxation is augmented by all comb inequalities of bounded size. This implies that large classes of branch-and-cut algorithms take exponential time for the Euclidean TSP, even on random inputs.

cs.DS↗

arXiv2022

Unit sphere fibrations in Euclidean space

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit $n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of $n$, we construct unit $n$-sphere fibrations in $\mathbb{R}^{2n+1}$.

math.GT↗

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

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

The paradoxical nature of easily improvable evidence

Established frameworks to understand problems with reproducibility in science begin with the relationship between our understanding of the prior probability of a claim and the statistical certainty that should be demanded of it, and explore the ways in which independent investigations, biases in study design and publication bias interact with these considerations. We propose a complementary perspective; namely, that to improve reproducibility in science, our interpretation of the persuasiveness of evidence (e.g., statistical significance thresholds) should be responsive to our understanding of the effort that would be required to improve that evidence. We will quantify this notion in some formal settings. Indeed, we will demonstrate that even simplistic models of evidence publication can exhibit an improvable evidence paradox, where the publication of easily improvable evidence in favor of a claim can best seen as evidence the claim is false.

stat.OT↗

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

A note on sampling biases in the Bangladesh mask trial

A recent cluster trial in Bangladesh randomized 600 villages into 300 treatment/control pairs, to evaluate the impact of an intervention to increase mask-wearing. Data was analyzed in a generalized linear model and significance asserted with parametric tests for the rate of the primary outcome (symptomatic and seropositive for COVID-19) between treatment and control villages. In this short note we re-analyze the data from this trial using standard non-parametric paired statistics tests on treatment/village pairs. With this approach, we find that behavioral outcomes like physical distancing are highly significant, while the primary outcome of the study is not. Importantly, we find that the behavior of unblinded staff when enrolling study participants is one of the most highly significant differences between treatment and control groups, contributing to a significant imbalance in denominators between treatment and control groups. The potential bias leading to this imbalance suggests caution is warranted when evaluating rates rather than counts. More broadly, the significant impacts on staff and participant behavior urge caution in interpreting small differences in the study outcomes that depended on survey response.

stat.AP↗

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↗

arXiv2020

Semi-bandit Optimization in the Dispersed Setting

The goal of data-driven algorithm design is to obtain high-performing algorithms for specific application domains using machine learning and data. Across many fields in AI, science, and engineering, practitioners will often fix a family of parameterized algorithms and then optimize those parameters to obtain good performance on example instances from the application domain. In the online setting, we must choose algorithm parameters for each instance as they arrive, and our goal is to be competitive with the best fixed algorithm in hindsight. There are two major challenges in online data-driven algorithm design. First, it can be computationally expensive to evaluate the loss functions that map algorithm parameters to performance, which often require the learner to run a combinatorial algorithm to measure its performance. Second, the losses can be extremely volatile and have sharp discontinuities. However, we show that in many applications, evaluating the loss function for one algorithm choice can sometimes reveal the loss for a range of similar algorithms, essentially for free. We develop online optimization algorithms capable of using this kind of extra information by working in the semi-bandit feedback setting. Our algorithms achieve regret bounds that are essentially as good as algorithms under full-information feedback and are significantly more computationally efficient. We apply our semi-bandit results to obtain the first provable guarantees for data-driven algorithm design for linkage-based clustering and we improve the best regret bounds for designing greedy knapsack algorithms.

cs.LG↗

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↗

arXiv2020

Failure of monotonicity in epidemic models

We discuss the failure of monotonicity properties for even simple compartmental epidemic models, for the case where transmission rates are non-constant. We also identify a special case in which monotonicity holds.

q-bio.PE↗

arXiv2020

Modeling strict age-targeted mitigation strategies for COVID-19

We use a simple SIR-like epidemic model which integrates known age-contact patterns for the United States to model the effect of age-targeted mitigation strategies for a COVID-19-like epidemic. We find that, among strategies which end with population immunity, strict age-targeted mitigation strategies have the potential to greatly reduce mortalities and ICU utilization for natural parameter choices.

q-bio.PE↗

arXiv2020

Minimum-weight combinatorial structures under random cost-constraints

Recall that Janson showed that if the edges of the complete graph $K_n$ are assigned exponentially distributed independent random weights, then the expected length of a shortest path between a fixed pair of vertices is asymptotically equal to $(\log n)/n$. We consider analogous problems where edges have not only a random length but also a random cost, and we are interested in the length of the minimum-length structure whose total cost is less than some cost budget. For several classes of structures, we determine the correct minimum length structure as a function of the cost-budget, up to constant factors. Moreover, we achieve this even in the more general setting where the distribution of weights and costs are arbitrary, so long as the density $f(x)$ as $x\to 0$ behaves like $cx^γ$ for some $γ\geq 0$; previously, this case was not understood even in the absence of cost constraints. We also handle the case where each edge has several independent costs associated to it, and we must simultaneously satisfy budgets on each cost. In this case, we show that the minimum-length structure obtainable is essentially controlled by the product of the cost thresholds.

math.CO↗

arXiv2020

A note on the rank of a sparse random matrix

Let $\mathbf{A}_{n,m;k}$ be a random $n \times m$ matrix with entries from some field $\mathbb{F}$ where there are exactly $k$ non-zero entries in each column, whose locations are chosen independently and uniformly at random from the set of all ${n \choose k}$ possibilities. In a previous paper (arXiv:1806.04988), we considered the rank of a random matrix in this model when the field is $\mathbb{F}=GF(2)$. In this note, we point out that with minimal modifications, the arguments from that paper actually allow analogous results when the field $\mathbb{F}$ is arbitrary. In particular, for any field $\mathbb{F}$ and any fixed $k\geq 3$, we determine an asymptotically correct estimate for the rank of $\mathbf{A}_{n,m;k}$ in terms of $c,n,k$ where $m=cn/k$, and $c$ is a constant. This formula works even when the values of the nonzero elements are adversarially chosen. When $\mathbb{F}$ is a finite field, we also determine the threshold for having full row rank, when the values of the nonzero elements are randomly chosen.

math.CO↗

arXiv2020

Stability of patterns in the Abelian sandpile

We show that the patterns in the Abelian sandpile are stable. The proof combines the structure theory for the patterns with the regularity machinery for non-divergence form elliptic equations. The stability results allows one to improve weak-* convergence of the Abelian sandpile to pattern convergence for certain classes of solutions.

math.AP↗