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Wesley Pegden

Publications and source records attributed to Wesley Pegden.

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

(Auto)formalization is supposed to be easy: Trellis process semantics for spelling out rigorous proofs

We present Trellis: an autoformalization system that leverages LLM agents in a deterministically constrained workflow to enforce incremental progress in Lean autoformalization tasks through iterative refinement of natural language proofs. Our approach is motivated by the common mathematician's notion of what it means to have a rigorous proof in the first place: namely, that it would be routine to elaborate any part of the proof in further detail. The result is a system which aims to achieve reliable autoformalization on a modest budget and with generalist agents, with specialization to autoformalization coming not from any task-specific agent training but instead from a meaning-of-rigor inspired workflow enforced by process semantics. We link to an end-to-end Lean formalization of a recent Ramsey theory breakthrough produced by the process.

cs.AI↗
arXiv2025

Competitively Constructed Planar Graphs

We introduce and study two Maker-Breaker-like games for constructing planar graphs: the edge drawing game, where two players take turns drawing non-intersecting edges between points in the plane, and the circle packing game, where the players take turns placing disjoint circles in the plane. Both games produce planar graphs: the edge drawing game results in a plane graph drawing, and the circle packing game yields a planar graph via the contact graph of the packing. For both games, we give necessary conditions under which a given planar graph can be constructed. We also show that the two games are indeed different by giving a class of graphs which can be constructed in one but not the other.

math.CO↗
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

Sampling Tree-Weighted Partitions Without Sampling Trees

This paper gives a new algorithm for sampling tree-weighted partitions of a large class of planar graphs. Formally, the tree-weighted distribution on $k$-partitions of a graph weights $k$-partitions proportional to the product of the number of spanning trees of each partition class. Recent work on computational redistricting analysis has driven special interest in the conditional distribution where all partition classes have the same size (balanced partitions). One class of Markov chains in wide use aims to sample from balanced tree-weighted $k$-partitions using a sampler for balanced tree-weighted 2-partitions. Previous implementations of this 2-partition sampler would draw a random spanning tree and check whether it contains an edge whose removal produces a balanced 2-component forest, rejecting if not. In practice, this is a significant computational bottleneck. We show that in fact it is possible to sample from the balanced tree-weighted 2-partition distribution directly, without first sampling a spanning tree; the acceptance and rejection rates are the same as in previous samplers. We prove that on a wide class of planar graphs encompassing network structures typically arising from the geographic data used in computational redistricting, our algorithm takes expected linear time $O(n)$. Notably, this is asymptotically faster than the best known method to generate random trees, which is $O(n \log^2 n)$ for approximate sampling and $O(n^{1 + \log \log \log n / \log \log n})$ for exact sampling. Additionally, we show that a variant of our algorithm also gives a speedup to $O(n \log n)$ for exact sampling of uniformly random trees on these families of graphs, improving the bounds for both exact and approximate sampling. We implement our algorithm and benchmark it on grid graphs, finding that it outperforms the standard bipartitioning method in the widely-used GerryChain library.

cs.DS↗
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

Structure-biased Maker-Breaker Games

In classical Maker-Breaker games on graphs, Maker and Breaker take turns claiming edges; Maker's goal is to claim all of some structure (e.g., a spanning tree, Hamilton cycle, etc.), while Breaker aims to stop her. The standard question considered is how powerful a Breaker Maker can defeat; i.e., for the $(1:b)$-biased game where Breaker takes $b$ edges per turn, how large can $b$ be for Maker to still have a winning strategy, for various possible goal sets? We introduce a variant of this question in which Breaker is required to choose their multiple edges as the edges of (a subgraph of) a given structure (e.g., a matching, clique, etc.) on each turn. We establish the order of magnitude of the threshold biases for triangle games, connectivity games, and Hamiltonicity games under clique, matching, and star biases respectively. We conclude that in many cases structure imposes major obstruction to Breaker, opening up a set of games whose strategies deviate from the classical biased Maker-Breaker game strategies, and shedding light on the types of Breaker strategies that may or may not work to prove tighter bounds in the classical setting.

math.CO↗
arXiv2025

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

Aspects of a randomly growing cluster in $\reals^d,d\geq 2

We consider a simple model of a growing cluster of points in $\Re^d,d\geq 2$. Beginning with a point $X_1$ located at the origin, we generate a random sequence of points $X_1,X_2,\ldots,X_i,\ldots,$. To generate $X_{i},i\geq 2$ we choose a uniform integer $j$ in $[i-1]=\{1,2,\ldots,i-1\}$ and then let $X_{i}=X_j+D_i$ where $D_i=(\delta_1,\ldots,\delta_d)$. Here the $\delta_j$ are independent copies of the Normal distribution $N(0,\sigma_i)$, where $\sigma_i=i^{-\alpha}$ for some $\alpha>0$. We prove that for any $\alpha>0$ the resulting point set is bounded a.s., and moreover, that the points generated look like samples from a $\beta$-dimensional subset of $\Re^d$ from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where $\beta=\min(d,1/\alpha)$.

math.PR↗
arXiv2024

The intersection of a random geometric graph with an Erd\H{o}s-R\'enyi graph

We study the intersection of a random geometric graph with an Erd\H{o}s-R\'enyi graph. Specifically, we generate the random geometric graph $G(n, r)$ by choosing $n$ points uniformly at random from $D=[0, 1]^2$ and joining any two points whose Euclidean distance is at most $r$. We let $G(n, p)$ be the classical Erd\H{o}s-R\'enyi graph, i.e. it has $n$ vertices and every pair of vertices is adjacent with probability $p$ independently. In this note we study $G(n, r, p):=G(n, r) \cap G(n, p)$. One way to think of this graph is that we take $G(n, r)$ and then randomly delete edges with probability $1-p$ independently. We consider the clique number, independence number, connectivity, Hamiltonicity, chromatic number, and diameter of this graph where both $p(n)\to 0$ and $r(n)\to 0$; the same model was studied by Kahle, Tian and Wang (2023) for $r(n)\to 0$ but $p$ fixed.

math.CO↗
arXiv2024

The Moran process on a random graph

We study the fixation probability for two versions of the Moran process on the random graph $G_{n,p}$ at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughtout the process there are vertices of two types, mutants and non-mutants. Mutants have fitness $s$ and non-mutants have fitness 1. The process starts with a unique individual mutant located at the vertex $v_0$. In the Birth-Death version of the process a random vertex is chosen proportional to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants $X$ is empty or $[n]$. In the Death-Birth version a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants $X$ is empty or $[n]$. The {\em fixation probability} is the probability that the process ends with $X=\emptyset$. We give asymptotically correct estimates of the fixation probability that depend on degree of $v_0$ and its neighbors.,

math.PR↗
arXiv2024

Youden's Demon is Sylvester's Problem

If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $\frac 6 \pi \arcsin\left(\frac{1}{3}\right)\approx .649$. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$. Our main tool is the observation that the Gale dual of independent samples in $\mathbb{R}^d$ itself can be taken to be a set of independent points (conditioned on barycenter at the origin) when the distribution of the points is Gaussian.

math.PR↗
arXiv2023

Sampling Balanced Forests of Grids in Polynomial Time

We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each $k$-partition according to the product, across its $k$ pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces. Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative $(1 \pm \varepsilon)$ constant with constant probability, and up to an additive constant with polynomial probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. These results imply polynomial time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs. Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into $k$ population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.

cs.DM↗
arXiv2023

The bright side of simple heuristics for the TSP

The greedy and nearest-neighbor TSP heuristics can both have $\log n$ approximation factors from optimal in worst case, even just for $n$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $d$-Ahlfor's regular metric space (which includes any $d$-manifold like the $d$-cube $[0,1]^d$ in the case $d$ is an integer but also fractals of dimension $d$ when $d$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have \emph{additive} errors from optimal on the order of the \emph{optimal} tour length through \emph{random} points in the same space, for $d>1$.

cs.DM↗
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 $n+O(n^{1-\delta})$ 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

Sequentially constrained Hamilton cycles in random graphs

We discuss the existence of Hamilton cycles in the random graph $G_{n,p}$ where there are restrictions caused by (i) coloring sequences, (ii) a subset of vertices must occur in a specific order and (iii) there is a bound on the number of inversions in the associated permutation.

math.CO↗
arXiv2023

On the intersecting family process

We study the intersecting family process initially studied in \cite{BCFMR}. Here $k=k(n)$ and $E_1,E_2,\ldots,E_m$ is a random sequence of $k$-sets from $\binom{[n]}{k}$ where $E_{r+1}$ is uniformly chosen from those $k$-sets that are not already chosen and that meet $E_i,i=1,2,\ldots,r$. We prove some new results for the case where $k=cn^{1/3}$ and for the case where $k\gg n^{1/2}$.

math.CO↗