arXiv subjects

Amanda Redlich

Publications and source records attributed to Amanda Redlich.

14 recordsLinked to original sources

arXiv2026

Power laws and power-of-two-choices

This paper analyzes a variation on the well-known "power of two choices" allocation algorithms. Classically, the smallest of $d$ randomly-chosen options is selected. We investigate what happens when the largest of $d$ randomly-chosen options is selected. This process generates a power-law-like distribution: the $i^{th}$-smallest value scales with $i^{d-1}$, where $d$ is the number of randomly-chosen options, with high probability. We give a formula for the expectation and show the distribution is concentrated around the expectation

cs.DS↗

arXiv2026

Broadcasting Agents and Adversary: A new variation on Cops and Robbers

We introduce a new game played on graphs, ``Agents and Adversary". This game is reminiscent of ``Cops and Robbers" but has some fundamental differences. We classify infinite families of graphs as Agents-win and Adversary-win. We then define a new type of graph symmetry and use it to define a winning strategy for Adversary. Finally, we give tight upper and lower bounds for Agents' time-to-win on several infinite families of graphs.

math.CO↗

arXiv2025

Hoffman-London graphs: When paths minimize $H$-colorings among trees

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. The number of $H$-colorings of $G$, $\hom(G,H)$, has been studied for many classes of $G$ and $H$. In particular, extremal questions of maximizing and minimizing $\hom(G,H)$ have been considered when $H$ is a clique or $G$ is a tree. In this paper, we develop a new technique using automorphisms of $H$ to show that $\hom(T,H)$ is minimized by paths as $T$ varies over trees on a fixed number of vertices. We introduce the term Hoffman-London to refer to graphs that are minimal in this sense. In particular, we define an automorphic similarity matrix which is used to compute $\hom(T,H)$ and give matrix conditions under which $H$ is Hoffman-London. We then apply this technique to identify several families of graphs that are Hoffman-London, including loop threshold graphs and some with applications in statistical physics (e.g. the Widom-Rowlinson model). By combining our approach with a few other observations, we fully characterize the minimizing trees for all graphs $H$ on three or fewer vertices.

math.CO↗

arXiv2025

Long paths need not minimize $H$-colorings among trees

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. By appropriate choice of $H$, these colorings can express, for instance, the independent sets or proper vertex colorings of $G$. Sidorenko proved that for any $H$, the $n$-vertex star admits at least as many $H$-colorings as any other $n$-vertex tree, but the minimization question remains open in general. For many graphs $H$, path graphs are among the trees with the fewest $H$-colorings, but work of Leontovich and subsequently Csikv\'ari and Lin shows that there is a graph $E_7$ on seven vertices and a target graph $H$ for which there are strictly fewer $H$-colorings of $E_7$ than of the path on seven vertices. We introduce a new strategy for enumerating homomorphisms from path-like trees to highly symmetric target graphs that allows us to make the previous observations completely explicit and extend them to infinitely many $n$ beyond $n=7$. In particular, we exhibit a target graph $H$ with the property that for each sufficiently large $n$, there is a tree $E_n$ on $n$ vertices that admits strictly fewer $H$-colorings than the path on $n$ vertices.

math.CO↗

arXiv2025

Characterizing Graphs as Algebraic Squares

Graphs that are squares under the gluing algebra arise in the study of homomorphism density inequalities such as Sidorenko's conjecture. Recent work has focused on these homomorphism density applications. This paper takes a new perspective and focuses on the graph properties of arbitrary square graphs, not only those relevant to homomorphism conjectures and theorems. We develop a set of necessary and/or sufficient conditions for a graph to be square. We apply these conditions to categorize several classical families of graphs as square or not. In addition, we create infinite families of square graphs by proving that joins and Cartesian, direct, strong, and lexicographic products of square graphs with arbitrary graphs are square.

math.CO↗

arXiv2025

Broadcast via Mobile Agents in a Dynamic Network: Interplay of Graph Properties & Agents

We revisit the problem of \textsc{Broadcast}, introduced by Das, Giachoudis, Luccio, and Markou [OPODIS, 2020], where $k+1$ agents are initially placed on an $n$ node dynamic graph, where $1$ agent has a message that must be broadcast to the remaining $k$ ignorant agents. The original paper studied the relationship between the number of agents needed to solve the problem and the edge density of the graph. The paper presented strong evidence that edge density of a graph, or the number of redundant edges within the graph, may be the correct graph property to accurately differentiate whether $k= o(n)$ ignorant agents (low edge density) or $k = \Omega(n)$ ignorant agents (high edge density) are needed to solve the problem. In this paper, we show that surprisingly, edge density may not in fact be the correct differentiating property. The original paper presents graphs with edge density $1.1\overline{6}$ that require $\Omega(n)$ agents, however, we construct graphs with edge density $> 1.1\overline{6}$ and develop an algorithm to solve the problem on those graphs using only $o(n)$ agents. We subsequently show that the relationship between edge density and number of agents is fairly weak by first constructing graphs with edge density tending to $1$ from above that require $\Omega(n/f(n))$ agents to solve, for any function $f(n) \to \infty$ as $n \to \infty$. We then construct an infinite family of graphs with edge density $< \rho$ requiring at least $k$ ignorant agents to solve \textsc{Broadcast}, for any $k>1$ and $\rho>1$.

cs.DM↗

arXiv2024

Relational Weight Optimization for Enhancing Team Performance in Multi-Agent Multi-Armed Bandits

We introduce an approach to improve team performance in a Multi-Agent Multi-Armed Bandit (MAMAB) framework using Fastest Mixing Markov Chain (FMMC) and Fastest Distributed Linear Averaging (FDLA) optimization algorithms. The multi-agent team is represented using a fixed relational network and simulated using the Coop-UCB2 algorithm. The edge weights of the communication network directly impact the time taken to reach distributed consensus. Our goal is to shrink the timescale on which the convergence of the consensus occurs to achieve optimal team performance and maximize reward. Through our experiments, we show that the convergence to team consensus occurs slightly faster in large constrained networks.

cs.MA↗

arXiv2023

Dispersion, Capacitated Nodes, and the Power of a Trusted Shepherd

In this paper, we look at and expand the problems of dispersion and Byzantine dispersion of mobile robots on a graph, introduced by Augustine and Moses~Jr.~[ICDCN~2018] and by Molla, Mondal, and Moses~Jr.~[ALGOSENSORS~2020], respectively, to graphs where nodes have variable capacities. We use the idea of a single shepherd, a more powerful robot that will never act in a Byzantine manner, to achieve fast Byzantine dispersion, even when other robots may be strong Byzantine in nature. We also show the benefit of a shepherd for dispersion on capacitated graphs when no Byzantine robots are present.

cs.DC↗

arXiv2022

Non-trivial squares and Sidorenko's conjecture

Let $t(H;G)$ be the homomorphism density of a graph $H$ into a graph $G$. Sidorenko's conjecture states that for any bipartite graph $H$, $t(H;G)\geq t(K_2;G)^{|E(H)|}$ for all graphs $G$. It is already known that such inequalities cannot be certified through the sums of squares method when $H$ is a so-called trivial square. In this paper, we investigate recent results about Sidorenko's conjecture and classify those involving trivial versus non-trivial squares. We then present some computational results. In particular, we categorize the bipartite graphs $H$ on at most 7 edges for which $t(H;G)\geq t(K_2;G)^{|E(H)|}$ has a sum of squares certificate. We then discuss other limitations for sums of squares proofs beyond trivial squares.

math.CO↗

arXiv2016

Balanced Allocation: Patience is not a Virtue

Load balancing is a well-studied problem, with balls-in-bins being the primary framework. The greedy algorithm $\mathsf{Greedy}[d]$ of Azar et al. places each ball by probing $d > 1$ random bins and placing the ball in the least loaded of them. With high probability, the maximum load under $\mathsf{Greedy}[d]$ is exponentially lower than the result when balls are placed uniformly randomly. V\"ocking showed that a slightly asymmetric variant, $\mathsf{Left}[d]$, provides a further significant improvement. However, this improvement comes at an additional computational cost of imposing structure on the bins. Here, we present a fully decentralized and easy-to-implement algorithm called $\mathsf{FirstDiff}[d]$ that combines the simplicity of $\mathsf{Greedy}[d]$ and the improved balance of $\mathsf{Left}[d]$. The key idea in $\mathsf{FirstDiff}[d]$ is to probe until a different bin size from the first observation is located, then place the ball. Although the number of probes could be quite large for some of the balls, we show that $\mathsf{FirstDiff}[d]$ requires only at most $d$ probes on average per ball (in both the standard and the heavily-loaded settings). Thus the number of probes is no greater than either that of $\mathsf{Greedy}[d]$ or $\mathsf{Left}[d]$. More importantly, we show that $\mathsf{FirstDiff}[d]$ closely matches the improved maximum load ensured by $\mathsf{Left}[d]$ in both the standard and heavily-loaded settings. We further provide a tight lower bound on the maximum load up to $O(\log \log \log n)$ terms. We additionally give experimental data that $\mathsf{FirstDiff}[d]$ is indeed as good as $\mathsf{Left}[d]$, if not better, in practice.

cs.DS↗

arXiv2014

Modular statistics for subgraph counts in sparse random graphs

Answering a question of Kolaitis and Kopparty, we show that, for given integer $q>1$ and pairwise nonisomorphic connected graphs $G_1...G_k$, if $p=p(n) $ is such that $\Pr(G_{n,p}\supseteq G_i)\to 1$ $\forall i$, then, with $\xi_i$ the number of copies of $G_i$ in $G_{n,p}$, $(\xi_1...\xi_k)$ is asymptotically uniformly distributed on ${\bf Z}_q^k$.

math.CO↗

arXiv2013

Unbalanced Allocations

We consider the unbalanced allocation of $m$ balls into $n$ bins by a randomized algorithm using the "power of two choices". For each ball, we select a set of bins at random, then place the ball in the fullest bin within the set. Applications of this generic algorithm range from cost minimization to condensed matter physics. In this paper, we analyze the distribution of the bin loads produced by this algorithm, considering, for example, largest and smallest loads, loads of subsets of the bins, and the likelihood of bins having equal loads.

cs.DM↗

arXiv2013

Remarks on Combinatorial Aspects of the KP Equation

We survey several results connecting combinatorics and Wronskian solutions of the KP equation, contextualizing the successes of a recent approach introduced by Kodama, et. al. We include the necessary combinatorial and analytical background to present a formula for generalized KP solitons, compute several explicit examples, and indicate how such a perspective could be used to extend previous research relating line-soliton solutions of the KP equation with Grassmannians.

nlin.SI↗

arXiv2012

Graph decomposition and parity

Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph $G(n,p)$. We use an idea of graph decompositions to give a sufficient condition for this distribution to tend to uniform modulo $q$. We determine the asymptotic distribution of all fixed two-component graphs in $G(n,p)$ for all $q$, and we give infinite families of many-component graphs with a uniform asymptotic distribution for all $q$. We also prove a negative result, that no simple proof of uniform asymptotic distribution for arbitrary graphs exists.

math.CO↗