arXiv subjects

Sara Krehbiel

Publications and source records attributed to Sara Krehbiel.

8 recordsLinked to original sources

arXiv2026

Coloring graphs as complete graph invariants

We investigate the extent to which the $k$-coloring graph $\mathcal{C}_{k}(G)$ uniquely determines the base graph $G$ and the number of colors $k$. The vertices of $\mathcal{C}_{k}(G)$ are the proper $k$-colorings of $G$, and edges connect colorings that differ on exactly one vertex. There are nonisomorphic graphs $G_1$ and $G_2$ with isomorphic coloring graphs, so $\mathcal{C}_{k}(G)$ is not a complete invariant in general. However, for color palettes with surplus colors (when the number of colors $k$ is greater than the chromatic number), we prove that the coloring graph is a complete invariant. Specifically, provided that $k_1 > χ(G_1)$, we show that $\mathcal{C}_{k_1}(G_1)\cong \mathcal{C}_{k_2}(G_2)$ implies $G_1\cong G_2$ and $k_1=k_2$. Thus, there is a natural bijection between pairs $(G, k)$ with $k > χ(G)$ and their coloring graphs $\mathcal{C}_k(G)$. Furthermore, no coloring graph of the form $\mathcal{C}_{χ(G)}(G)$ is isomorphic to a coloring graph with surplus colors. Our constructive proof provides a method to decide whether a coloring graph is generated with surplus colors, although the resulting algorithms are inefficient.

math.CO↗

arXiv2026

Extremal diameters of 3-coloring graphs of trees

Given a tree $T$, its 3-coloring graph $\mathcal{C}_3(T)$ has as vertices the proper 3-colorings of $T$, with edges joining colorings that differ at exactly one vertex. We call the diameter of $\mathcal{C}_3(T)$ the 3-coloring diameter of $T$. We introduce the notion of balanced labelings of $T$ and show that the 3-coloring diameter equals the maximum $L_1$-norm of a balanced labeling. Using this equivalence, we determine the maximum and minimum values of the 3-coloring diameter over all trees on $n$ vertices and characterize the extremal trees.

math.CO↗

arXiv2025

Bell coloring graphs: realizability and reconstruction

Given a graph $G$, the Bell $k$-coloring graph $\mathcal{B}_k(G)$ has vertices given by partitions of $V(G)$ into $k$ independent sets (allowing empty parts), with two partitions adjacent if they differ only in the placement of a single vertex. We first give a structural classification of cliques in Bell coloring graphs. We then show that all trees and all cycles arise as Bell coloring graphs, while $K_4-e$ is not a Bell coloring graph and, more generally, $K_n-e$ is not an induced subgraph of any Bell coloring graph whenever $n \geq 6$. We also prove two reconstruction results: the Bell $3$-coloring graph is a complete invariant for trees, and the Bell $n$-coloring multigraph determines any graph up to universal vertices.

math.CO↗

arXiv2025

Counting subgraphs of coloring graphs

The chromatic polynomial $π_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$. These coloring graphs can be understood as a reconfiguration system. We generalize the chromatic polynomial to $π_G^{(H)}(k)$, counting occurrences of arbitrary induced subgraphs $H$ in these coloring graphs, and we prove that these functions are polynomial in $k$. In particular, we study the chromatic pairs polynomial $π_{G}^{(P_2)}(k)$, which counts the number of edges in coloring graphs, corresponding to the number of pairs of colorings that differ on a single vertex. We show two trees share a chromatic pairs polynomial if and only if they have the same degree sequence, and we conjecture that the chromatic pairs polynomial refines the chromatic polynomial in general. We also instantiate our polynomials with other choices of $H$ to generate new graph invariants.

math.CO↗

arXiv2020

Privately detecting changes in unknown distributions

The change-point detection problem seeks to identify distributional changes in streams of data. Increasingly, tools for change-point detection are applied in settings where data may be highly sensitive and formal privacy guarantees are required, such as identifying disease outbreaks based on hospital records, or IoT devices detecting activity within a home. Differential privacy has emerged as a powerful technique for enabling data analysis while preventing information leakage about individuals. Much of the prior work on change-point detection---including the only private algorithms for this problem---requires complete knowledge of the pre-change and post-change distributions. However, this assumption is not realistic for many practical applications of interest. This work develops differentially private algorithms for solving the change-point problem when the data distributions are unknown. Additionally, the data may be sampled from distributions that change smoothly over time, rather than fixed pre-change and post-change distributions. We apply our algorithms to detect changes in the linear trends of such data streams. Finally, we also provide experimental results to empirically validate the performance of our algorithms.

math.ST↗

arXiv2018

Differentially Private Change-Point Detection

The change-point detection problem seeks to identify distributional changes at an unknown change-point k* in a stream of data. This problem appears in many important practical settings involving personal data, including biosurveillance, fault detection, finance, signal detection, and security systems. The field of differential privacy offers data analysis tools that provide powerful worst-case privacy guarantees. We study the statistical problem of change-point detection through the lens of differential privacy. We give private algorithms for both online and offline change-point detection, analyze these algorithms theoretically, and provide empirical validation of our results.

math.ST↗

arXiv2018

Differential Privacy for Growing Databases

We study the design of differentially private algorithms for adaptive analysis of dynamically growing databases, where a database accumulates new data entries while the analysis is ongoing. We provide a collection of tools for machine learning and other types of data analysis that guarantee differential privacy and accuracy as the underlying databases grow arbitrarily large. We give both a general technique and a specific algorithm for adaptive analysis of dynamically growing databases. Our general technique is illustrated by two algorithms that schedule black box access to some algorithm that operates on a fixed database to generically transform private and accurate algorithms for static databases into private and accurate algorithms for dynamically growing databases. These results show that almost any private and accurate algorithm can be rerun at appropriate points of data growth with minimal loss of accuracy, even when data growth is unbounded. Our specific algorithm directly adapts the private multiplicative weights algorithm to the dynamic setting, maintaining the accuracy guarantee of the static setting through unbounded data growth. Along the way, we develop extensions of several other differentially private algorithms to the dynamic setting, which may be of independent interest for future work on the design of differentially private algorithms for growing databases.

cs.DS↗

arXiv2011

Near Optimality in Covering and Packing Games by Exposing Global Information

Covering and packing problems can be modeled as games to encapsulate interesting social and engineering settings. These games have a high Price of Anarchy in their natural formulation. However, existing research applicable to specific instances of these games has only been able to prove fast convergence to arbitrary equilibria. This paper studies general classes of covering and packing games with learning dynamics models that incorporate a central authority who broadcasts weak, socially beneficial signals to agents that otherwise only use local information in their decision-making. Rather than illustrating convergence to an arbitrary equilibrium that may have very high social cost, we show that these systems quickly achieve near-optimal performance. In particular, we show that in the public service advertising model, reaching a small constant fraction of the agents is enough to bring the system to a state within a log n factor of optimal in a broad class of set cover and set packing games or a constant factor of optimal in the special cases of vertex cover and maximum independent set, circumventing social inefficiency of bad local equilibria that could arise without a central authority. We extend these results to the learn-then-decide model, in which agents use any of a broad class of learning algorithms to decide in a given round whether to behave according to locally optimal behavior or the behavior prescribed by the broadcast signal. The new techniques we use for analyzing these games could be of broader interest for analyzing more general classic optimization problems in a distributed fashion.

cs.GT↗