By a $z$-coloring of a graph $G$ we mean any proper vertex coloring consisting of the color classes $C_1, \ldots, C_k$ such that $(i)$ for any two colors $i$ and $j$ with $1 \leq i < j \leq k$, any vertex of color $j$ is adjacent to a vertex of color $i$, $(ii)$ there exists a set $\{u_1, \ldots, u_k\}$ of vertices of $G$ such that $u_j \in C_j$ for any $j \in \{1, \ldots, k\}$ and $u_k$ is adjacent to $u_j$ for each $1 \leq j \leq k$ with $j \not=k$, and $(iii)$ for each $i$ and $j$ with $i \not= j$, the vertex $u_j$ has a neighbor in $C_i$. Denote by $z(G)$ the maximum number of colors used in any $z$-coloring of $G$. Denote the Grundy and {\rm b}-chromatic number of $G$ by $\Gamma(G)$ and ${\rm b}(G)$, respectively. The $z$-coloring is an improvement over both the Grundy and b-coloring of graphs. We prove that $z(G)$ is much better than $\min\{\Gamma(G), {\rm b}(G)\}$ for infinitely many graphs $G$ by obtaining an infinite sequence $\{G_n\}_{n=3}^{\infty}$ of graphs such that $z(G_n)=n$ but $\Gamma(G_n)={\rm b}(G_n)=2n-1$ for each $n\geq 3$. We show that acyclic graphs are $z$-monotonic and $z$-continuous. Then it is proved that to decide whether $z(G)=\Delta(G)+1$ is $NP$-complete even for bipartite graphs $G$. We finally prove that to recognize graphs $G$ satisfying $z(G)=\chi(G)$ is $coNP$-complete, improving a previous result for the Grundy number.
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Past work exploring adversarial vulnerability have focused on situations where an adversary can perturb all dimensions of model input. On the other hand, a range of recent works consider the case where either (i) an adversary can perturb a limited number of input parameters or (ii) a subset of modalities in a multimodal problem. In both of these cases, adversarial examples are effectively constrained to a subspace $V$ in the ambient input space $\mathcal{X}$. Motivated by this, in this work we investigate how adversarial vulnerability depends on $\dim(V)$. In particular, we show that the adversarial success of standard PGD attacks with $\ell^p$ norm constraints behaves like a monotonically increasing function of $\epsilon (\frac{\dim(V)}{\dim \mathcal{X}})^{\frac{1}{q}}$ where $\epsilon$ is the perturbation budget and $\frac{1}{p} + \frac{1}{q} =1$, provided $p > 1$ (the case $p=1$ presents additional subtleties which we analyze in some detail). This functional form can be easily derived from a simple toy linear model, and as such our results land further credence to arguments that adversarial examples are endemic to locally linear models on high dimensional spaces.
In this paper, we propose IMA-GNN as an In-Memory Accelerator for centralized and decentralized Graph Neural Network inference, explore its potential in both settings and provide a guideline for the community targeting flexible and efficient edge computation. Leveraging IMA-GNN, we first model the computation and communication latencies of edge devices. We then present practical case studies on GNN-based taxi demand and supply prediction and also adopt four large graph datasets to quantitatively compare and analyze centralized and decentralized settings. Our cross-layer simulation results demonstrate that on average, IMA-GNN in the centralized setting can obtain ~790x communication speed-up compared to the decentralized GNN setting. However, the decentralized setting performs computation ~1400x faster while reducing the power consumption per device. This further underlines the need for a hybrid semi-decentralized GNN approach.
For any graph $G$ and any set $\mathcal{F}$ of graphs, let $\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$ does not contain a copy of a graph in $\mathcal{F}$. Thus, $\iota(G,\{K_1\})$ is the domination number of $G$. For any integer $k \geq 1$, let $\mathcal{F}_{0,k} = \{K_{1,k}\}$, let $\mathcal{F}_{1,k}$ be the set of regular graphs of degree at least $k-1$, let $\mathcal{F}_{2,k}$ be the set of graphs whose chromatic number is at least $k$, and let $\mathcal{F}_{3,k}$ be the union of $\mathcal{F}_{0,k}$, $\mathcal{F}_{1,k}$ and $\mathcal{F}_{2,k}$. We prove that if $G$ is a connected $n$-vertex graph and $\mathcal{F} = \mathcal{F}_{0,k} \cup \mathcal{F}_{1,k}$, then $\iota(G, \mathcal{F}) \leq \frac{n}{k+1}$ unless $G$ is a $k$-clique or $k = 2$ and $G$ is a $5$-cycle. This generalizes a bound of Caro and Hansberg on the $\{K_{1,k}\}$-isolation number, a bound of the author on the cycle isolation number, and a bound of Fenech, Kaemawichanurat and the author on the $k$-clique isolation number. By Brooks' Theorem, the same holds if $\mathcal{F} = \mathcal{F}_{3,k}$. The bounds are sharp.
Relational verification encompasses information flow security, regression verification, translation validation for compilers, and more. Effective alignment of the programs and computations to be related facilitates use of simpler relational invariants and relational procedure specs, which in turn enables automation and modular reasoning. Alignment has been explored in terms of trace pairs, deductive rules of relational Hoare logics (RHL), and several forms of product automata. This article shows how a simple extension of Kleene Algebra with Tests (KAT), called BiKAT, subsumes prior formulations, including alignment witnesses for forall-exists properties, which brings to light new RHL-style rules for such properties. Alignments can be discovered algorithmically or devised manually but, in either case, their adequacy with respect to the original programs must be proved; an explicit algebra enables constructive proof by equational reasoning. Furthermore our approach inherits algorithmic benefits from existing KAT-based techniques and tools, which are applicable to a range of semantic models.
We develop a linear time algorithm for finding the diameter of an asteroidal triple-free (AT-free) graph. Furthermore, we update the definition of polar pairs and develop new properties of polar pairs for (weak) dominating pair graphs. We prove that the problem of computing a simplicial vertex in a general graph can be accomplished in O(n^2) based on an existing reduction to the problem of finding diameter in an AT-free graph. We improve the best-known run-time complexities of several graph theoretical problems.
Empirical studies of the loss landscape of deep networks have revealed that many local minima are connected through low-loss valleys. Yet, little is known about the theoretical origin of such valleys. We present a general framework for finding continuous symmetries in the parameter space, which carve out low-loss valleys. Our framework uses equivariances of the activation functions and can be applied to different layer architectures. To generalize this framework to nonlinear neural networks, we introduce a novel set of nonlinear, data-dependent symmetries. These symmetries can transform a trained model such that it performs similarly on new samples, which allows ensemble building that improves robustness under certain adversarial attacks. We then show that conserved quantities associated with linear symmetries can be used to define coordinates along low-loss valleys. The conserved quantities help reveal that using common initialization methods, gradient flow only explores a small part of the global minimum. By relating conserved quantities to convergence rate and sharpness of the minimum, we provide insights on how initialization impacts convergence and generalizability.
We consider the influence maximization problem over a temporal graph, where there is a single fixed source. We deviate from the standard model of influence maximization, where the goal is to choose the set of most influential vertices. Instead, in our model we are given a fixed vertex, or source, and the goal is to find the best time steps to transmit so that the influence of this vertex is maximized. We frame this problem as a spreading process that follows a variant of the susceptible-infected-susceptible (SIS) model and we focus on four objective functions. In the MaxSpread objective, the goal is to maximize the total number of vertices that get infected at least once. In the MaxViral objective, the goal is to maximize the number of vertices that are infected at the same time step. In the MaxViralTstep objective, the goal is to maximize the number of vertices that are infected at a given time step. Finally, in MinNonViralTime, the goal is to maximize the total number of vertices that get infected every $d$ time steps. We perform a thorough complexity theoretic analysis for these four objectives over three different scenarios: (1) the unconstrained setting where the source can transmit whenever it wants; (2) the window-constrained setting where the source has to transmit at either a predetermined, or a shifting window; (3) the periodic setting where the temporal graph has a small period. We prove that all of these problems, with the exception of MaxSpread for periodic graphs, are intractable even for very simple underlying graphs.
Given an undirected unweighted graph $G = (V, E)$ on $n$ vertices and $m$ edges, a subgraph $H\subseteq G$ is a spanner of $G$ with stretch function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, iff for every pair $s, t$ of vertices in $V$, $\textsf{dist}_{H}(s, t)\le f(\textsf{dist}_{G}(s, t))$. When $f(d) = d + o(d)$, $H$ is called a sublinear additive spanner; when $f(d) = d + o(n)$, $H$ is called an additive spanner, and $f(d) - d$ is usually called the additive stretch of $H$. As our primary result, we show that for any constant $\delta>0$ and constant integer $k\geq 2$, every graph on $n$ vertices has a sublinear additive spanner with stretch function $f(d)=d+O(d^{1-1/k})$ and $O\big(n^{1+\frac{1+\delta}{2^{k+1}-1}}\big)$ edges. When $k = 2$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d^{1/2})$ and $\tilde{O}(n^{1+3/17})$ edges [Chechik, 2013]; for any constant integer $k\geq 3$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d^{1-1/k})$ and $O\bigg(n^{1+\frac{(3/4)^{k-2}}{7 - 2\cdot (3/4)^{k-2}}}\bigg)$ edges [Pettie, 2009]. Most importantly, the size of our spanners almost matches the lower bound of $\Omega\big(n^{1+\frac{1}{2^{k+1}-1}}\big)$ [Abboud, Bodwin, Pettie, 2017]. As our second result, we show a new construction of additive spanners with stretch $O(n^{0.403})$ and $O(n)$ edges, which slightly improves upon the previous stretch bound of $O(n^{3/7+\epsilon})$ achieved by linear-size spanners [Bodwin and Vassilevska Williams, 2016]. An additional advantage of our spanner is that it admits a subquadratic construction runtime of $\tilde{O}(m + n^{13/7})$, while the previous construction in [Bodwin and Vassilevska Williams, 2016] requires all-pairs shortest paths computation which takes $O(\min\{mn, n^{2.373}\})$ time.
We study the complexity of high-dimensional approximation in the $L_2$-norm when different classes of information are available; we compare the power of function evaluations with the power of arbitrary continuous linear measurements. Here, we discuss the situation when the number of linear measurements required to achieve an error $\varepsilon \in (0,1)$ in dimension $d\in\mathbb{N}$ depends only poly-logarithmically on $\varepsilon^{-1}$. This corresponds to an exponential order of convergence of the approximation error, which often happens in applications. However, it does not mean that the high-dimensional approximation problem is easy, the main difficulty usually lies within the dependence on the dimension $d$. We determine to which extent the required amount of information changes, if we allow only function evaluation instead of arbitrary linear information. It turns out that in this case we only lose very little, and we can even restrict to linear algorithms. In particular, several notions of tractability hold simultaneously for both types of available information.
Graph Neural Networks (GNNs) have been studied from the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs' training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice.
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