An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval $t$-coloring} if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. In 1990, Kamalian proved that if a graph $G$ with at least one edge has an interval $t$-coloring, then $t\leq 2|V(G)|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{11}{6}|V(G)|$. In the same paper Axenovich suggested a conjecture that if a planar graph $G$ has an interval $t$-coloring, then $t\leq \frac{3}{2}|V(G)|$. In this paper we confirm the conjecture by showing that if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{3|V(G)|-4}{2}$. We also prove that if an outerplanar graph $G$ has an interval $t$-coloring, then $t\leq |V(G)|-1$. Moreover, all these upper bounds are sharp.
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In order to advance academic research, it is important to assess and evaluate the academic influence of researchers and the findings they produce. Citation metrics are universally used methods to evaluate researchers. Amongst the several variations of citation metrics, the h-index proposed by Hirsch has become the leading measure. Recent work shows that h-index is not an effective measure to determine scientific impact - due to changing authorship patterns. This can be mitigated by using h-index of a paper to compute h- index of an author. We show that using fractional allocation of h-index gives better results. In this work, we reapply two indices based on the h-index of a single paper. The indices are referred to as: hp-index and hp-frac-index. We run large-scale experiments in three different fields with about a million publications and 3,000 authors. We also compare h-index of a paper with nine h-index like metrics. Our experiments show that hp-frac-index provides a unique ranking when compared to h-index. It also performs better than h-index in providing higher ranks to the awarded researcher.
We explore the analytic properties of the density function $ h(x;\gamma,\alpha) $, $ x \in (0,\infty) $, $ \gamma > 0 $, $ 0 < \alpha < 1 $ which arises from the domain of attraction problem for a statistic interpolating between the supremum and sum of random variables. The parameter $ \alpha $ controls the interpolation between these two cases, while $ \gamma $ parametrises the type of extreme value distribution from which the underlying random variables are drawn from. For $ \alpha = 0 $ the Fr\'echet density applies, whereas for $ \alpha = 1 $ we identify a particular Fox H-function, which are a natural extension of hypergeometric functions into the realm of fractional calculus. In contrast for intermediate $ \alpha $ an entirely new function appears, which is not one of the extensions to the hypergeometric function considered to date. We derive series, integral and continued fraction representations of this latter function.
In this paper, we study the problem of computing an edge-coloring in the (one-pass) W-streaming model. In this setting, the edges of an $n$-node graph arrive in an arbitrary order to a machine with a relatively small space, and the goal is to design an algorithm that outputs, as a stream, a proper coloring of the edges using the fewest possible number of colors. Behnezhad et al. [Behnezhad et al., 2019] devised the first non-trivial algorithm for this problem, which computes in $\tilde{O}(n)$ space a proper $O(\Delta^2)$-coloring w.h.p. (here $\Delta$ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] shows that it is possible to deterministically compute an $O(\Delta^2/s)$-coloring in $O(ns)$ space. However, none of the improvements, succeeded in reducing the number of colors to $O(\Delta^{2-\epsilon})$ while keeping the same space bound of $\tilde{O}(n)$. In particular, no progress was made on the question of whether computing an $O(\Delta)$-coloring is possible with roughly $O(n)$ space, which was stated in [Behnezhad et al., 2019] to be a major open problem. In this paper we bypass the quadratic bound by presenting a new randomized $\tilde{O}(n)$-space algorithm that uses $\tilde{O}(\Delta^{1.5})$ colors.
Dedicated model transformation languages are claimed to provide many benefits over the use of general purpose languages for developing model transformations. However, the actual advantages associated with the use of MTLs are poorly understood empirically. There is little knowledge and empirical assessment about what advantages and disadvantages hold and where they originate from. In a prior interview study, we elicited expert opinions on what advantages result from what factors and a number of factors that moderate the influence. We aim to quantitatively asses the interview results to confirm or reject the effects posed by different factors. We intend to gain insights into how valuable different factors are so that future studies can draw on these data for designing targeted and relevant studies. We gather data on the factors and quality attributes using an online survey. To analyse the data, we use universal structure modelling based on a structure model. We use significance values and path coefficients produced bz USM for each hypothesised interdependence to confirm or reject correlation and to weigh the strength of influence present. We analyzed 113 responses. The results show that the Tracing and Reuse Mechanisms are most important overall. Though the observed effects were generally 10 times lower than anticipated. Additionally, we found that a more nuanced view of moderation effects is warranted. Their moderating influence differed significantly between the different influences, with the strongest effects being 1000 times higher than the weakest. The empirical assessment of MTLs is a complex topic that cannot be solved by looking at a single stand-alone factor. Our results provide clear indication that evaluation should consider transformations of different sizes and use-cases. Language development should focus on providing transformation specific reuse mechanisms .
Spectral hypergraph sparsification, an attempt to extend well-known spectral graph sparsification to hypergraphs, has been extensively studied over the past few years. For undirected hypergraphs, Kapralov, Krauthgamer, Tardos, and Yoshida~(2022) have proved an $\varepsilon$-spectral sparsifier of the optimal $O^*(n)$ size, where $n$ is the number of vertices and $O^*$ suppresses the $\varepsilon^{-1}$ and $\log n$ factors. For directed hypergraphs, however, the optimal sparsifier size has not been known. Our main contribution is the first algorithm that constructs an $O^*(n^2)$-size $\varepsilon$-spectral sparsifier for a weighted directed hypergraph. Our result is optimal up to the $\varepsilon^{-1}$ and $\log n$ factors since there is a lower bound of $\Omega(n^2)$ even for directed graphs. We also show the first non-trivial lower bound of $\Omega(n^2/\varepsilon)$ for general directed hypergraphs. The basic idea of our algorithm is borrowed from the spanner-based sparsification for ordinary graphs by Koutis and Xu~(2016). Their iterative sampling approach is indeed useful for designing sparsification algorithms in various circumstances. To demonstrate this, we also present a similar iterative sampling algorithm for undirected hypergraphs that attains one of the best size bounds, enjoys parallel implementation, and can be transformed to be fault-tolerant.
The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory, the Laplace-Beltrami operator can be pre-/post-conditioned with an integral operator whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard~$1/r$ kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace-Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.
Let $G$ be a graph on $n$ vertices of maximum degree $\Delta$. We show that, for any $\delta > 0$, the down-up walk on independent sets of size $k \leq (1-\delta)\alpha_c(\Delta)n$ mixes in time $O_{\Delta,\delta}(k\log{n})$, thereby resolving a conjecture of Davies and Perkins in an optimal form. Here, $\alpha_{c}(\Delta)n$ is the NP-hardness threshold for the problem of counting independent sets of a given size in a graph on $n$ vertices of maximum degree $\Delta$. Our mixing time has optimal dependence on $k,n$ for the entire range of $k$; previously, even polynomial mixing was not known. In fact, for $k = \Omega_{\Delta}(n)$ in this range, we establish a log-Sobolev inequality with optimal constant $\Omega_{\Delta,\delta}(1/n)$. At the heart of our proof are three new ingredients, which may be of independent interest. The first is a method for lifting $\ell_\infty$-independence from a suitable distribution on the discrete cube -- in this case, the hard-core model -- to the slice by proving stability of an Edgeworth expansion using a multivariate zero-free region for the base distribution. The second is a generalization of the Lee-Yau induction to prove log-Sobolev inequalities for distributions on the slice with considerably less symmetry than the uniform distribution. The third is a sharp decomposition-type result which provides a lossless comparison between the Dirichlet form of the original Markov chain and that of the so-called projected chain in the presence of a contractive coupling.
Federated learning (FL) allows multiple parties to cooperatively learn a federated model without sharing private data with each other. The need of protecting such federated models from being plagiarized or misused, therefore, motivates us to propose a provable secure model ownership verification scheme using zero-knowledge proof, named FedZKP. It is shown that the FedZKP scheme without disclosing credentials is guaranteed to defeat a variety of existing and potential attacks. Both theoretical analysis and empirical studies demonstrate the security of FedZKP in the sense that the probability for attackers to breach the proposed FedZKP is negligible. Moreover, extensive experimental results confirm the fidelity and robustness of our scheme.
We consider $t$-Lee-error-correcting codes of length $n$ over the residue ring $\mathbb{Z}_m := \mathbb{Z}/m\mathbb{Z}$ and determine upper and lower bounds on the number of $t$-Lee-error-correcting codes. We use two different methods, namely estimating isolated nodes on bipartite graphs and the graph container method. The former gives density results for codes of fixed size and the latter for any size. This confirms some recent density results for linear Lee metric codes and provides new density results for nonlinear codes. To apply a variant of the graph container algorithm we also investigate some geometrical properties of the balls in the Lee metric.
In modern machine learning, inner product attention computation is a fundamental task for training large language models such as Transformer, GPT-1, BERT, GPT-2, GPT-3 and ChatGPT. Formally, in this problem, one is given as input three matrices $Q, K, V \in [-B,B]^{n \times d}$, and the goal is to construct the matrix $\mathrm{Att}(Q,K,V) := \mathrm{diag}(A {\bf 1}_n)^{-1} A V \in \mathbb{R}^{n \times d}$, where $A = \exp(QK^\top/d)$ is the `attention matrix', and $\exp$ is applied entry-wise. Straightforward methods for this problem explicitly compute the $n \times n$ attention matrix $A$, and hence require time $\Omega(n^2)$ even when $d = n^{o(1)}$ is small. In this paper, we investigate whether faster algorithms are possible by implicitly making use of the matrix $A$. We present two results, showing that there is a sharp transition at $B = \Theta(\sqrt{\log n})$. $\bullet$ If $d = O(\log n)$ and $B = o(\sqrt{\log n})$, there is an $n^{1+o(1)}$ time algorithm to approximate $\mathrm{Att}(Q,K,V)$ up to $1/\mathrm{poly}(n)$ additive error. $\bullet$ If $d = O(\log n)$ and $B = \Theta (\sqrt{\log n})$, assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory, it is impossible to approximate $\mathrm{Att}(Q,K,V)$ up to $1/\mathrm{poly}(n)$ additive error in truly subquadratic time $n^{2 - \Omega(1)}$. This gives a theoretical explanation for the phenomenon observed in practice that attention computation is much more efficient when the input matrices have smaller entries.
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