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This paper studies $k$-claw-free graphs, exploring the connection between an extremal combinatorics question and the power of a convex program in approximating the maximum-weight independent set in this graph class. For the extremal question, we consider the notion, that we call \textit{conditional $\chi$-boundedness} of a graph: Given a graph $G$ that is assumed to contain an independent set of a certain (constant) size, we are interested in upper bounding the chromatic number in terms of the clique number of $G$. This question, besides being interesting on its own, has algorithmic implications (which have been relatively neglected in the literature) on the performance of SDP relaxations in estimating the value of maximum-weight independent set. For $k=3$, Chudnovsky and Seymour (JCTB 2010) prove that any $3$-claw-free graph $G$ with an independent set of size three must satisfy $\chi(G) \leq 2 \omega(G)$. Their result implies a factor $2$-estimation algorithm for the maximum weight independent set via an SDP relaxation (providing the first non-trivial result for maximum-weight independent set in such graphs via a convex relaxation). An obvious open question is whether a similar conditional $\chi$-boundedness phenomenon holds for any $k$-claw-free graph. Our main result answers this question negatively. We further present some evidence that our construction could be useful in studying more broadly the power of convex relaxations in the context of approximating maximum weight independent set in $k$-claw free graphs. In particular, we prove a lower bound on families of convex programs that are stronger than known convex relaxations used algorithmically in this context.

For a graph $G$, a $D$-diameter-reducing exact hopset is a small set of additional edges $H$ that, when added to $G$, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most $D$ edges. A shortcut set is the analogous concept for reachability. These objects have been studied since the early '90s due to applications in parallel, distributed, dynamic, and streaming graph algorithms. For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA '22] and Bernstein and Wein [SODA '23] have finally improved over the folklore diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+\epsilon)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued? We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetilde{\Omega}(n^{1/2})$. This improves on the previous lower bound of $\widetilde{\Omega}(n^{1/3})$ by Kogan and Parter [FOCS '22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetilde{\Omega}(n^{1/4})$. This improves on the previous lower bound of $\Omega(n^{1/6})$ by Huang and Pettie [SIAM J. Disc. Math. '18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of $p$; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of $p \in [1, n^2]$.

An independent set in a graph $G$ is a set $S$ of pairwise non-adjacent vertices in $G$. A family $\mathcal{F}$ of independent sets in $G$ is called a $k$-independence covering family if for every independent set $I$ in $G$ of size at most $k$, there exists an $S \in \mathcal{F}$ such that $I \subseteq S$. Lokshtanov et al. [ACM Transactions on Algorithms, 2018] showed that graphs of degeneracy $d$ admit $k$-independence covering families of size $\binom{k(d+1)}{k} \cdot 2^{o(kd)} \cdot \log n$, and used this result to design efficient parameterized algorithms for a number of problems, including STABLE ODD CYCLE TRANSVERSAL and STABLE MULTICUT. In light of the results of Lokshtanov et al. it is quite natural to ask whether even more general families of graphs admit $k$-independence covering families of size $f(k)n^{O(1)}$. Graphs that exclude a complete bipartite graph $K_{d+1,d+1}$ with $d+1$ vertices on both sides as a subgraph, called $K_{d+1,d+1}$-free graphs, are a frequently considered generalization of $d$-degenerate graphs. This motivates the question whether $K_{d,d}$-free graphs admit $k$-independence covering families of size $f(k,d)n^{O(1)}$. Our main result is a resounding "no" to this question -- specifically we prove that even $K_{2,2}$-free graphs (or equivalently $C_4$-free graphs) do not admit $k$-independence covering families of size $f(k)n^{\frac{k}{4}-\epsilon}$.

Over the past decade, previous balanced datasets have been used to advance deep learning algorithms for industrial applications. In urban infrastructures and living environments, damage data mining cannot avoid imbalanced data issues because of rare unseen events and the high-quality status of improved operations. For visual inspection, the deteriorated class acquired from the surface of concrete and steel components are occasionally imbalanced. From numerous related surveys, we conclude that imbalanced data problems can be categorised into four types: 1) missing range of target and label valuables, 2) majority-minority class imbalance, 3) foreground background of spatial imbalance, and 4) long-tailed class of pixel-wise imbalance. Since 2015, many imbalanced studies have been conducted using deep-learning approaches, including regression, image classification, object detection, and semantic segmentation. However, anomaly detection for imbalanced data is not well known. In this study, we highlight a one-class anomaly detection application, whether anomalous class or not, and demonstrate clear examples of imbalanced vision datasets: medical disease, hazardous behaviour, material deterioration, plant disease, river sludge, and disaster damage. We provide key results on the advantage of damage-vision mining, hypothesising that the more effective the range of the positive ratio, the higher the accuracy gain of the anomalies feedback. In our imbalanced studies, compared with the balanced case with a positive ratio of $1/1$, we find that there is an applicable positive ratio $1/a$ where the accuracy is consistently high. However, the extremely imbalanced range is from one shot to $1/2a$, the accuracy of which is inferior to that of the applicable ratio. In contrast, with a positive ratio ranging over $2/a$, it shifts in the over-mining phase without an effective gain in accuracy.

A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix $A$ of dimension $n$. The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by $O(n^3 (\log n + \log ||A||)^2(\log n)^2)$ bit operations, where $||A||= \max_{ij} |A_{ij}|$ denotes the largest entry of $A$ in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time $(n^3 \log ||A||)^{1+o(1)}$ bit operations, where the exponent $"+o(1)"$ captures additional factors $c_1 (\log n)^{c_2} (\log \log ||A||)^{c_3}$ for positive real constants $c_1,c_2,c_3$.

This paper presents a Multiple Kernel Learning (abbreviated as MKL) framework for the Support Vector Machine (SVM) with the $(0, 1)$ loss function. Some KKT-like first-order optimality conditions are provided and then exploited to develop a fast ADMM algorithm to solve the nonsmooth nonconvex optimization problem. Numerical experiments on synthetic and real datasets show that the performance of our MKL-$L_{0/1}$-SVM is comparable with the one of the leading approaches called SimpleMKL developed by Rakotomamonjy, Bach, Canu, and Grandvalet [Journal of Machine Learning Research, vol.~9, pp.~2491--2521, 2008].

We study $L_2$-approximation problems in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameters $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss the strongly polynomial tractability (SPT), polynomial tractability (PT), and $(t_1,t_2)$-weak tractability ($(t_1,t_2)$-WT) for all $t_1>1$ and $t_2>0$ under the absolute or normalized error criterion. In particular, we obtain the matching necessary and sufficient condition for SPT or PT in terms of the parameters.

For a skew polynomial ring $R=A[X;\theta,\delta]$ where $A$ is a commutative frobenius ring, $\theta$ an endomorphism of $A$ and $\delta$ a $\theta$-derivation of $A$, we consider cyclic left module codes $\mathcal{C}=Rg/Rf\subset R/Rf$ where $g$ is a left and right divisor of $f$ in $R$. In this paper we derive a parity check matrix when $A$ is a finite commutative frobenius ring using only the framework of skew polynomial rings. We consider rings $A=B[a_1,\ldots,a_s]$ which are free $B$-algebras where the restriction of $\delta$ and $\theta$ to $B$ are polynomial maps. If a Gr\"obner basis can be computed over $B$, then we show that all Euclidean and Hermitian dual-containing codes $\mathcal{C}=Rg/Rf\subset R/Rf$ can be computed using a Gr\"obner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order $4$ with non-trivial endomorphism and the Galois ring of characteristic $4$.

Preconditioning is essential in iterative methods for solving linear systems of equations. We study a nonclassic matrix condition number, the $\omega$-condition number, in the context of optimal conditioning for low rank updating of positive definite matrices. For a positive definite matrix, this condition measure is the ratio of the arithmetic and geometric means of the eigenvalues. In particular, we concentrate on linear systems with low rank updates of positive definite matrices which are close to singular. These systems arise in the contexts of nonsmooth Newton methods using generalized Jacobians. We derive an explicit formula for the optimal $\omega$-preconditioned update in this framework. Evaluating or estimating the classical condition number $\kappa$ can be expensive. We show that the $\omega$-condition number can be evaluated exactly following a Cholesky or LU factorization and it estimates the actual condition of a linear system significantly better. Moreover, our empirical results show a significant decrease in the number of iterations required for a requested accuracy in the residual during an iterative method, i.e., these results confirm the efficacy of using the $\omega$-condition number compared to the classical condition number.

The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skii-Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of $o(1/k)$ while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.