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We devise coresets for kernel $k$-Means with a general kernel, and use them to obtain new, more efficient, algorithms. Kernel $k$-Means has superior clustering capability compared to classical $k$-Means, particularly when clusters are non-linearly separable, but it also introduces significant computational challenges. We address this computational issue by constructing a coreset, which is a reduced dataset that accurately preserves the clustering costs. Our main result is a coreset for kernel $k$-Means that works for a general kernel and has size $\mathrm{poly}(k\epsilon^{-1})$. Our new coreset both generalizes and greatly improves all previous results; moreover, it can be constructed in time near-linear in $n$. This result immediately implies new algorithms for kernel $k$-Means, such as a $(1+\epsilon)$-approximation in time near-linear in $n$, and a streaming algorithm using space and update time $\mathrm{poly}(k \epsilon^{-1} \log n)$. We validate our coreset on various datasets with different kernels. Our coreset performs consistently well, achieving small errors while using very few points. We show that our coresets can speed up kernel $k$-Means++ (the kernelized version of the widely used $k$-Means++ algorithm), and we further use this faster kernel $k$-Means++ for spectral clustering. In both applications, we achieve up to 1000x speedup while the error is comparable to baselines that do not use coresets.

The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter. In the process we introduce the problem of isomorphism testing for higher-order hypergraphs and show that finding the automorphism group of order-$k$ hypergraphs with vertex color classes of size $b$ is fixed parameter tractable for any constant $k$ and $b$ as fixed parameter.

Consider the task of performing a sequence of searches in a binary search tree. After each search, we allow an algorithm to arbitrarily restructure the tree. The cost of executing the task is the sum of the time spent searching and the time spent optimizing the searches with restructuring operations. Sleator and Tarjan introduced this notion in 1985, along with an algorithm and a conjecture. The algorithm, Splay, is an elegant procedure for performing adjustments that move searched items to the top of the tree. The conjecture, called dynamic optimality, is that the cost of splaying is always within a constant factor of the optimal algorithm for performing searches. We lay a foundation for proving the dynamic optimality conjecture. Central to our method is approximate monotonicity. Approximately monotone algorithms are those whose cost does not increase by more than a fixed multiple after removing searches from the sequence. As we shall see, Splay is dynamically optimal if and only if it is approximately monotone. This result extends to a weaker form of approximate monotonicity as well as insertion, deletion, and related algorithms. We prove that a lower bound on optimal execution cost is approximately monotone and outline how to adapt this proof from the lower bound to Splay, and how to overcome the remaining barriers to establishing dynamic optimality.

We show that some natural problems that are XNLP-hard (which implies W[t]-hardness for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds (which is strongly NP-complete, as shown by Itai), Minimum Maximum Outdegree (for which W[1]-hardness for treewidth was proven by Szeider), and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set (for which W[1]-hardness was proven by Dom, Lokshtanov, Saurabh and Villanger). Our hardness proofs (that beat existing results) use reduction to a recent XNLP-complete problem (Accepting Non-deterministic Checking Counter Machine). The new easy parameterized algorithms use a novel notion of weighted tree partition with an associated parameter that we call treebreadth, inspired by Seese's notion of tree-partite graphs, as well as techniques from dynamical programming and integer linear programming.

We consider classes of arbitrary (finite or infinite) graphs of bounded shrub-depth, specifically the classes $\mathrm{TM}_r(d)$ of arbitrary graphs that have tree models of height $d$ and $r$ labels. We show that the graphs of $\mathrm{TM}_r(d)$ are $\mathrm{MSO}$-pseudo-finite relative to the class $\mathrm{TM}^{\text{f}}_r(d)$ of finite graphs of $\mathrm{TM}_r(d)$; that is, that every $\mathrm{MSO}$ sentence true in a graph of $\mathrm{TM}_r(d)$ is also true in a graph of $\mathrm{TM}^{\text{f}}_r(d)$. We also show that $\mathrm{TM}_r(d)$ is closed under ultraproducts and ultraroots. These results have two consequences. The first is that the index of the $\mathrm{MSO}[m]$-equivalence relation on graphs of $\mathrm{TM}_r(d)$ is bounded by a $(d+1)$-fold exponential in $m$. The second is that $\mathrm{TM}_r(d)$ is exactly the class of all graphs that are $\mathrm{MSO}$-pseudo-finite relative to $\mathrm{TM}^{\text{f}}_r(d)$.

In this paper we study temporal design problems of undirected temporally connected graphs. The basic setting of these optimization problems is as follows: given an undirected graph $G$, what is the smallest number $|\lambda|$ of time-labels that we need to add to the edges of $G$ such that the resulting temporal graph $(G,\lambda)$ is temporally connected? As we prove, this basic problem, called MINIMUM LABELING, can be optimally solved in polynomial time, thus resolving an open question. The situation becomes however more complicated if we strengthen, or even if we relax a bit, the requirement of temporal connectivity of $(G,\lambda)$. One way to strengthen the temporal connectivity requirements is to upper-bound the allowed age (i.e., maximum label) of the obtained temporal graph $(G,\lambda)$. On the other hand, we can relax temporal connectivity by only requiring that there exists a temporal path between any pair of ``important'' vertices which lie in a subset $R\subseteq V$, which we call the terminals. This relaxed problem, called MINIMUM STEINER LABELING, resembles the problem STEINER TREE in static (i.e., non-temporal) graphs; however, as it turns out, STEINER TREE is not a special case of MINIMUM STEINER LABELING. We prove that MINIMUM STEINER LABELING is NP-hard and in FPT with respect to the number $|R|$ of terminals. Moreover, we prove that, adding the age restriction makes the above problems strictly harder (unless P=NP or W[1]=FPT). More specifically, we prove that the age-restricted version of MINIMUM LABELING becomes NP-complete on undirected graphs, while the age-restricted version of MINIMUM STEINER LABELING becomes W[1]-hard with respect to the number $|R|$ of terminals.

Evaluating predictive models is a crucial task in predictive analytics. This process is especially challenging with time series data where the observations show temporal dependencies. Several studies have analysed how different performance estimation methods compare with each other for approximating the true loss incurred by a given forecasting model. However, these studies do not address how the estimators behave for model selection: the ability to select the best solution among a set of alternatives. We address this issue and compare a set of estimation methods for model selection in time series forecasting tasks. We attempt to answer two main questions: (i) how often is the best possible model selected by the estimators; and (ii) what is the performance loss when it does not. We empirically found that the accuracy of the estimators for selecting the best solution is low, and the overall forecasting performance loss associated with the model selection process ranges from 1.2% to 2.3%. We also discovered that some factors, such as the sample size, are important in the relative performance of the estimators.

We prove that the stack-number of the strong product of three $n$-vertex paths is $\Theta(n^{1/3})$. The best previously known upper bound was $O(n)$. No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest $\Delta_0$ such that there exist a graph family with unbounded stack-number, bounded queue-number and maximum degree $\Delta_0$. We show that $\Delta_0\in \{6,7\}$.

Existing few-shot learning (FSL) methods assume that there exist sufficient training samples from source classes for knowledge transfer to target classes with few training samples. However, this assumption is often invalid, especially when it comes to fine-grained recognition. In this work, we define a new FSL setting termed few-shot fewshot learning (FSFSL), under which both the source and target classes have limited training samples. To overcome the source class data scarcity problem, a natural option is to crawl images from the web with class names as search keywords. However, the crawled images are inevitably corrupted by large amount of noise (irrelevant images) and thus may harm the performance. To address this problem, we propose a graph convolutional network (GCN)-based label denoising (LDN) method to remove the irrelevant images. Further, with the cleaned web images as well as the original clean training images, we propose a GCN-based FSL method. For both the LDN and FSL tasks, a novel adaptive aggregation GCN (AdarGCN) model is proposed, which differs from existing GCN models in that adaptive aggregation is performed based on a multi-head multi-level aggregation module. With AdarGCN, how much and how far information carried by each graph node is propagated in the graph structure can be determined automatically, therefore alleviating the effects of both noisy and outlying training samples. Extensive experiments show the superior performance of our AdarGCN under both the new FSFSL and the conventional FSL settings.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.