We bound the smoothed running time of the FLIP algorithm for local Max-Cut as a function of $\alpha$, the arboricity of the input graph. We show that, with high probability, the following holds (where $n$ is the number of nodes and $\phi$ is the smoothing parameter): 1) When $\alpha = O(\sqrt{\log n})$ FLIP terminates in $\phi poly(n)$ iterations. Previous to our results the only graph families for which FLIP was known to achieve a smoothed polynomial running time were complete graphs and graphs with logarithmic maximum degree. 2) For arbitrary values of $\alpha$ we get a running time of $\phi n^{O(\frac{\alpha}{\log n} + \log \alpha)}$. This improves over the best known running time for general graphs of $\phi n^{O(\sqrt{ \log n })}$ for $\alpha = o(\log^{1.5} n)$. Specifically, when $\alpha = O(\log n)$ we get a significantly faster running time of $\phi n^{O(\log \log n)}$.
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We consider the problems of testing and learning quantum $k$-junta channels, which are $n$-qubit to $n$-qubit quantum channels acting non-trivially on at most $k$ out of $n$ qubits and leaving the rest of qubits unchanged. We show the following. 1. An $O\left(k\right)$-query algorithm to distinguish whether the given channel is $k$-junta channel or is far from any $k$-junta channels, and a lower bound $\Omega\left(\sqrt{k}\right)$ on the number of queries; 2. An $\widetilde{O}\left(4^k\right)$-query algorithm to learn a $k$-junta channel, and a lower bound $\Omega\left(4^k/k\right)$ on the number of queries. This gives the first junta channel testing and learning results, and partially answers an open problem raised by Chen et al. (2023). In order to settle these problems, we develop a Fourier analysis framework over the space of superoperators and prove several fundamental properties, which extends the Fourier analysis over the space of operators introduced in Montanaro and Osborne (2010). Besides, we introduce $\textit{Influence-Sample}$ to replace $\textit{Fourier-Sample}$ proposed in Atici and Servedio (2007). Our $\textit{Influence-Sample}$ includes only single-qubit operations and results in only constant-factor decrease in efficiency.
We study a class of functional problems reducible to computing $f^{(n)}(x)$ for inputs $n$ and $x$, where $f$ is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its definition, and in whether we require $f$ to have a polynomial-time inverse or to be computible by a reversible logic circuit. These problems are characterized by the complexity class $\mathsf{FP}^{\mathsf{PSPACE}}$, and include natural $\mathsf{FP}^{\mathsf{PSPACE}}$-complete problems in circuit complexity, cellular automata, graph algorithms, and the dynamical systems described by piecewise-linear transformations.
A pervasive methodological error is the post-hoc interpretation of $p$-values. A $p$-value $p$ is not the level at which we reject the null, it is the level at which we would have rejected the null had we chosen level $p$. We introduce the notion of a post-hoc $p$-value, that does admit this interpretation. We show that $p$ is a post-hoc $p$-value if and only if $1/p$ is an $e$-value. Among other things, this implies that the product of independent post-hoc $p$-values is a post-hoc $p$-value. Moreover, we generalize post-hoc validity to a sequential setting and find that $(p_t)_{t \geq 1}$ is a post-hoc anytime valid $p$-process if and only if $(1/p_t)_{t \geq 1}$ is an $e$-process. In addition, we show that if we admit randomized procedures, any non-randomized post-hoc $p$-value can be trivially improved. In fact, we find that this in some sense characterizes non-randomized post-hoc $p$-values. Finally, we argue that we need to go beyond $e$-values if we want to consider randomized post-hoc inference in its full generality.
We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to 16,384 CPU cores.
We present a randomized algorithm that computes single-source shortest paths (SSSP) in $O(m\log^8(n)\log W)$ time when edge weights are integral and can be negative. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are $\tilde O((m+n^{1.5})\log W)$ [BLNPSSSW FOCS'20] and $m^{4/3+o(1)}\log W$ [AMV FOCS'20]. Near-linear time algorithms were known previously only for the special case of planar directed graphs [Fakcharoenphol and Rao FOCS'01]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is simple: it requires only a simple graph decomposition and elementary combinatorial tools. In fact, ours is the first combinatorial algorithm for negative-weight SSSP to break through the classic $\tilde O(m\sqrt{n}\log W)$ bound from over three decades ago [Gabow and Tarjan SICOMP'89].
We study the task of $(\epsilon, \delta)$-differentially private online convex optimization (OCO). In the online setting, the release of each distinct decision or iterate carries with it the potential for privacy loss. This problem has a long history of research starting with Jain et al. [2012] and the best known results for the regime of {\epsilon} being very small are presented in Agarwal et al. [2023]. In this paper we improve upon the results of Agarwal et al. [2023] in terms of the dimension factors as well as removing the requirement of smoothness. Our results are now the best known rates for DP-OCO in this regime. Our algorithms builds upon the work of [Asi et al., 2023] which introduced the idea of explicitly limiting the number of switches via rejection sampling. The main innovation in our algorithm is the use of sampling from a strongly log-concave density which allows us to trade-off the dimension factors better leading to improved results.
For a permutation $\pi: [K]\rightarrow [K]$, a sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $\pi$-pattern of size $K$, if there is a sequence of indices $(i_1, i_2, \cdots, i_K)$ ($i_1<i_2<\cdots<i_K$), satisfying that $f(i_a)<f(i_b)$ if $\pi(a)<\pi(b)$, for $a,b\in [K]$. Otherwise, $f$ is referred to as $\pi$-free. For the special case where $\pi = (1,2,\cdots, K)$, it is referred to as the monotone pattern. \cite{newman2017testing} initiated the study of testing $\pi$-freeness with one-sided error. They focused on two specific problems, testing the monotone permutations and the $(1,3,2)$ permutation. For the problem of testing monotone permutation $(1,2,\cdots,K)$, \cite{ben2019finding} improved the $(\log n)^{O(K^2)}$ non-adaptive query complexity of \cite{newman2017testing} to $O((\log n)^{\lfloor \log_{2} K\rfloor})$. Further, \cite{ben2019optimal} proposed an adaptive algorithm with $O(\log n)$ query complexity. However, no progress has yet been made on the problem of testing $(1,3,2)$-freeness. In this work, we present an adaptive algorithm for testing $(1,3,2)$-freeness. The query complexity of our algorithm is $O(\epsilon^{-2}\log^4 n)$, which significantly improves over the $O(\epsilon^{-7}\log^{26}n)$-query adaptive algorithm of \cite{newman2017testing}. This improvement is mainly achieved by the proposal of a new structure embedded in the patterns.
We study the two-player communication problem of determining whether two vertices $x, y$ are nearby in a graph $G$, with the goal of determining the graph structures that allow the problem to be solved with a constant-cost randomized protocol. Equivalently, we consider the problem of assigning constant-size random labels (sketches) to the vertices of a graph, which allow adjacency, exact distance thresholds, or approximate distance thresholds to be computed with high probability from the labels. Our main results are that, for monotone classes of graphs: constant-size adjacency sketches exist if and only if the class has bounded arboricity; constant-size sketches for exact distance thresholds exist if and only if the class has bounded expansion; constant-size approximate distance threshold (ADT) sketches imply that the class has bounded expansion; any class of constant expansion (i.e. any proper minor closed class) has constant-size ADT sketches; and a class may have arbitrarily small expansion without admitting constant-size ADT sketches.
Given a graph $G$, an integer $k\geq 0$, and a non-negative integral function $f:V(G) \rightarrow \mathcal{N}$, the {\sc Vector Domination} problem asks whether a set $S$ of vertices, of cardinality $k$ or less, exists in $G$ so that every vertex $v \in V(G)-S$ has at least $f(v)$ neighbors in $S$. The problem generalizes several domination problems and it has also been shown to generalize Bounded-Degree Vertex Deletion. In this paper, the parameterized version of Vector Domination is studied when the input graph is planar. A linear problem kernel is presented.
Cold-start problems are long-standing challenges for practical recommendations. Most existing recommendation algorithms rely on extensive observed data and are brittle to recommendation scenarios with few interactions. This paper addresses such problems using few-shot learning and meta learning. Our approach is based on the insight that having a good generalization from a few examples relies on both a generic model initialization and an effective strategy for adapting this model to newly arising tasks. To accomplish this, we combine the scenario-specific learning with a model-agnostic sequential meta-learning and unify them into an integrated end-to-end framework, namely Scenario-specific Sequential Meta learner (or s^2 meta). By doing so, our meta-learner produces a generic initial model through aggregating contextual information from a variety of prediction tasks while effectively adapting to specific tasks by leveraging learning-to-learn knowledge. Extensive experiments on various real-world datasets demonstrate that our proposed model can achieve significant gains over the state-of-the-arts for cold-start problems in online recommendation. Deployment is at the Guess You Like session, the front page of the Mobile Taobao.
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