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When rows of an $n \times d$ matrix $A$ are given in a stream, we study algorithms for approximating the top eigenvector of the matrix ${A}^TA$ (equivalently, the top right singular vector of $A$). We consider worst case inputs $A$ but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter $R = \sigma_1(A)^2/\sigma_2(A)^2 = \Omega(1)$, then there is a randomized algorithm that uses $O(h \cdot d \cdot \operatorname{polylog}(d))$ bits of space and outputs a unit vector $v$ that has a correlation $1 - O(1/\sqrt{R})$ with the top eigenvector $v_1$. Here $h$ denotes the number of \emph{heavy rows} in the matrix, defined as the rows with Euclidean norm at least $\|{A}\|_F/\sqrt{d \cdot \operatorname{polylog}(d)}$. We also provide a lower bound showing that any algorithm using $O(hd/R)$ bits of space can obtain at most $1 - \Omega(1/R^2)$ correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the $R = \Omega(\log n \cdot \log d)$ requirement in a recent work of Price and Xun (FOCS 2024). We note that the algorithm of Price and Xun works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in their analysis can be brought down to $R = \Omega(\log^2 d)$ for arbitrary order streams and $R = \Omega(\log d)$ for random order streams. The requirement of $R = \Omega(\log d)$ for random order streams is nearly tight for their analysis as we obtain a simple instance with $R = \Omega(\log d/\log\log d)$ for which their algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector $v_1$.

We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $k\in\{3,4,5\}$, and by Fraigniaud et al. [PODC 2024] for $k\geq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what is done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the ''local density'' of graphs without $2k$-cycles, which we believe is interesting on its own.

We consider an economic environment with one buyer and one seller. For a bundle $(t,q)\in [0,\infty[\times [0,1]=\mathbb{Z}$, $q$ refers to the winning probability of an object, and $t$ denotes the payment that the buyer makes. We consider continuous and monotone preferences on $\mathbb{Z}$ as the primitives of the buyer. These preferences can incorporate both quasilinear and non-quasilinear preferences, and multidimensional pay-off relevant parameters. We define rich single-crossing subsets of this class and characterize strategy-proof mechanisms by using monotonicity of the mechanisms and continuity of the indirect preference correspondences. We also provide a computationally tractable optimization program to compute the optimal mechanism for mechanisms with finite range. We do not use revenue equivalence and virtual valuations as tools in our proofs. Our proof techniques bring out the geometric interaction between the single-crossing property and the positions of bundles $(t,q)$s in the space $\mathbb{Z}$. We also provide an extension of our analysis to an $n-$buyer environment, and to the situation where $q$ is a qualitative variable.

We study frequency domain electromagnetic scattering at a bounded, penetrable, and inhomogeneous obstacle $ \Omega \subset \mathbb{R}^3 $. From the Stratton-Chu integral representation, we derive a new representation formula when constant reference coefficients are given for the interior domain. The resulting integral representation contains the usual layer potentials, but also volume potentials on $\Omega$. Then it is possible to follow a single-trace approach to obtain boundary integral equations perturbed by traces of compact volume integral operators with weakly singular kernels. The coupled boundary and volume integral equations are discretized with a Galerkin approach with usual Curl-conforming and Div-conforming finite elements on the boundary and in the volume. Compression techniques and special quadrature rules for singular integrands are required for an efficient and accurate method. Numerical experiments provide evidence that our new formulation enjoys promising properties.

Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension $n$ [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, then the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper gives another supportive result for this conjecture by showing that translational tiling of the $4$-dimensional space with a set of three connected tiles is undecidable.

Let $P$ be a set of $n$ points in $\mathbb{R}^d$, and let $\varepsilon,\psi \in (0,1)$ be parameters. Here, we consider the task of constructing a $(1+\varepsilon)$-spanner for $P$, where every edge might fail (independently) with probability $1-\psi$. For example, for $\psi=0.1$, about $90\%$ of the edges of the graph fail. Nevertheless, we show how to construct a spanner that survives such a catastrophe with near linear number of edges. The measure of reliability of the graph constructed is how many pairs of vertices lose $(1+\varepsilon)$-connectivity. Surprisingly, despite the spanner constructed being of near linear size, the number of failed pairs is close to the number of failed pairs if the underlying graph was a clique. Specifically, we show how to construct such an exact dependable spanner in one dimension of size $O(\tfrac{n}{\psi} \log n)$, which is optimal. Next, we build an $(1+\varepsilon)$-spanners for a set $P \subseteq \mathbb{R}^d$ of $n$ points, of size $O( C n \log n )$, where $C \approx 1/\bigl(\varepsilon^{d} \psi^{4/3}\bigr)$. Surprisingly, these new spanners also have the property that almost all pairs of vertices have a $\leq 4$-hop paths between them realizing this short path.

We study the problem of privately releasing an approximate minimum spanning tree (MST). Given a graph $G = (V, E, \vec{W})$ where $V$ is a set of $n$ vertices, $E$ is a set of $m$ undirected edges, and $ \vec{W} \in \mathbb{R}^{|E|} $ is an edge-weight vector, our goal is to publish an approximate MST under edge-weight differential privacy, as introduced by Sealfon in PODS 2016, where $V$ and $E$ are considered public and the weight vector is private. Our neighboring relation is $\ell_\infty$-distance on weights: for a sensitivity parameter $\Delta_\infty$, graphs $ G = (V, E, \vec{W}) $ and $ G' = (V, E, \vec{W}') $ are neighboring if $\|\vec{W}-\vec{W}'\|_\infty \leq \Delta_\infty$. Existing private MST algorithms face a trade-off, sacrificing either computational efficiency or accuracy. We show that it is possible to get the best of both worlds: With a suitable random perturbation of the input that does not suffice to make the weight vector private, the result of any non-private MST algorithm will be private and achieves a state-of-the-art error guarantee. Furthermore, by establishing a connection to Private Top-k Selection [Steinke and Ullman, FOCS '17], we give the first privacy-utility trade-off lower bound for MST under approximate differential privacy, demonstrating that the error magnitude, $\tilde{O}(n^{3/2})$, is optimal up to logarithmic factors. That is, our approach matches the time complexity of any non-private MST algorithm and at the same time achieves optimal error. We complement our theoretical treatment with experiments that confirm the practicality of our approach.

We consider maximizing an unknown monotonic, submodular set function $f: 2^{[n]} \rightarrow [0,1]$ with cardinality constraint under stochastic bandit feedback. At each time $t=1,\dots,T$ the learner chooses a set $S_t \subset [n]$ with $|S_t| \leq k$ and receives reward $f(S_t) + \eta_t$ where $\eta_t$ is mean-zero sub-Gaussian noise. The objective is to minimize the learner's regret with respect to an approximation of the maximum $f(S_*)$ with $|S_*| = k$, obtained through robust greedy maximization of $f$. To date, the best regret bound in the literature scales as $k n^{1/3} T^{2/3}$. And by trivially treating every set as a unique arm one deduces that $\sqrt{ {n \choose k} T }$ is also achievable using standard multi-armed bandit algorithms. In this work, we establish the first minimax lower bound for this setting that scales like $\tilde{\Omega}(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $\tilde{\Omega}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. Moreover, we propose an algorithm Sub-UCB that achieves regret $\tilde{\mathcal{O}}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$ capable of matching the lower bound on regret for the restricted class up to logarithmic factors.

We consider temporal numeric planning problems $\Pi$ expressed in PDDL2.1 level 3, and show how to produce SMT formulas $(i)$ whose models correspond to valid plans of $\Pi$, and $(ii)$ that extend the recently proposed planning with patterns approach from the numeric to the temporal case. We prove the correctness and completeness of the approach and show that it performs very well on 10 domains with required concurrency.

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove that for every prime $p$ there exists some $\delta = \delta(p)>0$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k$, there exists a $k$-nearly orthogonal set of at least $d^{\delta \cdot k/\log k}$ vectors of $\mathbb{F}^d$. The size of the set is optimal up to the $\log k$ term in the exponent. We further prove two extensions of this result. In the first, we provide a large set ${\cal A}$ of non-self-orthogonal vectors of $\mathbb{F}^d$ such that for every two subsets of ${\cal A}$ of size $k+1$ each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every $k+1$ vectors of the produced set ${\cal A}$ include $\ell+1$ pairwise orthogonal vectors for an arbitrary fixed integer $1 \leq \ell \leq k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.