Uniform sampling from the set $\mathcal{G}(\mathbf{d})$ of graphs with a given degree-sequence $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ is a classical problem in the study of random graphs. We consider an analogue for temporal graphs in which the edges are labeled with integer timestamps. The input to this generation problem is a tuple $\mathbf{D} = (\mathbf{d}, T) \in \mathbb N^n \times \mathbb N_{>0}$ and the task is to output a uniform random sample from the set $\mathcal{G}(\mathbf{D})$ of temporal graphs with degree-sequence $\mathbf{d}$ and timestamps in the interval $[1, T]$. By allowing repeated edges with distinct timestamps, $\mathcal{G}(\mathbf{D})$ can be non-empty even if $\mathcal{G}(\mathbf{d})$ is, and as a consequence, existing algorithms are difficult to apply. We describe an algorithm for this generation problem which runs in expected time $O(M)$ if $\Delta^{2+\epsilon} = O(M)$ for some constant $\epsilon > 0$ and $T - \Delta = \Omega(T)$ where $M = \sum_i d_i$ and $\Delta = \max_i d_i$. Our algorithm applies the switching method of McKay and Wormald $[1]$ to temporal graphs: we first generate a random temporal multigraph and then remove self-loops and duplicated edges with switching operations which rewire the edges in a degree-preserving manner.
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Let $\Omega = [0,1]^d$ be the unit cube in $\mathbb{R}^d$. We study the problem of how efficiently, in terms of the number of parameters, deep neural networks with the ReLU activation function can approximate functions in the Sobolev spaces $W^s(L_q(\Omega))$ and Besov spaces $B^s_r(L_q(\Omega))$, with error measured in the $L_p(\Omega)$ norm. This problem is important when studying the application of neural networks in a variety of fields, including scientific computing and signal processing, and has previously been solved only when $p=q=\infty$. Our contribution is to provide a complete solution for all $1\leq p,q\leq \infty$ and $s > 0$ for which the corresponding Sobolev or Besov space compactly embeds into $L_p$. The key technical tool is a novel bit-extraction technique which gives an optimal encoding of sparse vectors. This enables us to obtain sharp upper bounds in the non-linear regime where $p > q$. We also provide a novel method for deriving $L_p$-approximation lower bounds based upon VC-dimension when $p < \infty$. Our results show that very deep ReLU networks significantly outperform classical methods of approximation in terms of the number of parameters, but that this comes at the cost of parameters which are not encodable.
The Euclidean algorithm is one of the oldest algorithms known to mankind. Given two integral numbers $a_1$ and $a_2$, it computes the greatest common divisor (gcd) of $a_1$ and $a_2$ in a very elegant way. From a lattice perspective, it computes a basis of the sum of two one-dimensional lattices $a_1 \mathbb{Z}$ and $a_2 \mathbb{Z}$ as $\gcd(a_1,a_2) \mathbb{Z} = a_1 \mathbb{Z} + a_2 \mathbb{Z}$. In this paper, we show that the classical Euclidean algorithm can be adapted in a very natural way to compute a basis of a general lattice $L(a_1, \ldots , a_m)$ given vectors $a_1, \ldots , a_m \in \mathbb{Z}^n$ with $m> \mathrm{rank}(a_1, \ldots ,a_m)$. Similar to the Euclidean algorithm, our algorithm is very easy to describe and implement and can be written within 12 lines of pseudocode. While the Euclidean algorithm halves the largest number in every iteration, our generalized algorithm halves the determinant of a full rank subsystem leading to at most $\log (\det B)$ many iterations, for some initial subsystem $B$. Therefore, we can compute a basis of the lattice using at most $\tilde{O}((m-n)n\log(\det B) + mn^{\omega-1}\log(||A||_\infty))$ arithmetic operations, where $\omega$ is the matrix multiplication exponent and $A = (a_1, \ldots, a_m)$. Even using the worst case Hadamard bound for the determinant, our algorithm improves upon existing algorithm. Another major advantage of our algorithm is that we can bound the entries of the resulting lattice basis by $\tilde{O}(n^2\cdot ||A||_{\infty})$ using a simple pivoting rule. This is in contrast to the typical approach for computing lattice basis, where the Hermite normal form (HNF) is used. In the HNF, entries can be as large as the determinant and hence can only be bounded by an exponential term.
Given two $n$-element structures, $\mathcal{A}$ and $\mathcal{B}$, which can be distinguished by a sentence of $k$-variable first-order logic ($\mathcal{L}^k$), what is the minimum $f(n)$ such that there is guaranteed to be a sentence $\phi \in \mathcal{L}^k$ with at most $f(n)$ quantifiers, such that $\mathcal{A} \models \phi$ but $\mathcal{B} \not \models \phi$? We will present various results related to this question obtained by using the recently introduced QVT games. In particular, we show that when we limit the number of variables, there can be an exponential gap between the quantifier depth and the quantifier number needed to separate two structures. Through the lens of this question, we will highlight some difficulties that arise in analysing the QVT game and some techniques which can help to overcome them. We also show, in the setting of the existential-positive fragment, how to lift quantifier depth lower bounds to quantifier number lower bounds. This leads to almost tight bounds.
Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where $\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of $\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$ such that each subset in $\mathcal{S}$ contains exactly one element in $\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as exactly hittable interval graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.
We consider geometric problems on planar $n^2$-point sets in the congested clique model. Initially, each node in the $n$-clique network holds a batch of $n$ distinct points in the Euclidean plane given by $O(\log n)$-bit coordinates. In each round, each node can send a distinct $O(\log n)$-bit message to each other node in the clique and perform unlimited local computations. We show that the convex hull of the input $n^2$-point set can be constructed in $O(\min\{ h,\log n\})$ rounds, where $h$ is the size of the hull, on the congested clique. We also show that a triangulation of the input $n^2$-point set can be constructed in $O(\log^2n)$ rounds on the congested clique. Finally, we demonstrate that the Voronoi diagram of $n^2$ points with $O(\log n)$-bit coordinates drawn uniformly at random from a unit square can be computed within the square with high probability in $O(1)$ rounds on the congested clique.
For an angle $\alpha\in (0,\pi)$, we consider plane graphs and multigraphs in which the edges are either (i) one-bend polylines with an angle $\alpha$ between the two edge segments, or (ii) circular arcs of central angle $2(\pi-\alpha)$. We derive upper and lower bounds on the maximum density of such graphs in terms of $\alpha$. As an application, we improve upon bounds for the number of edges in $\alpha AC_1^=$ graphs (i.e., graphs that can be drawn in the plane with one-bend edges such that any two crossing edges meet at angle $\alpha$). This is the first improvement on the size of $\alpha AC_1^=$ graphs in over a decade.
Linear regression is one of the most fundamental linear algebra problems. Given a dense matrix $A \in \mathbb{R}^{n \times d}$ and a vector $b$, the goal is to find $x'$ such that $ \| Ax' - b \|_2^2 \leq (1+\epsilon) \min_{x} \| A x - b \|_2^2 $. The best classical algorithm takes $O(nd) + \mathrm{poly}(d/\epsilon)$ time [Clarkson and Woodruff STOC 2013, Nelson and Nguyen FOCS 2013]. On the other hand, quantum linear regression algorithms can achieve exponential quantum speedups, as shown in [Wang Phys. Rev. A 96, 012335, Kerenidis and Prakash ITCS 2017, Chakraborty, Gily{\'e}n and Jeffery ICALP 2019]. However, the running times of these algorithms depend on some quantum linear algebra-related parameters, such as $\kappa(A)$, the condition number of $A$. In this work, we develop a quantum algorithm that runs in $\widetilde{O}(\epsilon^{-1}\sqrt{n}d^{1.5}) + \mathrm{poly}(d/\epsilon)$ time. It provides a quadratic quantum speedup in $n$ over the classical lower bound without any dependence on data-dependent parameters. In addition, we also show our result can be generalized to multiple regression and ridge linear regression.
The \emph{local edge-length ratio} of a planar straight-line drawing $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$ that share a common vertex. The \emph{global edge-length ratio} of $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$. The local (global) edge-length ratio of a planar graph is the infimum over all local (global) edge-length ratios of its planar straight-line drawings. We show that there exist planar graphs with $n$ vertices whose local edge-length ratio is $\Omega(\sqrt{n})$. We then show a technique to establish upper bounds on the global (and hence local) edge-length ratio of planar graphs and~apply~it to Halin graphs and to other families of graphs having outerplanarity two.
A set of vectors $S \subseteq \mathbb{R}^d$ is $(k_1,\varepsilon)$-clusterable if there are $k_1$ balls of radius $\varepsilon$ that cover $S$. A set of vectors $S \subseteq \mathbb{R}^d$ is $(k_2,\delta)$-far from being clusterable if there are at least $k_2$ vectors in $S$, with all pairwise distances at least $\delta$. We propose a probabilistic algorithm to distinguish between these two cases. Our algorithm reaches a decision by only looking at the extreme values of a scalar valued hash function, defined by a random field, on $S$; hence, it is especially suitable in distributed and online settings. An important feature of our method is that the algorithm is oblivious to the number of vectors: in the online setting, for example, the algorithm stores only a constant number of scalars, which is independent of the stream length. We introduce random field hash functions, which are a key ingredient in our paradigm. Random field hash functions generalize locality-sensitive hashing (LSH). In addition to the LSH requirement that ``nearby vectors are hashed to similar values", our hash function also guarantees that the ``hash values are (nearly) independent random variables for distant vectors". We formulate necessary conditions for the kernels which define the random fields applied to our problem, as well as a measure of kernel optimality, for which we provide a bound. Then, we propose a method to construct kernels which approximate the optimal one.
Given an undirected possibly weighted $n$-vertex graph $G=(V,E)$ and a set $\mathcal{P}\subseteq V^2$ of pairs, a subgraph $S=(V,E')$ is called a ${\cal P}$-pairwise $\alpha$-spanner of $G$, if for every pair $(u,v)\in\mathcal{P}$ we have $d_S(u,v)\leq\alpha\cdot d_G(u,v)$. The parameter $\alpha$ is called the stretch of the spanner, and its size overhead is define as $\frac{|E'|}{|{\cal P}|}$. A surprising connection was recently discussed between the additive stretch of $(1+\epsilon,\beta)$-spanners, to the hopbound of $(1+\epsilon,\beta)$-hopsets. A long sequence of works showed that if the spanner/hopset has size $\approx n^{1+1/k}$ for some parameter $k\ge 1$, then $\beta\approx\left(\frac1\epsilon\right)^{\log k}$. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if $|{\cal P}|\approx n^{1+1/k}$, then a ${\cal P}$-pairwise $(1+\epsilon)$-spanner must have size at least $\beta\cdot |{\cal P}|$ with $\beta\approx\left(\frac1\epsilon\right)^{\log k}$ (a near matching upper bound was recently shown in \cite{ES23}). We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for ${\cal P}$-pairwise $\alpha$-spanners. In particular, we show that if $|{\cal P}|\approx n^{1+1/k}$, then the size overhead is $\beta\approx\frac k\alpha$. A source-wise spanner is a special type of pairwise spanner, for which ${\cal P}=A\times V$ for some $A\subseteq V$. A prioritized spanner is given also a ranking of the vertices $V=(v_1,\dots,v_n)$, and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions, we improve on the state-of-the-art results for source-wise and prioritized spanners.
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