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Relaxation methods such as Jacobi or Gauss-Seidel are often applied as smoothers in algebraic multigrid. Incomplete factorizations can also be employed, however, direct triangular solves are comparatively slow on GPUs. Previous work by Antz et al. \cite{Anzt2015} proposed an iterative approach for solving such sparse triangular systems. However, when using the stationary Jacobi iteration, if the upper or lower triangular factor is highly non-normal, the iterations will diverge. An ILUT smoother is introduced for classical Ruge-St\"uben C-AMG that applies Ruiz scaling to mitigate the non-normality of the upper triangular factor. Our approach facilitates the use of Jacobi iteration in place of the inherently sequential triangular solve. Because the scaling is applied to the upper triangular factor as opposed to the global matrix, it can be done locally on an MPI-rank for a diagonal block of the global matrix. A performance model is provided along with numerical results for matrices extracted from the PeleLM \cite{PeleLM} pressure continuity solver.

In 1961, Gomory and Hu showed that the max-flow values of all $n\choose 2$ pairs of vertices in an undirected graph can be computed using only $n-1$ calls to any max-flow algorithm, and succinctly represented them in a tree (called the Gomory-Hu tree later). Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$ for this problem; with current max-flow algorithms, the running time is $\tilde{O}(mn + n^{5/2})$. We break this 60-year old barrier by giving an $\tilde{O}(n^{2})$-time algorithm for the Gomory-Hu tree problem, already with current max-flow algorithms. For unweighted graphs, our techniques show equivalence (up to poly-logarithmic factors in running time) between Gomory-Hu tree (i.e., all-pairs max-flow values) and a single-pair max-flow.

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.

Comparator circuits are a natural circuit model for studying bounded fan-out computation whose power sits between nondeterministic branching programs and general circuits. Despite having been studied for nearly three decades, the first superlinear lower bound against comparator circuits was proved only recently by G\'al and Robere (ITCS 2020), who established a $\Omega((n/\log n)^{1.5})$ lower bound on the size of comparator circuits computing an explicit function of $n$ bits. In this paper, we initiate the study of average-case complexity and circuit analysis algorithms for comparator circuits. Departing from previous approaches, we exploit the technique of shrinkage under random restrictions to obtain a variety of new results for this model. Among them, we show - Average-case Lower Bounds. For every $k = k(n)$ with $k \geq \log n$, there exists a polynomial-time computable function $f_k$ on $n$ bits such that, for every comparator circuit $C$ with at most $n^{1.5}/O(k\cdot \sqrt{\log n})$ gates, we have \[ \text{Pr}_{x\in\left\{ 0,1 \right\}^n}\left[C(x)=f_k(x)\right]\leq \frac{1}{2} + \frac{1}{2^{\Omega(k)}}. \] This average-case lower bound matches the worst-case lower bound of G\'al and Robere by letting $k=O(\log n)$. - #SAT Algorithms. There is an algorithm that counts the number of satisfying assignments of a given comparator circuit with at most $n^{1.5}/O\!\left(k\cdot \sqrt{\log n}\right)$ gates, in time $2^{n-k}\cdot\text{poly}(n)$, for any $k\leq n/4$. The running time is non-trivial when $k=\omega(\log n)$. - Pseudorandom Generators and MCSP Lower Bounds. There is a pseudorandom generator of seed length $s^{2/3+o(1)}$ that fools comparator circuits with $s$ gates. Also, using this PRG, we obtain an $n^{1.5-o(1)}$ lower bound for MCSP against comparator circuits.

We consider the product of determinantal point processes (DPPs), a point process whose probability mass is proportional to the product of principal minors of multiple matrices, as a natural, promising generalization of DPPs. We study the computational complexity of computing its normalizing constant, which is among the most essential probabilistic inference tasks. Our complexity-theoretic results (almost) rule out the existence of efficient algorithms for this task unless the input matrices are forced to have favorable structures. In particular, we prove the following: (1) Computing $\sum_S\det({\bf A}_{S,S})^p$ exactly for every (fixed) positive even integer $p$ is UP-hard and Mod$_3$P-hard, which gives a negative answer to an open question posed by Kulesza and Taskar. (2) $\sum_S\det({\bf A}_{S,S})\det({\bf B}_{S,S})\det({\bf C}_{S,S})$ is NP-hard to approximate within a factor of $2^{O(|I|^{1-\epsilon})}$ or $2^{O(n^{1/\epsilon})}$ for any $\epsilon>0$, where $|I|$ is the input size and $n$ is the order of the input matrix. This result is stronger than the #P-hardness for the case of two matrices derived by Gillenwater. (3) There exists a $k^{O(k)}n^{O(1)}$-time algorithm for computing $\sum_S\det({\bf A}_{S,S})\det({\bf B}_{S,S})$, where $k$ is the maximum rank of $\bf A$ and $\bf B$ or the treewidth of the graph formed by nonzero entries of $\bf A$ and $\bf B$. Such parameterized algorithms are said to be fixed-parameter tractable. These results can be extended to the fixed-size case. Further, we present two applications of fixed-parameter tractable algorithms given a matrix $\bf A$ of treewidth $w$: (4) We can compute a $2^{\frac{n}{2p-1}}$-approximation to $\sum_S\det({\bf A}_{S,S})^p$ for any fractional number $p>1$ in $w^{O(wp)}n^{O(1)}$ time. (5) We can find a $2^{\sqrt n}$-approximation to unconstrained MAP inference in $w^{O(w\sqrt n)}n^{O(1)}$ time.

We present a general technique, based on parametric search with some twist, for solving a variety of optimization problems on a set of points in the plane or in higher dimensions. These problems include (i) the reverse shortest path problem in unit-disk graphs, recently studied by Wang and Zhao, (ii) the same problem for weighted unit-disk graphs, with a decision procedure recently provided by Wang and Xue, (iii) extensions of these problems to three and higher dimensions, (iv) the discrete Fr\'echet distance with one-sided shortcuts in higher dimensions, extending the study by Ben Avraham et al., and (v) the maximum-height independent towers problem, in which we want to erect vertical towers of maximum height over a 1.5-dimensional terrain so that no pair of tower tips are mutually visible. We obtain significantly improved solutions for problems (i) and (ii), and new efficient solutions to problems (iii), (iv) and (v), which do not appear to have been studied earlier.

Jump functions are the {most-studied} non-unimodal benchmark in the theory of randomized search heuristics, in particular, evolutionary algorithms (EAs). They have significantly improved our understanding of how EAs escape from local optima. However, their particular structure -- to leave the local optimum one can only jump directly to the global optimum -- raises the question of how representative such results are. For this reason, we propose an extended class $\textsc{Jump}_{k,\delta}$ of jump functions that contain a valley of low fitness of width $\delta$ starting at distance $k$ from the global optimum. We prove that several previous results extend to this more general class: for all {$k \le \frac{n^{1/3}}{\ln{n}}$} and $\delta < k$, the optimal mutation rate for the $(1+1)$~EA is $\frac{\delta}{n}$, and the fast $(1+1)$~EA runs faster than the classical $(1+1)$~EA by a factor super-exponential in $\delta$. However, we also observe that some known results do not generalize: the randomized local search algorithm with stagnation detection, which is faster than the fast $(1+1)$~EA by a factor polynomial in $k$ on $\textsc{Jump}_k$, is slower by a factor polynomial in $n$ on some $\textsc{Jump}_{k,\delta}$ instances. Computationally, the new class allows experiments with wider fitness valleys, especially when they lie further away from the global optimum.

We show that if a disc triangulation has all internal vertex degrees at least 6, then the full triangulation may be determined from the pairwise graph distance between boundary vertices. A similar result holds for quadrangulations with all internal degrees at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local non-positive curvature. However, we show that a natural conjecture for a "mixed" version of the two results is not true.

We consider mixed finite element methods with exact symmetric stress tensors. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.

This paper studies distributed binary test of statistical independence under communication (information bits) constraints. While testing independence is very relevant in various applications, distributed independence test is particularly useful for event detection in sensor networks where data correlation often occurs among observations of devices in the presence of a signal of interest. By focusing on the case of two devices because of their tractability, we begin by investigating conditions on Type I error probability restrictions under which the minimum Type II error admits an exponential behavior with the sample size. Then, we study the finite sample-size regime of this problem. We derive new upper and lower bounds for the gap between the minimum Type II error and its exponential approximation under different setups, including restrictions imposed on the vanishing Type I error probability. Our theoretical results shed light on the sample-size regimes at which approximations of the Type II error probability via error exponents became informative enough in the sense of predicting well the actual error probability. We finally discuss an application of our results where the gap is evaluated numerically, and we show that exponential approximations are not only tractable but also a valuable proxy for the Type II probability of error in the finite-length regime.