We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover ($cc$) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem $cc$ is at most the number of taxa, in fractional hypertreewidth $cc$ is at most the number of hyperedges, and in treewidth of Bayesian networks $cc$ is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most $2^{cc}$, the number of potential maximal cliques is at most $3^{cc}$, and these objects can be listed in times $O^*(2^{cc})$ and $O^*(3^{cc})$, respectively, even when no edge clique cover is given as input; the $O^*(\cdot)$ notation omits factors polynomial in the input size. These enumeration algorithms imply $O^*(3^{cc})$ time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give $O^*(4^m)$ time and $O^*(3^m)$ time algorithms, respectively, where $m$ is the number of hyperedges. When an edge clique cover of size $cc'$ is given as a part of the input we give $O^*(2^{cc'})$ time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an $O^*(2^n)$ time algorithm for perfect phylogeny, where $n$ is the number of taxa. We also give polynomial space algorithms with time complexities $O^*(9^{cc'})$ and $O^*(9^{cc + O(\log^2 cc)})$ for problems in this framework.
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We consider the problem of enumerating optimal solutions for two hypergraph $k$-partitioning problems -- namely, Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. The input in hypergraph $k$-partitioning problems is a hypergraph $G=(V, E)$ with positive hyperedge costs along with a fixed positive integer $k$. The goal is to find a partition of $V$ into $k$ non-empty parts $(V_1, V_2, \ldots, V_k)$ -- known as a $k$-partition -- so as to minimize an objective of interest. 1. If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-$k$-Partition. A subset of hyperedges is a minmax-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Minmax-Hypergraph-$k$-Partition. 2. If the objective of interest is the total cost of hyperedges crossing the $k$-partition, then the problem is known as Hypergraph-$k$-Cut. A subset of hyperedges is a min-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Hypergraph-$k$-Cut. We give the first polynomial bound on the number of minmax-$k$-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed $k$. Our technique is strong enough to also enable an $n^{O(k)}p$-time deterministic algorithm to enumerate all min-$k$-cut-sets in hypergraphs, thus improving on the previously known $n^{O(k^2)}p$-time deterministic algorithm, where $n$ is the number of vertices and $p$ is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).
The Schrijver graph $S(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $\{1,2,\ldots,n\}$ that do not include two consecutive elements modulo $n$, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of $S(n,k)$ is $n-2k+2$ (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of $S(n,k)$ with $n-2k+1$ colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class $\mathsf{PPA}$. We prove that it can be solved by a randomized algorithm with running time $n^{O(1)} \cdot k^{O(k)}$, hence it is fixed-parameter tractable with respect to the parameter $k$.
We formulate the quadratic eigenvalue problem underlying the mathematical model of a linear vibrational system as an eigenvalue problem of a diagonal-plus-low-rank matrix $A$. The eigenvector matrix of $A$ has a Cauchy-like structure. Optimal viscosities are those for which $trace(X)$ is minimal, where $X$ is the solution of the Lyapunov equation $AX+XA^{*}=GG^{*}$. Here $G$ is a low-rank matrix which depends on the eigenfrequencies that need to be damped. After initial eigenvalue decomposition of linearized problem which requires $O(n^3)$ operations, our algorithm computes optimal viscosities for each choice of external dampers in $O(n^2)$ operations, provided that the number of dampers is small. Hence, the subsequent optimization is order of magnitude faster than in the standard approach which solves Lyapunov equation in each step, thus requiring $O(n^3)$ operations. Our algorithm is based on $O(n^2)$ eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast $O(n^2)$ multiplication of linked Cauchy-like matrices.
We investigate optimal execution problems with instantaneous price impact and stochastic resilience. First, in the setting of linear price impact function we derive a closed-form recursion for the optimal strategy, generalizing previous results with deterministic transient price impact. Second, we develop a numerical algorithm for the case of nonlinear price impact. We utilize an actor-critic framework that constructs two neural-network surrogates for the value function and the feedback control. One advantage of such functional approximators is the ability to do parametric learning, i.e. to incorporate some of the model parameters as part of the input space. Precise calibration of price impact, resilience, etc., is known to be extremely challenging and hence it is critical to understand sensitivity of the strategy to these parameters. Our parametric neural network (NN) learner organically scales across 3-6 input dimensions and is shown to accurately approximate optimal strategy across a range of parameter configurations. We provide a fully reproducible Jupyter Notebook with our NN implementation, which is of independent pedagogical interest, demonstrating the ease of use of NN surrogates in (parametric) stochastic control problems.
The success of large-scale models in recent years has increased the importance of statistical models with numerous parameters. Several studies have analyzed over-parameterized linear models with high-dimensional data that may not be sparse; however, existing results depend on the independent setting of samples. In this study, we analyze a linear regression model with dependent time series data under over-parameterization settings. We consider an estimator via interpolation and developed a theory for excess risk of the estimator under multiple dependence types. This theory can treat infinite-dimensional data without sparsity and handle long-memory processes in a unified manner. Moreover, we bound the risk in our theory via the integrated covariance and nondegeneracy of autocorrelation matrices. The results show that the convergence rate of risks with short-memory processes is identical to that of cases with independent data, while long-memory processes slow the convergence rate. We also present several examples of specific dependent processes that can be applied to our setting.
Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.
In this short note, we show that for any $\epsilon >0$ and $k<n^{0.5-\epsilon}$ the choice number of the Kneser graph $KG_{n,k}$ is $\Theta (n\log n)$.
Extracting non-Gaussian information from the non-linear regime of structure formation is key to fully exploiting the rich data from upcoming cosmological surveys probing the large-scale structure of the universe. However, due to theoretical and computational complexities, this remains one of the main challenges in analyzing observational data. We present a set of summary statistics for cosmological matter fields based on 3D wavelets to tackle this challenge. These statistics are computed as the spatial average of the complex modulus of the 3D wavelet transform raised to a power $q$ and are therefore known as invariant wavelet moments. The 3D wavelets are constructed to be radially band-limited and separable on a spherical polar grid and come in three types: isotropic, oriented, and harmonic. In the Fisher forecast framework, we evaluate the performance of these summary statistics on matter fields from the Quijote suite, where they are shown to reach state-of-the-art parameter constraints on the base $\Lambda$CDM parameters, as well as the sum of neutrino masses. We show that we can improve constraints by a factor 5 to 10 in all parameters with respect to the power spectrum baseline.
For a given nonnegative matrix $A=(A_{ij})$, the matrix scaling problem asks whether $A$ can be scaled to a doubly stochastic matrix $XAY$ for some positive diagonal matrices $X,Y$. The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization $A_{ij} \leftarrow A_{ij}/\sum_{j}A_{ij}$ and column-normalization $A_{ij} \leftarrow A_{ij}/\sum_{i}A_{ij}$ alternatively. By this algorithm, $A$ converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with $A$ has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph $G$, which is identified with the $0,1$-matrix $A_G$. Linial, Samorodnitsky, and Wigderson showed that a polynomial number of the Sinkhorn iterations for $A_G$ decides whether $G$ has a perfect matching. In this paper, we show an extension of this result: If $G$ has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker -- a certificate of the nonexistence of a perfect matching. Our analysis is based on an interpretation of the Sinkhorn algorithm as alternating KL-divergence minimization (Csisz\'{a}r and Tusn\'{a}dy 1984, Gietl and Reffel 2013) and its limiting behavior for a nonscalable matrix (Aas 2014). We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage-Mendelsohn decomposition of a bipartite graph.
This paper takes a different approach for the distributed linear parameter estimation over a multi-agent network. The parameter vector is considered to be stochastic with a Gaussian distribution. The sensor measurements at each agent are linear and corrupted with additive white Gaussian noise. Under such settings, this paper presents a novel distributed estimation algorithm that fuses the the concepts of consensus and innovations by incorporating the consensus terms (of neighboring estimates) into the innovation terms. Under the assumption of distributed parameter observability, introduced in this paper, we design the optimal gain matrices such that the distributed estimates are consistent and achieves fast convergence.
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