We study a fundamental model of online preference aggregation, where an algorithm maintains an ordered list of $n$ elements. An input is a stream of preferred sets $R_1, R_2, \dots, R_t, \dots$. Upon seeing $R_t$ and without knowledge of any future sets, an algorithm has to rerank elements (change the list ordering), so that at least one element of $R_t$ is found near the list front. The incurred cost is a sum of the list update costs (the number of swaps of neighboring list elements) and access costs (position of the first element of $R_t$ on the list). This scenario occurs naturally in applications such as ordering items in an online shop using aggregated preferences of shop customers. The theoretical underpinning of this problem is known as Min-Sum Set Cover. Unlike previous work (Fotakis et al., ICALP 2020, NIPS 2020) that mostly studied the performance of an online algorithm ALG against the static optimal solution (a single optimal list ordering), in this paper, we study an arguably harder variant where the benchmark is the provably stronger optimal dynamic solution OPT (that may also modify the list ordering). In terms of an online shop, this means that the aggregated preferences of its user base evolve with time. We construct a computationally efficient randomized algorithm whose competitive ratio (ALG-to-OPT cost ratio) is $O(r^2)$ and prove the existence of a deterministic $O(r^4)$-competitive algorithm. Here, $r$ is the maximum cardinality of sets $R_t$. This is the first algorithm whose ratio does not depend on $n$: the previously best algorithm for this problem was $O(r^{3/2} \cdot \sqrt{n})$-competitive and $\Omega(r)$ is a lower bound on the performance of any deterministic online algorithm.
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For a finite set of balls of radius $r$, the $k$-fold cover is the space covered by at least $k$ balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the $k$-fold filtration of the centers. For $k=1$, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger $k$, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the $k$-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case $k=1$, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points. Our method also extends to the multicover bifiltration, composed of the $k$-fold filtrations for several values of $k$, with the same size and complexity bounds.
In probabilistic structural mechanics, the Hasofer-Lind reliability index problem is a paradigmatic equality constrained problem of searching for the minimum distance from a point to a surface. In practical engineering problems, such surface is defined implicitly, requiring the solution of a boundary-value problem. Recently, it was proposed in the literature a hybrid micro-genetic algorithm (HmGA), with mixed real-binary genotype and novel deterministic operators for equality-constraint handling, namely the Genetic Repair and Region Zooming mechanisms (G. das Neves Carneiro and C. Concei\c{c}\~ao Ant\'onio, "Global optimal reliability index of implicit composite laminate structures by evolutionary algorithms", Struct Saf, vol. 79, pp. 54-65, 2019). We investigate the limit-behavior of the HmGA and present the convergence theorems for the algorithm. It is proven that Genetic Repair is a conditionally stable mechanism, and its modes of convergence are discussed. Based on a Markov chain analysis, the conditions for the convergence with probability 1 of the HmGA are given and discussed.
Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to capture natural constraints of those optimization problems by combining the information provided by two colorings: a vertex coloring of a graph and the induced edge coloring on a subgraph of its complement; the goal is to minimize, among all proper colorings of the graph for a fixed number of colors, the number of edges in the subgraph with the endpoints of the same color. The study of the robust coloring model has been focused on the search for heuristics due to its NP-hard character when using at least three colors, but little progress has been made in other directions. We present a new approach on the problem obtaining the first collection of non-heuristic results for general graphs; among them, we prove that robust coloring is the model that better approaches the equitable partition of the vertex set, even when the graph does not admit a so-called \emph{equitable coloring}. We also show the NP-completeness of its decision problem for the unsolved case of two colors, obtain bounds on the associated robust coloring parameter, and solve a conjecture on paths that illustrates the complexity of studying this coloring model.
Given a set $P$ of $n$ weighted points and a set $S$ of $m$ disks in the plane, the hitting set problem is to compute a subset $P'$ of points of $P$ such that each disk contains at least one point of $P'$ and the total weight of all points of $P'$ is minimized. The problem is known to be NP-hard. In this paper, we consider a line-constrained version of the problem in which all disks are centered on a line $\ell$. We present an $O((m+n)\log(m+n)+\kappa \log m)$ time algorithm for the problem, where $\kappa$ is the number of pairs of disks that intersect. For the unit-disk case where all disks have the same radius, the running time can be reduced to $O((n + m)\log(m + n))$. In addition, we solve the problem in $O((m + n)\log(m + n))$ time in the $L_{\infty}$ and $L_1$ metrics, in which a disk is a square and a diamond, respectively. Our techniques can also be used to solve other geometric hitting set problems. For example, given in the plane a set $P$ of $n$ weighted points and a set $S$ of $n$ half-planes, we solve in $O(n^4\log n)$ time the problem of finding a minimum weight hitting set of $P$ for $S$. This improves the previous best algorithm of $O(n^6)$ time by nearly a quadratic factor.
This article presents an approach for modelling hysteresis in piezoelectric materials, that leverages recent advancements in machine learning, particularly in sparse-regression techniques. While sparse regression has previously been used to model various scientific and engineering phenomena, its application to nonlinear hysteresis modelling in piezoelectric materials has yet to be explored. The study employs the least-squares algorithm with a sequential threshold to model the dynamic system responsible for hysteresis, resulting in a concise model that accurately predicts hysteresis for both simulated and experimental piezoelectric material data. Several numerical experiments are performed, including learning butterfly-shaped hysteresis and modelling real-world hysteresis data for a piezoelectric actuator. The presented approach is compared to traditional regression-based and neural network methods, demonstrating its efficiency and robustness. Source code is available at //github.com/chandratue/SmartHysteresis
We introduce a new kernelization tool, called rainbow matching technique}, that is appropriate for the design of polynomial kernels for packing problems and their hitting counterparts. Our technique capitalizes on the powerful combinatorial results of [Graf, Harris, Haxell, SODA 2021]. We apply the rainbow matching technique on four (di)graph packing or hitting problems, namely the Triangle-Packing in Tournament problem (TPT), where we ask for a packing of $k$ directed triangles in a tournament, Directed Feedback Vertex Set in Tournament problem (FVST), where we ask for a (hitting) set of at most $k$ vertices which intersects all triangles of a tournament, the Induced 2-Path-Packing (IPP) where we ask for a packing of $k$ induced paths of length two in a graph and Induced 2-Path Hitting Set problem (IPHS), where we ask for a (hitting) set of at most $k$ vertices which intersects all induced paths of length two in a graph. The existence of a sub-quadratic kernels for these problems was proven for the first time in [Fomin, Le, Lokshtanov, Saurabh, Thomass\'e, Zehavi. ACM Trans. Algorithms, 2019], where they gave a kernel of $O(k^{3/2})$ vertices for the two first problems and $O(k^{5/3})$ vertices for the two last. In the same paper it was questioned whether these bounds can be (optimally) improved to linear ones. Motivated by this question, we apply the rainbow matching technique and prove that TPT and FVST admit (almost linear) kernels of $k^{1+\frac{O(1)}{\sqrt{\log{k}}}}$ vertices and that IPP and IPHS admit kernels of $O(k)$ vertices.
Discrepancy theory provides powerful tools for producing higher-quality objects which "beat the union bound" in fundamental settings throughout combinatorics and computer science. However, this quality has often come at the price of more expensive algorithms. We introduce a new framework for bridging this gap, by allowing for the efficient implementation of discrepancy-theoretic primitives. Our framework repeatedly solves regularized optimization problems to low accuracy to approximate the partial coloring method of [Rot17], and simplifies and generalizes recent work of [JSS23] on fast algorithms for Spencer's theorem. In particular, our framework only requires that the discrepancy body of interest has exponentially large Gaussian measure and is expressible as a sublevel set of a symmetric, convex function. We combine this framework with new tools for proving Gaussian measure lower bounds to give improved algorithms for a variety of sparsification and coloring problems. As a first application, we use our framework to obtain an $\widetilde{O}(m \cdot \epsilon^{-3.5})$ time algorithm for constructing an $\epsilon$-approximate spectral sparsifier of an $m$-edge graph, matching the sparsity of [BSS14] up to constant factors and improving upon the $\widetilde{O}(m \cdot \epsilon^{-6.5})$ runtime of [LeeS17]. We further give a state-of-the-art algorithm for constructing graph ultrasparsifiers and an almost-linear time algorithm for constructing linear-sized degree-preserving sparsifiers via discrepancy theory; in the latter case, such sparsifiers were not known to exist previously. We generalize these results to their analogs in sparsifying isotropic sums of positive semidefinite matrices. Finally, to demonstrate the versatility of our technique, we obtain a nearly-input-sparsity time constructive algorithm for Spencer's theorem (where we recover a recent result of [JSS23]).
Kernel two-sample tests have been widely used for multivariate data in testing equal distribution. However, existing tests based on mapping distributions into a reproducing kernel Hilbert space are mainly targeted at specific alternatives and do not work well for some scenarios when the dimension of the data is moderate to high due to the curse of dimensionality. We propose a new test statistic that makes use of a common pattern under moderate and high dimensions and achieves substantial power improvements over existing kernel two-sample tests for a wide range of alternatives. We also propose alternative testing procedures that maintain high power with low computational cost, offering easy off-the-shelf tools for large datasets. The new approaches are compared to other state-of-the-art tests under various settings and show good performance. The new approaches are illustrated on two applications: The comparison of musks and non-musks using the shape of molecules, and the comparison of taxi trips started from John F.Kennedy airport in consecutive months. All proposed methods are implemented in an R package kerTests.
As social issues related to gender bias attract closer scrutiny, accurate tools to determine the gender profile of large groups become essential. When explicit data is unavailable, gender is often inferred from names. Current methods follow a strategy whereby individuals of the group, one by one, are assigned a gender label or probability based on gender-name correlations observed in the population at large. We show that this strategy is logically inconsistent and has practical shortcomings, the most notable of which is the systematic underestimation of gender bias. We introduce a global inference strategy that estimates gender composition according to the context of the full list of names. The tool suffers from no intrinsic methodological effects, is robust against errors, easily implemented, and computationally light.
Erickson defined the fusible numbers as a set $\mathcal F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $\mathcal F$ is well ordered, with order type $\varepsilon_0$. They also investigated a recursively defined function $M\colon \mathbb{R}\to\mathbb{R}$. They showed that the set of points of discontinuity of $M$ is a subset of $\mathcal F$ of order type $\varepsilon_0$. They also showed that, although $M$ is a total function on $\mathbb R$, the fact that the restriction of $M$ to $\mathbb{Q}$ is total is not provable in first-order Peano arithmetic $\mathsf{PA}$. In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets $\mathcal F$ of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function $g:\mathbb R^n\to\mathbb R$. The most straightforward generalization of $\frac{x+y+1}{2}$ to an $n$-ary function is the function $\frac{x_1+\cdots+x_n+1}{n}$. We show that this function generates a set $\mathcal F_n$ whose order type is just $\varphi_{n-1}(0)$. For this, we develop recursively defined functions $M_n\colon \mathbb{R}\to\mathbb{R}$ naturally generalizing the function $M$. Furthermore, we prove that for any linear function $g:\mathbb R^n\to\mathbb R$, the order type of the resulting $\mathcal F$ is at most $\varphi_{n-1}(0)$. Finally, we show that there do exist continuous functions $g:\mathbb R^n\to\mathbb R$ for which the order types of the resulting sets $\mathcal F$ approach the small Veblen ordinal.
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