The approximability of non-Boolean satisfiability problems and restricted integer programming

AbstractIn this paper we present improved approximation algorithms for two classes of maximization problems defined in Barland et al. (J. Comput. System Sci. 57(2) (1998) 144). Our factors of approximation substantially improve the previous known results and are close to the best possible. On the other hand, we show that the approximation results in the framework of Barland et al. hold also in the parallel setting, and thus we have a new common framework for both computational settings. We prove almost tight non-approximability results, thus solving a main open question of Barland et al.We obtain the results through the constraint satisfaction problem over multi-valued domains, for which we develop approximation algorithms and show non-approximability results. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of probabilistically checkable proofs and multi-prover one-round proof systems

On sketching approximations for symmetric Boolean CSPs

A Boolean maximum constraint satisfaction problem, Max-CSP($f$), is specified by a predicate $f:\{-1,1\}^k\to\{0,1\}$. An $n$-variable instance of Max-CSP($f$) consists of a list of constraints, each of which applies $f$ to $k$ distinct literals drawn from the $n$ variables. For $k=2$, Chou, Golovnev, and Velusamy [CGV20, FOCS 2020] obtained explicit ratios characterizing the $\sqrt n$-space streaming approximability of every predicate. For $k \geq 3$, Chou, Golovnev, Sudan, and Velusamy [CGSV21, arXiv:2102.12351] proved a general dichotomy theorem for $\sqrt n$-space sketching algorithms: For every $f$, there exists $\alpha(f)\in (0,1]$ such that for every $\epsilon>0$, Max-CSP($f$) is $(\alpha(f)-\epsilon)$-approximable by an $O(\log n)$-space linear sketching algorithm, but $(\alpha(f)+\epsilon)$-approximation sketching algorithms require $\Omega(\sqrt{n})$ space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting $\alpha'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}$, we show that for odd $k \geq 3$, $\alpha(k$AND$) = \alpha'_k$, and for even $k \geq 2$, $\alpha(k$AND$) = 2\alpha'_{k+1}$. We also resolve the ratio for the "at-least-$(k-1)$-$1$'s" function for all even $k$; the "exactly-$\frac{k+1}2$-$1$'s" function for odd $k \in \{3,\ldots,51\}$; and fifteen other functions. We stress here that for general $f$, according to [CGSV21], closed-form expressions for $\alpha(f)$ need not have existed a priori. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [CGV20] while simplifying [CGSV21]. Finally, we investigate the $\sqrt n$-space streaming lower bounds in [CGSV21], and show that they are incomplete for $3$AND.Comment: 27 pages; same results but significant changes in presentatio

On streaming approximation algorithms for constraint satisfaction problems

In this thesis, we explore streaming algorithms for approximating constraint satisfaction problems (CSPs). The setup is roughly the following: A computer has limited memory space, sees a long "stream" of local constraints on a set of variables, and tries to estimate how many of the constraints may be simultaneously satisfied. The past ten years have seen a number of works in this area, and this thesis includes both expository material and novel contributions. Throughout, we emphasize connections to the broader theories of CSPs, approximability, and streaming models, and highlight interesting open problems. The first part of our thesis is expository: We present aspects of previous works that completely characterize the approximability of specific CSPs like Max-Cut and Max-Dicut with $\sqrt{n}$-space streaming algorithm (on $n$-variable instances), while characterizing the approximability of all CSPs in $\sqrt n$ space in the special case of "composable" (i.e., sketching) algorithms, and of a particular subclass of CSPs with linear-space streaming algorithms. In the second part of the thesis, we present two of our own joint works. We begin with a work with Madhu Sudan and Santhoshini Velusamy in which we prove linear-space streaming approximation-resistance for all ordering CSPs (OCSPs), which are "CSP-like" problems maximizing over sets of permutations. Next, we present joint work with Joanna Boyland, Michael Hwang, Tarun Prasad, and Santhoshini Velusamy in which we investigate the $\sqrt n$-space streaming approximability of symmetric Boolean CSPs with negations. We give explicit $\sqrt n$-space sketching approximability ratios for several families of CSPs, including Max-$k$AND; develop simpler optimal sketching approximation algorithms for threshold predicates; and show that previous lower bounds fail to characterize the $\sqrt n$-space streaming approximability of Max-$3$AND.Comment: Harvard College senior thesis; 119 pages plus references; abstract shortened for arXiv; formatted with Dissertate template (feel free to copy!); exposits papers arXiv:2105.01782 (APPROX 2021) and arXiv:2112.06319 (APPROX 2022

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Numerical calculation model for the global analysis of concrete structures with masonry walls

The numerical simulation in the field of civil engineering, while widely used in structural design, has not benefited from the full potential offered by new technologies for the analysis and design of composite materials within the framework of the finite element, technologies that are already present in industries such as automotive, aerospace and shipbuilding. This thesis is based on the numerical simulation, and emerges as the need to combine and improve existing technologies in the field of finite element analysis for composite materials, to assess the overall structural behavior of reinforced concrete buildings with masonry in-fills, and consequently, to support the derivation of rational rules for analysis and design purposes. Prior to the beginning of this thesis, a huge concern was the large amount of computational resources needed for both solving systems of linear equations resulting from the use of the finite elements method, and for storing internal variables needed in the integration of constitutive models. Therefore, in this work, computational strategies used to enable the analysis of real life structures are also provided. The simplicity required to handle meshes with high amount of finite elements pushed us to develop a new layered finite element, that can reproduce the non-linear behavior of its constituent materials when there are out-of-plane stresses, this, without having to introduce additional degrees of freedom. The finite element proposed has been com- pared to finite element with different kinematics obtaining excellent results. The robustness and efficiency of the developed methodology for analysis of masonry and concrete buildings, is conditioned by the ability of using different patterns of steel reinforcement, which are typically presented in real life structures. That is why it has also been necessary to develop a computing program capable of reading both finite element meshes, and patterns of fibers represented with convex polygons, and as a result of areas intersections between polygons returns volumetric participation of fiber and matrix of constituents materials for each layer, in addition had to return the fiber orientation with respect to the local axis of the finite element. The numerical results obtained have been compared in some cases with experimental results available in the literature, in other cases, with numerical results obtained using Building Codes, in both cases, there have been good agreement between them. Finally, it has been possible to characterize a representative medium-rise building of Mexico City using the capacity spectrum method. This method is widely used nowadays for the assessment of building behavior, since using fragility curves can represent the ability of a building to resist an earthquake.La simulación numérica en el campo de la ingeniera civil, aunque es ampliamente utilizada en el diseño estructural, no se ha beneficiado de todo el potencial que ofrecen las nuevas tecnologías para el análisis y diseño de materiales compuestos dentro del marco de los elementos finitos, tecnologías que ya están presenten en industrias como la automotriz, aeroespacial o la naval. Este trabajo de tesis esta basado en la simulación numérica, y surge como la necesidad de combinar y mejorar tecnologías existentes en el campo de los elementos finitos y de análisis de materiales compuestos, para conocer el comportamiento estructural global de los edificios de hormigón armado con rellenos de mampostería, y para apoyar en la derivación de reglas racionales con fines de diseño. Una preocupación previa al inicio de esta tesis doctoral, era la gran cantidad de recursos computacionales necesarios, tanto para la resolución de los sistemas de ecuaciones lineales resultantes con el uso del método de los elementos finitos, como para el almacenamiento de variables internas necesarias en la integración de modelos constitutivos. Por ello, dentro de este trabajo, también se proporcionan las estrategias computacionales usadas que permiten el análisis de estructuras de la vida real. La simplicidad requerida para el manejo de mallas con gran cantidad de elementos finitos lleva a desarrollar un elemento de l¿amina con diferentes capas, que pueda reproducir el comportamiento no lineal de sus materiales componentes cuando existen tensiones fuera del plano, sin que haya que introducir grados de libertad adicionales. El elemento finito propuesto ha sido comparado con elementos finitos de diferente cinemática obteniendo excelentes resultados. La robustez y eficiencia de la metodología desarrollada para el análisis de edificios de hormigón y mampostería, esta condicionada a la capacidad de utilizar los diferentes patrones de acero de refuerzo que típicamente se presentan en estructuras de la vida real. Es por ello que también ha sido necesario desarrollar un programa de computo capaz de leer tanto una malla de elementos finitos, como un patrón de fibras representado con polígonos convexos, y que mediante operaciones de intersecciones de ¿áreas entre polígonos, de como resultado la participación volumétrica de matriz y fibra de los materiales componentes por cada capa, además de la orientación de la fibra con respecto a los ejes locales del elemento finito. Los resultados numéricos obtenidos se han comparado con resultados experimentales presentes en la literatura, y con resultados numéricos obtenidos utilizando normas de construcción, en ambos casos, se han observado buenos ajustes entre ellos. Finalmente,ha sido posible caracterizar un edificio representativo ubicado en la Ciudad de México usando el método del espectro de capacidad. Dicho método es ampliamente utilizado hoy en día para el diseño y evaluación sismo-resistente de estructuras, ya que mediante el uso de curvas de fragilidad permite representar la susceptibilidad de una estructuraa ser dañada debido a un terremoto