73 research outputs found

    On largest volume simplices and sub-determinants

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    We show that the problem of finding the simplex of largest volume in the convex hull of nn points in Qd\mathbb{Q}^d can be approximated with a factor of O(logd)d/2O(\log d)^{d/2} in polynomial time. This improves upon the previously best known approximation guarantee of d(d1)/2d^{(d-1)/2} by Khachiyan. On the other hand, we show that there exists a constant c>1c>1 such that this problem cannot be approximated with a factor of cdc^d, unless P=NPP=NP. % This improves over the 1.091.09 inapproximability that was previously known. Our hardness result holds even if n=O(d)n = O(d), in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where cˉ\bar c is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d×nd\times n matrix

    Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration

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    Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors v1,,vmRdv_1,\ldots,v_m \in \mathbb{R}^d and a constraint family B2[m]{\cal B}\subseteq 2^{[m]}, find a set SBS \in \cal{B} that maximizes the squared volume of the simplex spanned by the vectors in SS. A motivating example is the data-summarization problem in machine learning where one is given a collection of vectors that represent data such as documents or images. The volume of a set of vectors is used as a measure of their diversity, and partition or matroid constraints over [m][m] are imposed in order to ensure resource or fairness constraints. Recently, Nikolov and Singh presented a convex program and showed how it can be used to estimate the value of the most diverse set when B{\cal B} corresponds to a partition matroid. This result was recently extended to regular matroids in works of Straszak and Vishnoi, and Anari and Oveis Gharan. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms -- that also output a set -- remained open. The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the subdeterminant maximization problem for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a nonconvex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that allows us to relate the optimal value of these nonconvex functions to their value at a random point. Unlike prior work on the constrained subdeterminant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity.Comment: in FOCS 201

    Approximate Convex Optimization by Online Game Playing

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    Lagrangian relaxation and approximate optimization algorithms have received much attention in the last two decades. Typically, the running time of these methods to obtain a ϵ\epsilon approximate solution is proportional to 1ϵ2\frac{1}{\epsilon^2}. Recently, Bienstock and Iyengar, following Nesterov, gave an algorithm for fractional packing linear programs which runs in 1ϵ\frac{1}{\epsilon} iterations. The latter algorithm requires to solve a convex quadratic program every iteration - an optimization subroutine which dominates the theoretical running time. We give an algorithm for convex programs with strictly convex constraints which runs in time proportional to 1ϵ\frac{1}{\epsilon}. The algorithm does NOT require to solve any quadratic program, but uses gradient steps and elementary operations only. Problems which have strictly convex constraints include maximum entropy frequency estimation, portfolio optimization with loss risk constraints, and various computational problems in signal processing. As a side product, we also obtain a simpler version of Bienstock and Iyengar's result for general linear programming, with similar running time. We derive these algorithms using a new framework for deriving convex optimization algorithms from online game playing algorithms, which may be of independent interest

    Practical Polytope Volume Approximation

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    International audienceWe experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytope's volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problem's geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B 11 ,. .. , B 15 , whereas exact methods have only computed that of B 10

    High-dimensional polytopes defined by oracles: algorithms, computations and applications

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    Η επεξεργασία και ανάλυση γεωμετρικών δεδομένων σε υψηλές διαστάσεις διαδραματίζει ένα θεμελιώδη ρόλο σε διάφορους κλάδους της επιστήμης και της μηχανικής. Τις τελευταίες δεκαετίες έχουν αναπτυχθεί πολλοί επιτυχημένοι γεωμετρικοί αλγόριθμοι σε 2 και 3 διαστάσεις. Ωστόσο, στις περισσότερες περιπτώσεις, οι επιδόσεις τους σε υψηλότερες διαστάσεις δεν είναι ικανοποιητικές. Αυτή η συμπεριφορά είναι ευρέως γνωστή ως κατάρα των μεγάλων διαστάσεων (curse of dimensionality). Δυο πλαίσια λύσης που έχουν υιοθετηθεί για να ξεπεραστεί αυτή η δυσκολία είναι η εκμετάλλευση της ειδικής δομής των δεδομένων, όπως σε περιπτώσεις αραιών (sparse) δεδομένων ή στην περίπτωση που τα δεδομένα βρίσκονται σε χώρο χαμηλότερης διάστασης, και ο σχεδιασμός προσεγγιστικών αλγορίθμων. Στη διατριβή αυτή μελετάμε προβλήματα μέσα σε αυτά τα πλαίσια. Το κύριο ερευνητικό πεδίο της παρούσας εργασίας είναι η διακριτή και υπολογιστικής γεωμετρία και οι σχέσεις της με τους κλάδους της επιστήμης των υπολογιστών και τα εφαρμοσμένα μαθηματικά, όπως είναι η θεωρία πολυτόπων, οι υλοποιήσεις αλγορίθμων, οι πιθανοθεωρητικοί γεωμετρικοί αλγόριθμοι, η υπολογιστική αλγεβρική γεωμετρία και η βελτιστοποίηση. Τα θεμελιώδη γεωμετρικά αντικείμενα της μελέτης μας είναι τα πολύτοπα, και οι βασικές τους ιδιότητες είναι η κυρτότητα και ότι ορίζονται από ένα μαντείο (oracle) σε ένα χώρο υψηλής διάστασης. Η επεξεργασία και ανάλυση γεωμετρικών δεδομένων σε υψηλές διαστάσεις διαδραματίζει ένα θεμελιώδη ρόλο σε διάφορους κλάδους της επιστήμης και της μηχανικής. Τις τελευταίες δεκαετίες έχουν αναπτυχθεί πολλοί επιτυχημένοι γεωμετρικοί αλγόριθμοι σε 2 και 3 διαστάσεις. Ωστόσο, στις περισσότερες περιπτώσεις, οι επιδόσεις τους σε υψηλότερες διαστάσεις δεν είναι ικανοποιητικές. Δυο πλαίσια λύσης που έχουν υιοθετηθεί για να ξεπεραστεί αυτή η δυσκολία είναι η εκμετάλλευση της ειδικής δομής των δεδομένων, όπως σε περιπτώσεις αραιών (sparse) δεδομένων ή στην περίπτωση που τα δεδομένα βρίσκονται σε χώρο χαμηλότερης διάστασης, και ο σχεδιασμός προσεγγιστικών αλγορίθμων. Το κύριο ερευνητικό πεδίο της παρούσας εργασίας είναι η διακριτή και υπολογιστικής γεωμετρία και οι σχέσεις της με τους κλάδους της επιστήμης των υπολογιστών και τα εφαρμοσμένα μαθηματικά. Η συμβολή αυτής της διατριβής είναι τριπλή. Πρώτον, στο σχεδιασμό και την ανάλυση των γεωμετρικών αλγορίθμων για προβλήματα σε μεγάλες διαστάσεις. Δεύτερον, θεωρητικά αποτελέσματα σχετικά με το συνδυαστικό χαρακτηρισμό βασικών οικογενειών πολυτόπων. Τρίτον, η εφαρμογή και πειραματική ανάλυση των προτεινόμενων αλγορίθμων και μεθόδων. Η ανάπτυξη λογισμικού ανοιχτού κώδικα, που είναι διαθέσιμο στο κοινό και βασίζεται και επεκτείνει διαδεδομένες γεωμετρικές και αλγεβρικές βιβλιοθήκες λογισμικού, όπως η CGAL και το polymake.The processing and analysis of high dimensional geometric data plays a fundamental role in disciplines of science and engineering. The last decades many successful geometric algorithms has been developed in 2 and 3 dimensions. However, in most cases their performance in higher dimensions is poor. This behavior is commonly called the curse of dimensionality. A solution framework adopted for the healing of the curse of dimensionality is the exploitation of the special structure of the data, such as sparsity or low intrinsic dimension and the design of approximation algorithms. The main research area of this thesis is discrete and computational geometry and its connections to branches of computer science and applied mathematics. The contribution of this thesis is threefold. First, the design and analysis of geometric algorithms for problems concerning high-dimensional, convex polytopes, such as convex hull and volume computation and their applications to computational algebraic geometry and optimization. Second, the establishment of combinatorial characterization results for essential polytope families. Third, the implementation and experimental analysis of the proposed algorithms and methods. The developed software is opensource, publicly available and builds on and extends state-of-the-art geometric and algebraic software libraries such as CGAL and polymake

    A Unifying Geometric Solution Framework and Complexity Analysis for Variational Inequalities

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    In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids, and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems. The analysis rests upon the assumption of strong-f-monotonicity, which is weaker than strict and strong monotonicity. Since linear programs satisfy this assumption, the general framework applies to linear programs
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