60 research outputs found

    On the Greedy Algorithm for the Shortest Common Superstring Problem with Reversals

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    We study a variation of the classical Shortest Common Superstring (SCS) problem in which a shortest superstring of a finite set of strings SS is sought containing as a factor every string of SS or its reversal. We call this problem Shortest Common Superstring with Reversals (SCS-R). This problem has been introduced by Jiang et al., who designed a greedy-like algorithm with length approximation ratio 44. In this paper, we show that a natural adaptation of the classical greedy algorithm for SCS has (optimal) compression ratio 12\frac12, i.e., the sum of the overlaps in the output string is at least half the sum of the overlaps in an optimal solution. We also provide a linear-time implementation of our algorithm.Comment: Published in Information Processing Letter

    On Approximability of Bounded Degree Instances of Selected Optimization Problems

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    In order to cope with the approximation hardness of an underlying optimization problem, it is advantageous to consider specific families of instances with properties that can be exploited to obtain efficient approximation algorithms for the restricted version of the problem with improved performance guarantees. In this thesis, we investigate the approximation complexity of selected NP-hard optimization problems restricted to instances with bounded degree, occurrence or weight parameter. Specifically, we consider the family of dense instances, where typically the average degree is bounded from below by some function of the size of the instance. Complementarily, we examine the family of sparse instances, in which the average degree is bounded from above by some fixed constant. We focus on developing new methods for proving explicit approximation hardness results for general as well as for restricted instances. The fist part of the thesis contributes to the systematic investigation of the VERTEX COVER problem in k-hypergraphs and k-partite k-hypergraphs with density and regularity constraints. We design efficient approximation algorithms for the problems with improved performance guarantees as compared to the general case. On the other hand, we prove the optimality of our approximation upper bounds under the Unique Games Conjecture or a variant. In the second part of the thesis, we study mainly the approximation hardness of restricted instances of selected global optimization problems. We establish improved or in some cases the first inapproximability thresholds for the problems considered in this thesis such as the METRIC DIMENSION problem restricted to graphs with maximum degree 3 and the (1,2)-STEINER TREE problem. We introduce a new reductions method for proving explicit approximation lower bounds for problems that are related to the TRAVELING SALESPERSON (TSP) problem. In particular, we prove the best up to now inapproximability thresholds for the general METRIC TSP problem, the ASYMMETRIC TSP problem, the SHORTEST SUPERSTRING problem, the MAXIMUM TSP problem and TSP problems with bounded metrics

    Some comments on symmetric orbifolds of K3

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    We consider two dimensional N=(4,4)\mathcal{N}=(4,4) superconformal field theories in the moduli space of symmetric orbifolds of K3. We complete a classification of the discrete groups of symmetries of these models, conditional to a series of assumptions and with certain restrictions. Furthermore, we provide a partial classification of the set of twining genera, encoding the action of a discrete symmetry gg on a space of supersymmetric states in these models. These results suggest the existence of a number of surprising identities between seemingly different Borcherds products, representing Siegel modular forms of degree two and level N>1N>1. We also provide a critical review of various properties of the moduli space of these superconformal field theories, including the groups of dualities, the set of singular models and the locus of symmetric orbifold points, and describe some puzzles related to our (lack of) understanding of these properties.Comment: 54 pages; v3: various points clarified; appendix E added; matches with published versio

    (2,2) Superconformal Bootstrap in Two Dimensions

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    We find a simple relation between two-dimensional BPS N=2 superconformal blocks and bosonic Virasoro conformal blocks, which allows us to analyze the crossing equations for BPS 4-point functions in unitary (2,2) superconformal theories numerically with semidefinite programming. We constrain gaps in the non-BPS spectrum through the operator product expansion of BPS operators, in ways that depend on the moduli of exactly marginal deformations through chiral ring coefficients. In some cases, our bounds on the spectral gaps are observed to be saturated by free theories, by N=2 Liouville theory, and by certain Landau-Ginzburg models.Comment: 56 pages, 14 figure

    From Knowledge, Knowability and the Search for Objective Randomness to a New Vision of Complexity

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    Herein we consider various concepts of entropy as measures of the complexity of phenomena and in so doing encounter a fundamental problem in physics that affects how we understand the nature of reality. In essence the difficulty has to do with our understanding of randomness, irreversibility and unpredictability using physical theory, and these in turn undermine our certainty regarding what we can and what we cannot know about complex phenomena in general. The sources of complexity examined herein appear to be channels for the amplification of naturally occurring randomness in the physical world. Our analysis suggests that when the conditions for the renormalization group apply, this spontaneous randomness, which is not a reflection of our limited knowledge, but a genuine property of nature, does not realize the conventional thermodynamic state, and a new condition, intermediate between the dynamic and the thermodynamic state, emerges. We argue that with this vision of complexity, life, which with ordinary statistical mechanics seems to be foreign to physics, becomes a natural consequence of dynamical processes.Comment: Phylosophica
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