DOCSLIB.ORG

Approximation Hardness of Optimization Problems in Intersection Graphs of D-Dimensional Boxes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Approximation Hardness of Optimization Problems in Intersection Graphs of D-Dimensional Boxes Approximation Hardness of Optimization Problems in Intersection Graphs of d-dimensional Boxes Miroslav Chleb´ık¤ Janka Chleb´ıkov´a y Abstract Maximum Independent Set, in d-box intersection The Maximum Independent Set problem in d-box graphs, graphs for any fixed d ¸ 3. i.e., in the intersection graphs of axis-parallel rectangles in Rd, is a challenge open problem. For any fixed d ¸ 2 Overview. Many optimization problems like Max- the problem is NP-hard and no approximation algorithm imum Clique, Maximum Independent Set, and with ratio o(logd¡1 n) is known. In some restricted cases, Minimum (Vertex) Coloring are NP-hard for gen- e.g., for d-boxes with bounded aspect ratio, a PTAS exists eral graphs but solvable in polynomial time for interval [17]. In this paper we prove APX-hardness (and hence non- graphs [20]. Many of them are known to be NP-hard existence of a PTAS, unless P = NP), of the Maximum already in 2-dimensional models of geometric intersec- Independent Set problem in d-box graphs for any fixed tion graphs (e.g., in unit disk graphs). In most cases the 443 geometric restrictions allow us to obtain better approxi- d ¸ 3. We state also first explicit lower bound 442 on efficient approximability in such case. Additionally, we mation algorithms (or even in polynomial time solvabil- provide a generic method how to prove APX-hardness for ity) for problems that are in general graphs extremely many NP-hard graph optimization problems in d-box graphs hard to approximate. On the other hand, these geomet- for any fixed d ¸ 3. In 2-dimensional case we give a generic ric restrictions make the task to achieve some hardness approach to NP-hardness results for these problems in highly results more difficult. restricted intersection graphs of axis-parallel unit squares Among basic NP-hard graph optimization problems (alternatively, in unit disk graphs). in d-box graphs (d ¸ 2), only Maximum Clique is known to be exactly solvable in polynomial time ([5], 1 Introduction [29], [35]). Some optimization problems are known to be NP-hard in d-box intersection graphs for any fixed The intersection graph of a family of sets S , v 2 V , is v d ¸ 2, for example Maximum Independent Set [18], a graph with vertex set V such that u is adjacent to v [25], or Minimum Coloring [30]. if and only if S \ S 6= ;. The family fS ; v 2 V g is an u v v The challenging open problem, the Maximum In- intersection representation of this graph. Geometrical dependent Set problem (shortly, Max-IS), in d-box models of intersection graphs deal with families of intersection graphs (d ¸ 2) can be formulated as follows: subsets of Rd with some geometric properties. In this for a given set R of n axis-parallel d-dimensional boxes, paper we are mainly interested in families of axis- find a maximum cardinality subset R¤ ⊆ R of pairwise parallel d-dimensional boxes. Their intersection graphs disjoint boxes. The problem has attracted an attention are called d-box intersection graphs, or simply d-box of many researchers ([1], [11], [12], [17], [23], [26], [33]) graphs. Recall that a d-dimensional box (d-box) is a due to its applications in map labeling, data mining, subset of Rd that is a Cartesian product of d intervals VLSI design, image processing, and point location in in R. For convenience, terms an interval and a rectangle d-dimensional Euclidean space. As the problem is NP- are used for 1-box and 2-box, respectively. hard for any fixed d ¸ 2 ([18], [25]), its approximability In this paper we describe a generic approach to is intensively studied. approximation hardness results for many graph opti- Let us describe briefly known approximability re- mization problems as, e.g., Minimum Vertex Cover, sults for it, see [12] for more detailed overview. The earliest result was the shifting grid method based PTAS by Hochbaum and Maass [23] for the case of unit d- ¤Max Planck Institute for Mathematics in the Sciences, Insel- cubes. This method works for any collection of ‘fat’ straße 22-26, D-04103 Leipzig, Germany, [email protected] objects in Rd of roughly the same size. Their scheme y d Faculty of Mathematics, Physics and Informatics, Come- required nO(k ) time to guarantee an approximation fac- nius University, Mlynsk´a dolina, 842 48 Bratislava, Slovakia, 1 [email protected] tor of (1+ k ). By applying dynamic programming along one of the dimensions, running time has been reduced graphs of maximum degree 3 (Section 3). d¡1 to nO(k ) and generalized to objects, not necessarily ² Each graph obtained from another one by at least fat, but whose projections to the last (d¡1) coordinates 3-subdivision of each edge is an intersection graph are fat and of roughly the same size. This was essen- of d-boxes for any fixed d ¸ 3. Moreover, its tially established by Agarwal et al. [1] in their work on representation can be provided in time polynomial 1 in its size. (Section 2). unit-height rectangles in the plane. To achieve (1 + k )- approximation they need time O(n2k¡1 + n log n). Both these results are interesting for their general- Generalizing in another direction, Erlebach et al. ity and can be of independent interest. The same ap- [17] obtained a PTAS for fat objects of possibly varying plies to a newly introduced notion of a shift-reduction sizes, such as arbitrary d-cubes or bounded aspect ratio (Section 3) used in the proof of APX-hardness for many 4 d-boxes. The running time of their algorithm (nO(k ) graph optimization problems in d-boxes (d ¸ 3), e.g., d¡1 for d = 2) has been improved to nO(k ) by Chan [12]. Minimum Vertex Cover, Minimum (Independent) For arbitrary d-boxes, even for d = 2, the existence Dominating Set, and Maximum Induced Match- of a PTAS or a constant factor approximation algo- ing. The methods also allow to provide explicit lower rithm, is largely open problem. As has been observed bounds on efficient approximability. This is demon- in several papers ([1], [26]), a logarithmic approxima- strated on the Maximum Independent Set prob- tion factor is possible in this case. For example, Agar- lem in d-box graphs (d ¸ 3), for which we prove NP- 1 wal et al. [1] described O(n log n)-time algorithm with hardness to achieve approximation factor of 1 + 442 . d¡1 Usually better approximation algorithms for optimiza- factor at most dlog2 ne. This generalizes to dlog2 ne - d¡1 tion problems in d-box graphs need a representation of approximation algorithms with O(n log2 n) running time, for general d. Nielsen [33] independently de- an input graph by d-boxes, not merely a graph. Our scribed algorithm with optimum-sensitive approxima- hardness results apply to this setting as well. Moreover, tion factor (1 + log (is(R)))d¡1, where is(R) is the they apply to the highly restricted case, in which no 2 d maximum number of independent boxes of R. Cur- point of R is simultaneously covered by more than two rently, no polynomial time algorithm is known with d-boxes, and each d-box intersects at most 3 others. o(logd¡1 n)-approximation factor, although Berman et Our methods do not apply to the planar case in d¡1 which there is still hope for a PTAS. All problems al. [11] have observed that a log2 n bound can be re- duced by arbitrary multiple constant. More precisely, studied in the paper are NP-hard in 2-dimensions as d¡1 O(bd¡1) well, even in very restricted setting of intersection a factor dlogb ne can be achieved in n time, for any fixed integer b ¸ 2. As it was already men- graphs of axis-parallel unit squares (or unit disks). tioned in [11], it remains open to understand the limits We provide a generic approach to NP-hardness results on the approximability of the Maximum Independent for many graph optimization problems in intersection Set problem in intersection graphs of d-box graphs for graphs of axis-parallel unit squares (alternatively, in d ¸ 2. unit disk graphs). We also contribute to another independent set Our results. In this paper we present the proof of problem connected with sets of d-boxes, introduced APX-hardness (with some explicit lower bounds) for the by Bafna et al. [4]. The problem is the Maximum Maximum Independent Set problem in axis-parallel Weighted Independent Set problem (Max-w-IS) d-dimensional boxes for any fixed d ¸ 3. It follows, in in proper d-union graphs, i.e., in graphs that can be particular, that the existence of a PTAS for the problem expressed as the union of d proper interval graphs. restricted to d-boxes with bounded aspect ratio [17] For d = 2 the problem is quite well understood. For cannot be generalized (unless P = NP) to arbitrary d- d ¸ 3 we improve significantly the upper bounds boxes for any fixed d ¸ 3. on approximability of this problem (see [4], [9], [10], We provide also a generic method how to prove [6]). Namely, the Max-w-IS problem in proper d-union 1 approximation hardness results in d-box graphs that graphs, d ¸ 2, can be approximated within (d + 2 ) in is applied simultaneously to several graph optimization polynomial time, and within (2d ¡ 1) in nearly linear problems such as covering, domination, and matching time. problems. The idea of our APX-hardness results is based on the proof of the following two facts: Definitions and notations.
Load more

Recommended publications
  • Advanced Complexity Theory
    Advanced Complexity Theory Markus Bl¨aser& Bodo Manthey Universit¨atdes Saarlandes Draft|February 9, 2011 and forever 2 1 Complexity of optimization prob- lems 1.1 Optimization problems The study of the complexity of solving optimization problems is an impor- tant practical aspect of complexity theory. A good textbook on this topic is the one by Ausiello et al. [ACG+99]. The book by Vazirani [Vaz01] is also recommend, but its focus is on the algorithmic side. Definition 1.1. An optimization problem P is a 4-tuple (IP ; SP ; mP ; goalP ) where ∗ 1. IP ⊆ f0; 1g is the set of valid instances of P , 2. SP is a function that assigns to each valid instance x the set of feasible ∗ 1 solutions SP (x) of x, which is a subset of f0; 1g . + 3. mP : f(x; y) j x 2 IP and y 2 SP (x)g ! N is the objective function or measure function. mP (x; y) is the objective value of the feasible solution y (with respect to x). 4. goalP 2 fmin; maxg specifies the type of the optimization problem. Either it is a minimization or a maximization problem. When the context is clear, we will drop the subscript P . Formally, an optimization problem is defined over the alphabet f0; 1g. But as usual, when we talk about concrete problems, we want to talk about graphs, nodes, weights, etc. In this case, we tacitly assume that we can always find suitable encodings of the objects we talk about. ∗ Given an instance x of the optimization problem P , we denote by SP (x) the set of all optimal solutions, that is, the set of all y 2 SP (x) such that mP (x; y) = goalfmP (x; z) j z 2 SP (x)g: (Note that the set of optimal solutions could be empty, since the maximum need not exist.
    [Show full text]
  • APX Radio Family Brochure
    APX MISSION-CRITICAL P25 COMMUNICATIONS BROCHURE APX P25 COMMUNICATIONS THE BEST OF WHAT WE DO Whether you’re a state trooper, firefighter, law enforcement officer or highway maintenance technician, people count on you to get the job done. There’s no room for error. This is mission critical. APX™ radios exist for this purpose. They’re designed to be reliable and to optimize your communications, specifically in extreme environments and during life-threatening situations. Even with the widest portfolio in the industry, APX continues to evolve. The latest APX NEXT smart radio series delivers revolutionary new capabilities to keep you safer and more effective. WE’VE PUT EVERYTHING WE’VE LEARNED OVER THE LAST 90 YEARS INTO APX. THAT’S WHY IT REPRESENTS THE VERY BEST OF THE MOTOROLA SOLUTIONS PORTFOLIO. THERE IS NO BETTER. BROCHURE APX P25 COMMUNICATIONS OUTLAST AND OUTPERFORM RELIABLE COMMUNICATIONS ARE NON-NEGOTIABLE APX two-way radios are designed for extreme durability, so you can count on them to work under the toughest conditions. From the rugged aluminum endoskeleton of our portable radios to the steel encasement of our mobile radios, APX is built to last. Pressure-tested HEAR AND BE HEARD tempered glass display CLEAR COMMUNICATION CAN MAKE A DIFFERENCE The APX family is designed to help you hear and be heard with unparalleled clarity, so you’re confident your message will always get through. Multiple microphones and adaptive windporting technology minimize noise from wind, crowds and sirens. And the loud and clear speaker ensures you can hear over background sounds. KEEP INFORMATION PROTECTED EVERYDAY, SECURITY IS BEING PUT TO THE TEST With the APX family, you can be sure that your calls stay private, secure, and confidential.
    [Show full text]
  • The Complexity Zoo
    The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/.
    [Show full text]
  • MODULES and ITS REVERSES 1. Introduction the Hölder Inequality
    Ann. Funct. Anal. A nnals of F unctional A nalysis ISSN: 2008-8752 (electronic) URL: www.emis.de/journals/AFA/ HOLDER¨ TYPE INEQUALITIES ON HILBERT C∗-MODULES AND ITS REVERSES YUKI SEO1 This paper is dedicated to Professor Tsuyoshi Ando Abstract. In this paper, we show Hilbert C∗-module versions of H¨older- McCarthy inequality and its complementary inequality. As an application, we obtain H¨oldertype inequalities and its reverses on a Hilbert C∗-module. 1. Introduction The H¨olderinequality is one of the most important inequalities in functional analysis. If a = (a1; : : : ; an) and b = (b1; : : : ; bn) are n-tuples of nonnegative numbers, and 1=p + 1=q = 1, then ! ! Xn Xn 1=p Xn 1=q ≤ p q aibi ai bi for all p > 1 i=1 i=1 i=1 and ! ! Xn Xn 1=p Xn 1=q ≥ p q aibi ai bi for all p < 0 or 0 < p < 1: i=1 i=1 i=1 Non-commutative versions of the H¨olderinequality and its reverses have been studied by many authors. T. Ando [1] showed the Hadamard product version of a H¨oldertype. T. Ando and F. Hiai [2] discussed the norm H¨olderinequality and the matrix H¨olderinequality. B. Mond and O. Shisha [15], M. Fujii, S. Izumino, R. Nakamoto and Y. Seo [7], and S. Izumino and M. Tominaga [11] considered the vector state version of a H¨oldertype and its reverses. J.-C. Bourin, E.-Y. Lee, M. Fujii and Y. Seo [3] showed the geometric operator mean version, and so on.
    [Show full text]
  • User's Guide for Complexity: a LATEX Package, Version 0.80
    User’s Guide for complexity: a LATEX package, Version 0.80 Chris Bourke April 12, 2007 Contents 1 Introduction 2 1.1 What is complexity? ......................... 2 1.2 Why a complexity package? ..................... 2 2 Installation 2 3 Package Options 3 3.1 Mode Options .............................. 3 3.2 Font Options .............................. 4 3.2.1 The small Option ....................... 4 4 Using the Package 6 4.1 Overridden Commands ......................... 6 4.2 Special Commands ........................... 6 4.3 Function Commands .......................... 6 4.4 Language Commands .......................... 7 4.5 Complete List of Class Commands .................. 8 5 Customization 15 5.1 Class Commands ............................ 15 1 5.2 Language Commands .......................... 16 5.3 Function Commands .......................... 17 6 Extended Example 17 7 Feedback 18 7.1 Acknowledgements ........................... 19 1 Introduction 1.1 What is complexity? complexity is a LATEX package that typesets computational complexity classes such as P (deterministic polynomial time) and NP (nondeterministic polynomial time) as well as sets (languages) such as SAT (satisfiability). In all, over 350 commands are defined for helping you to typeset Computational Complexity con- structs. 1.2 Why a complexity package? A better question is why not? Complexity theory is a more recent, though mature area of Theoretical Computer Science. Each researcher seems to have his or her own preferences as to how to typeset Complexity Classes and has built up their own personal LATEX commands file. This can be frustrating, to say the least, when it comes to collaborations or when one has to go through an entire series of files changing commands for compatibility or to get exactly the look they want (or what may be required).
    [Show full text]
  • Maximum Clique in Disk-Like Intersection Graphs
    Maximum Clique in Disk-Like Intersection Graphs Édouard Bonnet Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France [email protected] Nicolas Grelier Department of Computer Science, ETH Zürich [email protected] Tillmann Miltzow Utrecht University, Utrecht Netherlands [email protected] Abstract We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark ’90, Raghavan and Spinrad ’03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. ’18, Bonamy et al. ’18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes I follow the same road. They show that, for every graph G of a large-enough class C, the complement of an even subdivision of G belongs to the intersection class I. Then they conclude invoking the hardness of Maximum Independent Set on the class C, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. ’18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme.
    [Show full text]
  • The Trichotomy of HAVING Queries on a Probabilistic Database
    VLDBJ manuscript No. (will be inserted by the editor) The Trichotomy of HAVING Queries on a Probabilistic Database Christopher Re´ · Dan Suciu Received: date / Accepted: date Abstract We study the evaluation of positive conjunctive cant extension of a previous dichotomy result for Boolean queries with Boolean aggregate tests (similar to HAVING in conjunctive queries into safe and not safe. For each of the SQL) on probabilistic databases. More precisely, we study three classes we present novel techniques. For safe queries, conjunctive queries with predicate aggregates on probabilis- we describe an evaluation algorithm that uses random vari- tic databases where the aggregation function is one of MIN, ables over semirings. For apx-safe queries, we describe an MAX, EXISTS, COUNT, SUM, AVG, or COUNT(DISTINCT) and fptras that relies on a novel algorithm for generating a ran- the comparison function is one of =; ,; ≥; >; ≤, or < . The dom possible world satisfying a given condition. Finally, for complexity of evaluating a HAVING query depends on the ag- hazardous queries we give novel proofs of hardness of ap- gregation function, α, and the comparison function, θ. In this proximation. The results for safe queries were previously paper, we establish a set of trichotomy results for conjunc- announced [43], but all other results are new. tive queries with HAVING predicates parametrized by (α, θ). Keywords Probabilistic Databases · Query Evaluation · For such queries (without self joins), one of the following Sampling Algorithms · Semirings · Safe Plans three statements is true: (1) The exact evaluation problem has P-time data complexity. In this case, we call the query safe.
    [Show full text]
  • Another Problem with Variants on Either Side of P Vs. NP Divide
    Another problem with variants on either side of P vs. NP divide Leonid Chindelevitch 28 March 2019 Theory Seminar, Spring 2019, Simon Fraser University Background From mice to men through genome rearrangements Mouse X-Chromosome Human X-Chromosome 1 Ancestral Reconstruction M3 M1 M2 A B C D Input: Tree and genomes A; B; C; D Output: Ancestral genomes M1; M2; M3 2 The Median of Three A M B C Input: Genomes A; B; C Output: Genome M (the median, AKA the lowest common ancestor) which minimizes d(A; M) + d(B; M) + d(C; M) 3 Genome Elements Adjacencies: fah; bhg; fbt ; ct g; telomeres: at , ch 4 at ah bt bh ct ch 0 1 at 1 0 0 0 0 0 ah B 0 0 0 1 0 0 C B C b B 0 0 0 0 1 0 C t B C B C bh B 0 1 0 0 0 0 C B C ct @ 0 0 1 0 0 0 A ch 0 0 0 0 0 1 This is a genome matrix. Genome matrices can be represented by involutions: (ah bh)(bt ct ). Genome Representations ah at bh bt ch ct 5 Genome matrices can be represented by involutions: (ah bh)(bt ct ). Genome Representations ah at bh bt ch ct at ah bt bh ct ch 0 1 at 1 0 0 0 0 0 ah B 0 0 0 1 0 0 C B C b B 0 0 0 0 1 0 C t B C B C bh B 0 1 0 0 0 0 C B C ct @ 0 0 1 0 0 0 A ch 0 0 0 0 0 1 This is a genome matrix.
    [Show full text]
  • Quantum Complexity Theory
    QQuuaannttuumm CCoommpplelexxitityy TThheeooryry IIII:: QQuuaanntutumm InInterateracctivetive ProofProof SSysystemtemss John Watrous Department of Computer Science University of Calgary CCllaasssseess ooff PPrroobblleemmss Computational problems can be classified in many different ways. Examples of classes: • problems solvable in polynomial time by some deterministic Turing machine • problems solvable by boolean circuits having a polynomial number of gates • problems solvable in polynomial space by some deterministic Turing machine • problems that can be reduced to integer factoring in polynomial time CComommmonlyonly StudStudiedied CClasseslasses P class of problems solvable in polynomial time on a deterministic Turing machine NP class of problems solvable in polynomial time on some nondeterministic Turing machine Informally: problems with efficiently checkable solutions PSPACE class of problems solvable in polynomial space on a deterministic Turing machine CComommmonlyonly StudStudiedied CClasseslasses BPP class of problems solvable in polynomial time on a probabilistic Turing machine (with “reasonable” error bounds) L class of problems solvable by some deterministic Turing machine that uses only logarithmic work space SL, RL, NL, PL, LOGCFL, NC, SC, ZPP, R, P/poly, MA, SZK, AM, PP, PH, EXP, NEXP, EXPSPACE, . ThThee lliistst ggoesoes onon…… …, #P, #L, AC, SPP, SAC, WPP, NE, AWPP, FewP, CZK, PCP(r(n),q(n)), D#P, NPO, GapL, GapP, LIN, ModP, NLIN, k-BPB, NP[log] PP P , P , PrHSPACE(s), S2P, C=P, APX, DET, DisNP, EE, ELEMENTARY, mL, NISZK,
    [Show full text]
  • Class on Algorithmic Lower Bounds and Hardness Proofs, Lecture 11
    6.890: Algorithmic Lower Bounds: Fun With Hardness Proofs Fall 2014 Lecture 11 Scribe Notes Prof. Erik Demaine 1 Recap In the last lecture we looked at several different types of reductions for NPO problems as shown in the figure below. Reductions [Crescenzi 1997] Recall that one of the most commonly used reductions is the L-reduction, introduced by Papadim­ itriou and Yannakakis in [?]. This reduction consists of a pair of polynomial mappings (f(:); g(:)) 0 where f maps an instance x of the problem A we are reducing from to an instance x of the problem 0 0 B we are reducing to, and g maps a feasible solution y (of x i.e. instance of B) to a feasible solution y (of x) such that the following conditions hold: 0 1. OP TB (x ) ≤ α OP TA(x) 0 0 2. jcostA(y) − OP TA(x)j ≤ β jcostB(y ) − OP TB(x )j for some non-zero α; β . Thus an L-reduction implies a PTAS-reduction with δ(E) = E/(αβ) for the minimization case. As the figure above shows L ; AP-reduction because in the maximization case 1 δ(E) = (αβ(1+1/E)−1) which is a non-linear function. Also, we have discussed the following APX-complete problems so far (results introduced in [?, ?]): • Max 3-SAT-3 and Max3E-SAT-E5: Where we L-reduced from Max 3-SAT and used properties of expander graphs to account for the bound on occurences of a variable. • Independent set (for bounded degree graphs): By Strict reduction from the above problem 1 • Vertex cover (for bounded degree graphs): By L-reduction from the above since in bounded degree graphs both independednt set and vertex cover are Θ(jV j).
    [Show full text]
  • Master's Thesis
    Modular forms modulo p FRANCESC-XAVIER GISPERT SÁNCHEZ under the direction of ADRIAN IOVITA and MATTEO LONGO A thesis submitted to in partial fulfilment of the requirements for the ALGANT MASTER’S DEGREE IN MATHEMATICS Defence date: 16th July 2018 Biblatex information: @thesis{gispert2018modformsmodp, author={Gispert Sanchez, Francesc}, title={Modular forms modulo \(p\)}, date={2018-07-16}, institution={Universität Regensburg and Università degli Studi di Padova}, type={Master’s thesis}, pagetotal={99} } © 2018 by Francesc Gispert Sánchez. This Master’s degree thesis on modular forms modulo p is made available under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Licence. Visit http://creativecommons.org/licenses/by-nc-sa/4.0/ to view a copy of this licence. Dedicat a en Julian Assange, la Tamara Carrasco, en Roger Español, l’Adrià (d’Esplugues de Llobregat), l’Anna Gabriel, en Pablo Hasél, en Jordi Perelló, la Clara Ponsatí, en Josep Valtònyc i, en general, a tots els valents que són injustament perseguits per la seva incansable lluita per a l’alliberament nacional. Segarem cadenes fins a la vostra llibertat, que és també la nostra. Donec perficiam. Abstract This thesis presents the theory of modular forms in positive characteristic p. Under certain hypotheses, there are modular curves parametrizing elliptic curves with additional level structures of arithmetic interest. Katz defined modular forms as global sections of a family of line bundles over these modular curves. Moreover, the modular curves have a special kind of points called the cusps. The germs of a modular form at the cusps correspond to power series in one variable called the q–expansions of the modular form.
    [Show full text]
  • A Survey on the Structure of Approximation Classes
    COMPUTERSCIENCEREVIEW 4 (2010) 19±40 available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/cosrev Survey A survey on the structure of approximation classes Bruno Escoffier∗, Vangelis Th. Paschos LAMSADE, CNRS and Université Paris­Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France ARTICLEINFO ABSTRACT Article history: The structure of approximability classes by the introduction of approximation preserving Received 15 December 2008 reductions has been one of the main research programmes in theoretical computer science Received in revised form during the last thirty years. This paper surveys the main results achieved in this domain. 13 November 2009 c 2009 Elsevier Inc. All rights reserved. Accepted 13 November 2009 1. Introduction In a second time, the objective is to structure every such sub­class by exhibiting “hard” problems for each of them. The issue of polynomial approximation theory is the con­ This requires a processing of the same type as for NP­ struction of algorithms for NP­hard problems that compute completeness but adapted to the combinatorial characteris­ “good” (under some predefined quality criterion) feasible so­ tics of each of the approximability classes handled. This can lutions for them in polynomial time. This issue appears very be done by the introduction of notions of approximability pre­ early (in the seminal paper by Johnson [1]), only three years serving reductions. Sections 4–8 are dedicated to such reduc­ after the celebrated Cook’s theorem [2]. For over thirty years, tions. polynomial approximation has been a major research pro­ The technique of transforming a problem into another in gram in theoretical computer science that has motivated nu­ such a way that the solution of the latter entails, somehow, merous studies by researchers from all over the world.
    [Show full text]