NP-Completeness
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NP-Completeness (Chapter 8)
CSE 421" Algorithms NP-Completeness (Chapter 8) 1 What can we feasibly compute? Focus so far has been to give good algorithms for specific problems (and general techniques that help do this). Now shifting focus to problems where we think this is impossible. Sadly, there are many… 2 History 3 A Brief History of Ideas From Classical Greece, if not earlier, "logical thought" held to be a somewhat mystical ability Mid 1800's: Boolean Algebra and foundations of mathematical logic created possible "mechanical" underpinnings 1900: David Hilbert's famous speech outlines program: mechanize all of mathematics? http://mathworld.wolfram.com/HilbertsProblems.html 1930's: Gödel, Church, Turing, et al. prove it's impossible 4 More History 1930/40's What is (is not) computable 1960/70's What is (is not) feasibly computable Goal – a (largely) technology-independent theory of time required by algorithms Key modeling assumptions/approximations Asymptotic (Big-O), worst case is revealing Polynomial, exponential time – qualitatively different 5 Polynomial Time 6 The class P Definition: P = the set of (decision) problems solvable by computers in polynomial time, i.e., T(n) = O(nk) for some fixed k (indp of input). These problems are sometimes called tractable problems. Examples: sorting, shortest path, MST, connectivity, RNA folding & other dyn. prog., flows & matching" – i.e.: most of this qtr (exceptions: Change-Making/Stamps, Knapsack, TSP) 7 Why "Polynomial"? Point is not that n2000 is a nice time bound, or that the differences among n and 2n and n2 are negligible. Rather, simple theoretical tools may not easily capture such differences, whereas exponentials are qualitatively different from polynomials and may be amenable to theoretical analysis. -
Probabilistically Checkable Proofs Over the Reals
Electronic Notes in Theoretical Computer Science 123 (2005) 165–177 www.elsevier.com/locate/entcs Probabilistically Checkable Proofs Over the Reals Klaus Meer1 ,2 Department of Mathematics and Computer Science Syddansk Universitet, Campusvej 55, 5230 Odense M, Denmark Abstract Probabilistically checkable proofs (PCPs) have turned out to be of great importance in complexity theory. On the one hand side they provide a new characterization of the complexity class NP, on the other hand they show a deep connection to approximation results for combinatorial optimization problems. In this paper we study the notion of PCPs in the real number model of Blum, Shub, and Smale. The existence of transparent long proofs for the real number analogue NPR of NP is discussed. Keywords: PCP, real number model, self-testing over the reals. 1 Introduction One of the most important and influential results in theoretical computer science within the last decade is the PCP theorem proven by Arora et al. in 1992, [1,2]. Here, PCP stands for probabilistically checkable proofs, a notion that was further developed out of interactive proofs around the end of the 1980’s. The PCP theorem characterizes the class NP in terms of languages accepted by certain so-called verifiers, a particular version of probabilistic Turing machines. It allows to stabilize verification procedures for problems in NP in the following sense. Suppose for a problem L ∈ NP and a problem 1 partially supported by the EU Network of Excellence PASCAL Pattern Analysis, Statis- tical Modelling and Computational Learning and by the Danish Natural Science Research Council SNF. 2 Email: [email protected] 1571-0661/$ – see front matter © 2005 Elsevier B.V. -
Chapter 9. Coping with NP-Completeness
Chapter 9 Coping with NP-completeness You are the junior member of a seasoned project team. Your current task is to write code for solving a simple-looking problem involving graphs and numbers. What are you supposed to do? If you are very lucky, your problem will be among the half-dozen problems concerning graphs with weights (shortest path, minimum spanning tree, maximum flow, etc.), that we have solved in this book. Even if this is the case, recognizing such a problem in its natural habitat—grungy and obscured by reality and context—requires practice and skill. It is more likely that you will need to reduce your problem to one of these lucky ones—or to solve it using dynamic programming or linear programming. But chances are that nothing like this will happen. The world of search problems is a bleak landscape. There are a few spots of light—brilliant algorithmic ideas—each illuminating a small area around it (the problems that reduce to it; two of these areas, linear and dynamic programming, are in fact decently large). But the remaining vast expanse is pitch dark: NP- complete. What are you to do? You can start by proving that your problem is actually NP-complete. Often a proof by generalization (recall the discussion on page 270 and Exercise 8.10) is all that you need; and sometimes a simple reduction from 3SAT or ZOE is not too difficult to find. This sounds like a theoretical exercise, but, if carried out successfully, it does bring some tangible rewards: now your status in the team has been elevated, you are no longer the kid who can't do, and you have become the noble knight with the impossible quest. -
CS 561, Lecture 24 Outline All-Pairs Shortest Paths Example
Outline CS 561, Lecture 24 • All Pairs Shortest Paths Jared Saia • TSP Approximation Algorithm University of New Mexico 1 All-Pairs Shortest Paths Example • For the single-source shortest paths problem, we wanted to find the shortest path from a source vertex s to all the other vertices in the graph • We will now generalize this problem further to that of finding • For any vertex v, we have dist(v, v) = 0 and pred(v, v) = the shortest path from every possible source to every possible NULL destination • If the shortest path from u to v is only one edge long, then • In particular, for every pair of vertices u and v, we need to dist(u, v) = w(u → v) and pred(u, v) = u compute the following information: • If there’s no shortest path from u to v, then dist(u, v) = ∞ – dist(u, v) is the length of the shortest path (if any) from and pred(u, v) = NULL u to v – pred(u, v) is the second-to-last vertex (if any) on the short- est path (if any) from u to v 2 3 APSP Lots of Single Sources • The output of our shortest path algorithm will be a pair of |V | × |V | arrays encoding all |V |2 distances and predecessors. • Many maps contain such a distance matric - to find the • Most obvious solution to APSP is to just run SSSP algorithm distance from (say) Albuquerque to (say) Ruidoso, you look |V | times, once for every possible source vertex in the row labeled “Albuquerque” and the column labeled • Specifically, to fill in the subarray dist(s, ∗), we invoke either “Ruidoso” Dijkstra’s or Bellman-Ford starting at the source vertex s • In this class, we’ll focus -
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/. -
Approximability Preserving Reduction Giorgio Ausiello, Vangelis Paschos
Approximability preserving reduction Giorgio Ausiello, Vangelis Paschos To cite this version: Giorgio Ausiello, Vangelis Paschos. Approximability preserving reduction. 2005. hal-00958028 HAL Id: hal-00958028 https://hal.archives-ouvertes.fr/hal-00958028 Preprint submitted on 11 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024 CAHIER DU LAMSADE 227 Septembre 2005 Approximability preserving reductions Giorgio AUSIELLO, Vangelis Th. PASCHOS Approximability preserving reductions∗ Giorgio Ausiello1 Vangelis Th. Paschos2 1 Dipartimento di Informatica e Sistemistica Università degli Studi di Roma “La Sapienza” [email protected] 2 LAMSADE, CNRS UMR 7024 and Université Paris-Dauphine [email protected] September 15, 2005 Abstract We present in this paper a “tour d’horizon” of the most important approximation-pre- serving reductions that have strongly influenced research about structure in approximability classes. 1 Introduction The technique of transforming a problem into another in such a way that the solution of the latter entails, somehow, the solution of the former, is a classical mathematical technique that has found wide application in computer science since the seminal works of Cook [10] and Karp [19] who introduced particular kinds of transformations (called reductions) with the aim of study- ing the computational complexity of combinatorial decision problems. -
Computability (Section 12.3) Computability
Computability (Section 12.3) Computability • Some problems cannot be solved by any machine/algorithm. To prove such statements we need to effectively describe all possible algorithms. • Example (Turing Machines): Associate a Turing machine with each n 2 N as follows: n $ b(n) (the binary representation of n) $ a(b(n)) (b(n) split into 7-bit ASCII blocks, w/ leading 0's) $ if a(b(n)) is the syntax of a TM then a(b(n)) else (0; a; a; S,halt) fi • So we can effectively describe all possible Turing machines: T0; T1; T2;::: Continued • Of course, we could use the same technique to list all possible instances of any computational model. For example, we can effectively list all possible Simple programs and we can effectively list all possible partial recursive functions. • If we want to use the Church-Turing thesis, then we can effectively list all possible solutions (e.g., Turing machines) to every intuitively computable problem. Decidable. • Is an arbitrary first-order wff valid? Undecidable and partially decidable. • Does a DFA accept infinitely many strings? Decidable • Does a PDA accept a string s? Decidable Decision Problems • A decision problem is a problem that can be phrased as a yes/no question. Such a problem is decidable if an algorithm exists to answer yes or no to each instance of the problem. Otherwise it is undecidable. A decision problem is partially decidable if an algorithm exists to halt with the answer yes to yes-instances of the problem, but may run forever if the answer is no. -
Admm for Multiaffine Constrained Optimization Wenbo Gao†, Donald
ADMM FOR MULTIAFFINE CONSTRAINED OPTIMIZATION WENBO GAOy, DONALD GOLDFARBy, AND FRANK E. CURTISz Abstract. We propose an expansion of the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain easily verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-Lojasiewicz (K-L)property holds, this is strengthened to convergence to a single con- strained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnega- tive matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems, and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions. 1. Introduction The alternating direction method of multipliers (ADMM) is an iterative method that was initially proposed for solving linearly-constrained separable optimization problems having the form: ( inf f(x) + g(y) (P 0) x;y Ax + By − b = 0: The augmented Lagrangian L of the problem (P 0), for some penalty parameter ρ > 0, is defined to be ρ L(x; y; w) = f(x) + g(y) + hw; Ax + By − bi + kAx + By − bk2: 2 In iteration k, with the iterate (xk; yk; wk), ADMM takes the following steps: (1) Minimize L(x; yk; wk) with respect to x to obtain xk+1. -
Approximation Algorithms
Lecture 21 Approximation Algorithms 21.1 Overview Suppose we are given an NP-complete problem to solve. Even though (assuming P = NP) we 6 can’t hope for a polynomial-time algorithm that always gets the best solution, can we develop polynomial-time algorithms that always produce a “pretty good” solution? In this lecture we consider such approximation algorithms, for several important problems. Specific topics in this lecture include: 2-approximation for vertex cover via greedy matchings. • 2-approximation for vertex cover via LP rounding. • Greedy O(log n) approximation for set-cover. • Approximation algorithms for MAX-SAT. • 21.2 Introduction Suppose we are given a problem for which (perhaps because it is NP-complete) we can’t hope for a fast algorithm that always gets the best solution. Can we hope for a fast algorithm that guarantees to get at least a “pretty good” solution? E.g., can we guarantee to find a solution that’s within 10% of optimal? If not that, then how about within a factor of 2 of optimal? Or, anything non-trivial? As seen in the last two lectures, the class of NP-complete problems are all equivalent in the sense that a polynomial-time algorithm to solve any one of them would imply a polynomial-time algorithm to solve all of them (and, moreover, to solve any problem in NP). However, the difficulty of getting a good approximation to these problems varies quite a bit. In this lecture we will examine several important NP-complete problems and look at to what extent we can guarantee to get approximately optimal solutions, and by what algorithms. -
Homework 4 Solutions Uploaded 4:00Pm on Dec 6, 2017 Due: Monday Dec 4, 2017
CS3510 Design & Analysis of Algorithms Section A Homework 4 Solutions Uploaded 4:00pm on Dec 6, 2017 Due: Monday Dec 4, 2017 This homework has a total of 3 problems on 4 pages. Solutions should be submitted to GradeScope before 3:00pm on Monday Dec 4. The problem set is marked out of 20, you can earn up to 21 = 1 + 8 + 7 + 5 points. If you choose not to submit a typed write-up, please write neat and legibly. Collaboration is allowed/encouraged on problems, however each student must independently com- plete their own write-up, and list all collaborators. No credit will be given to solutions obtained verbatim from the Internet or other sources. Modifications since version 0 1. 1c: (changed in version 2) rewored to showing NP-hardness. 2. 2b: (changed in version 1) added a hint. 3. 3b: (changed in version 1) added comment about the two loops' u variables being different due to scoping. 0. [1 point, only if all parts are completed] (a) Submit your homework to Gradescope. (b) Student id is the same as on T-square: it's the one with alphabet + digits, NOT the 9 digit number from student card. (c) Pages for each question are separated correctly. (d) Words on the scan are clearly readable. 1. (8 points) NP-Completeness Recall that the SAT problem, or the Boolean Satisfiability problem, is defined as follows: • Input: A CNF formula F having m clauses in n variables x1; x2; : : : ; xn. There is no restriction on the number of variables in each clause. -
Complexity-Adjustable SC Decoding of Polar Codes for Energy Consumption Reduction
Complexity-adjustable SC decoding of polar codes for energy consumption reduction Citation for published version (APA): Zheng, H., Chen, B., Abanto-Leon, L. F., Cao, Z., & Koonen, T. (2019). Complexity-adjustable SC decoding of polar codes for energy consumption reduction. IET Communications, 13(14), 2088-2096. https://doi.org/10.1049/iet-com.2018.5643 DOI: 10.1049/iet-com.2018.5643 Document status and date: Published: 27/08/2019 Document Version: Accepted manuscript including changes made at the peer-review stage Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
Simple Doubly-Efficient Interactive Proof Systems for Locally
Electronic Colloquium on Computational Complexity, Revision 3 of Report No. 18 (2017) Simple doubly-efficient interactive proof systems for locally-characterizable sets Oded Goldreich∗ Guy N. Rothblumy September 8, 2017 Abstract A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier’s strategy can be implemented in almost-linear-time. We present direct constructions of doubly-efficient interactive proof systems for problems in P that are believed to have relatively high complexity. Specifically, such constructions are presented for t-CLIQUE and t-SUM. In addition, we present a generic construction of such proof systems for a natural class that contains both problems and is in NC (and also in SC). The proof systems presented by us are significantly simpler than the proof systems presented by Goldwasser, Kalai and Rothblum (JACM, 2015), let alone those presented by Reingold, Roth- blum, and Rothblum (STOC, 2016), and can be implemented using a smaller number of rounds. Contents 1 Introduction 1 1.1 The current work . 1 1.2 Relation to prior work . 3 1.3 Organization and conventions . 4 2 Preliminaries: The sum-check protocol 5 3 The case of t-CLIQUE 5 4 The general result 7 4.1 A natural class: locally-characterizable sets . 7 4.2 Proof of Theorem 1 . 8 4.3 Generalization: round versus computation trade-off . 9 4.4 Extension to a wider class . 10 5 The case of t-SUM 13 References 15 Appendix: An MA proof system for locally-chracterizable sets 18 ∗Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel.