The Equivalence of Sampling and Searching
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Algorithms & Computational Complexity
Computational Complexity O ΩΘ Russell Buehler January 5, 2015 2 Chapter 1 Preface What follows are my personal notes created during my undergraduate algorithms course and later an independent study under Professor Holliday as a graduate student. I am not a complexity theorist; I am a graduate student with some knowledge who is–alas–quite fallible. Accordingly, this text is made available as a convenient reference, set of notes, and summary, but without even the slightest hint of a guarantee that everything contained within is factual and correct. This said, if you find an error, I would much appreciate it if you let me know so that it can be corrected. 3 4 CHAPTER 1. PREFACE Contents 1 Preface 3 2 A Brief Review of Algorithms 1 2.1 Algorithm Analysis............................................1 2.1.1 O-notation............................................1 2.2 Common Structures............................................3 2.2.1 Graphs..............................................3 2.2.2 Trees...............................................4 2.2.3 Networks.............................................4 2.3 Algorithms................................................6 2.3.1 Greedy Algorithms........................................6 2.3.2 Dynamic Programming...................................... 10 2.3.3 Divide and Conquer....................................... 12 2.3.4 Network Flow.......................................... 15 2.4 Data Structures.............................................. 17 3 Deterministic Computation 19 3.1 O-Notation............................................... -
Complexity Theory in Computer Science Pdf
Complexity theory in computer science pdf Continue Complexity is often used to describe an algorithm. One could hear something like my sorting algorithm working in n2n'2n2 time in complexity, it's written as O (n2)O (n'2)O (n2) and polynomial work time. Complexity is used to describe the use of resources in algorithms. In general, the resources of concern are time and space. The complexity of the algorithm's time is the number of steps it must take to complete. The cosmic complexity of the algorithm represents the amount of memory the algorithm needs to work. The complexity of the algorithm's time describes how many steps the algorithm should take with regard to input. If, for example, each of the nnn elements entered only works once, this algorithm will take time O'n)O(n)O(n). If, for example, the algorithm has to work on one input element (regardless of input size), it is a constant time, or O(1)O(1)O(1), the algorithm, because regardless of the size of the input, only one is done. If the algorithm performs nnn-operations for each of the nnn elements injected into the algorithm, then this algorithm is performed in time O'n2)O (n2). In the design and analysis of algorithms, there are three types of complexities that computer scientists think: the best case, the worst case, and the complexity of the average case. The best, worst and medium complexity complexity can describe the time and space this wiki will talk about in terms of the complexity of time, but the same concepts can be applied to the complexity of space. -
LSPACE VS NP IS AS HARD AS P VS NP 1. Introduction in Complexity
LSPACE VS NP IS AS HARD AS P VS NP FRANK VEGA Abstract. The P versus NP problem is a major unsolved problem in com- puter science. This consists in knowing the answer of the following question: Is P equal to NP? Another major complexity classes are LSPACE, PSPACE, ESPACE, E and EXP. Whether LSPACE = P is a fundamental question that it is as important as it is unresolved. We show if P = NP, then LSPACE = NP. Consequently, if LSPACE is not equal to NP, then P is not equal to NP. According to Lance Fortnow, it seems that LSPACE versus NP is easier to be proven. However, with this proof we show this problem is as hard as P versus NP. Moreover, we prove the complexity class P is not equal to PSPACE as a direct consequence of this result. Furthermore, we demonstrate if PSPACE is not equal to EXP, then P is not equal to NP. In addition, if E = ESPACE, then P is not equal to NP. 1. Introduction In complexity theory, a function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem [6]. A functional problem F is defined as a binary relation (x; y) 2 R over strings of an arbitrary alphabet Σ: R ⊂ Σ∗ × Σ∗: A Turing machine M solves F if for every input x such that there exists a y satisfying (x; y) 2 R, M produces one such y, that is M(x) = y [6]. -
Counting Below #P: Classes, Problems and Descriptive Complexity
Msc Thesis Counting below #P : Classes, problems and descriptive complexity Angeliki Chalki µΠλ8 Graduate Program in Logic, Algorithms and Computation National and Kapodistrian University of Athens Supervised by Aris Pagourtzis Advisory Committee Dimitris Fotakis Aris Pagourtzis Stathis Zachos September 2016 Abstract In this thesis, we study counting classes that lie below #P . One approach, the most regular in Computational Complexity Theory, is the machine-based approach. Classes like #L, span-L [1], and T otP ,#PE [38] are defined establishing space and time restrictions on Turing machine's computational resources. A second approach is Descriptive Complexity's approach. It characterizes complexity classes by the type of logic needed to express the languages in them. Classes deriving from this viewpoint, like #FO [44], #RHΠ1 [16], #RΣ2 [44], are equivalent to #P , the class of AP - interriducible problems to #BIS, and some subclass of the problems owning an FPRAS respectively. A great objective of such an investigation is to gain an understanding of how “efficient counting" relates to these already defined classes. By “efficient counting" we mean counting solutions of a problem using a polynomial time algorithm or an FPRAS. Many other interesting properties of the classes considered and their problems have been examined. For example alternative definitions of counting classes using relation-based op- erators, and the computational difficulty of complete problems, since complete problems capture the difficulty of the corresponding class. Moreover, in Section 3.5 we define the log- space analog of the class T otP and explore how and to what extent results can be transferred from polynomial time to logarithmic space computation. -
On Evaluating Human Problem Solving of Computationally Hard Problems
On Evaluating Human Problem Solving of Computationally Hard Problems Sarah Carruthers1 and Ulrike Stege1 Abstract This article is concerned with how computer science, and more exactly computational complexity theory, can inform cognitive science. In particular, we suggest factors to be taken into account when investigating how people deal with computational hardness. This discussion will address the two upper levels of Marr’s Level Theory: the computational level and the algorithmic level. Our reasons for believing that humans indeed deal with hard cognitive functions are threefold: (1) Several computationally hard functions are sug- gested in the literature, e.g., in the areas of visual search, visual perception and analogical reasoning, linguistic processing, and decision making. (2) People appear to be attracted to computationally hard recreational puzzles and games. Examples of hard puzzles include Sudoku, Minesweeper, and the 15-Puzzle. (3) A number of research articles in the area of human problem solving suggest that humans are capable of solving hard computational problems, like the Euclidean Traveling Salesperson Problem, quickly and near-optimally. This article gives a brief introduction to some theories and foundations of complexity theory and motivates the use of computationally hard problems in human problem solving with a short survey of known results of human performance, a review of some computation- ally hard games and puzzles, and the connection between complexity theory and models of cognitive functions. We aim to illuminate the role that computer science, in particular complexity theory, can play in the study of human problem solving. Theoretical computer science can provide a wealth of interesting problems for human study, but it can also help to provide deep insight into these problems. -
The Complexity of Counting Poset and Permutation Patterns
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 64(1) (2016), Pages 154–165 The complexity of counting poset and permutation patterns Joshua Cooper Anna Kirkpatrick Mathematics Department University of South Carolina 1523 Greene St., Columbia, SC 29208 U.S.A. Abstract We introduce a notion of pattern occurrence that generalizes both clas- sical permutation patterns as well as poset containment. Many ques- tions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems – particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension-2 posets is #P-hard; enumerating the lin- ear extensions that occur in realizers of dimension-2 posets can be done in polynomial time, while for unconstrained dimension it is GI-complete; counting not necessarily induced, injective occurrences among dimension- 2posetsis#P-hard; counting injective or not necessarily injective occur- rences of an arbitrary pattern in a dimension-1 text is #P-hard, although it is in FP if the pattern poset is constrained to have bounded intrinsic width; and counting injective occurrences of a dimension-1 pattern in an arbitrary text is #P-hard, while it is in FP for bounded-dimension texts. This framework naturally leads to a number of open questions, chief among which are (1) is it #P-hard to count the number of oc- currences of a dimension-2 pattern in a dimension-1 text, and (2) is it #P-hard to count the number of permutations avoiding a given pattern? 1 Introduction A tremendous amount of study has been dedicated to understanding occurrence or non-occurrence of combinatorial substructures: which substructures are avoidable and counting objects that avoid them, what substructures random and random- like objects possess, enumerating substructures in general objects, describing the subclass of objects that have particular substructure counts, etc. -
On Some Fine-Grained Questions in Algorithms and Complexity
On some fine-grained questions in algorithms and complexity Virginia Vassilevska Williams Massachusetts Institute of Technology, EECS and CSAIL Abstract In recent years, a new “fine-grained” theory of computational hardness has been developed, based on “fine-grained reductions” that focus on exact running times for problems. Mimicking NP-hardness, the approach is to (1) select a key problem X that for some function t, is conjectured to not be solvable by any O(t(n)1−") time algorithm for " > 0, and (2) reduce X in a fine-grained way to many impor- tant problems, thus giving tight conditional time lower bounds for them. This approach has led to the discovery of many meaningful relationships between problems, and to equivalence classes. The main key problems used to base hardness on have been: the 3-SUM problem, the CNF-SAT problem (based on the Strong Exponential Time Hypothesis (SETH)) and the All Pairs Shortest Paths Problem. Research on SETH-based lower bounds has flourished in particular in recent years showing that the classical algorithms are optimal for problems such as Approximate Diameter, Edit Distance, Frechet Distance and Longest Common Subsequence. This paper surveys the current progress in this area, and highlights some exciting new developments. 1 Introduction Arguably the main goal of the theory of algorithms is to study the worst case time complexity of fundamental computational problems. When considering a problem P , we fix a computational model, such as a Random Access Machine (RAM) or a Turing machine (TM). Then we strive to develop an efficient algorithm that solves P and to prove that for a (hopefully slow growing) function t(n), the algorithm solves P on instances of size n in O(t(n)) time in that computational model. -
The Computational Complexity of Games and Markets: an Introduction for Economists
The Computational Complexity of Games and Markets: An Introduction for Economists Andrew McLennan∗ University of Queensland December 2011 Abstract This is an expository survey of recent results in computer science related to the computation of fixed points, with the central one beingthat the problemof finding an approximate Nash equilibrium of a bimatrix game is PPAD-complete. This means that this problem is, in a certain sense, as hard as any fixed point problem. Subsequently many other problems have been shown to be PPAD- complete, including finding Walrasian equilibria in certain simple exchange economies. We also comment on the scientific consequences of complexity as a barrier to equilibration, and other sorts of complexity, for our understanding of how markets operate. It is argued that trading in complex systems of markets should be analogized to games such as chess, go, bridge, and poker, in which the very best players are much better than all but a small number of competitors. These traders make positive rents, and their presence is a marker of complexity. Consequences for the efficient markets hypothesis are sketched. Running Title: Complexity of Games and Markets Journal of Economic Literature Classification Numbers G12 and G14. Keywords: Computational complexity, two person games, Nash equilibrium, Scarf algorithm, NP, TFNP, PPAD, FPTAS, Lemke-Howson algorithm, Walrasian equilibrium, arbitrage, asset trading, efficient market hypothesis. ∗School of Economics, Level 6 Colin Clark Building, University of Queensland, QLD 4072 Australia, [email protected]. McLennan’s work was funded in part by Australian Research Council grant DP0773324. I would like to gratefully acknowledge the comments received at Games Toulouse 2011, and at seminar pre- sentations at the Australian National University, the Institute for Social and Economic Research at Osaka University, and the Kyoto Institute for Economic Research. -
In Search of the Densest Subgraph
algorithms Article In Search of the Densest Subgraph András Faragó and Zohre R. Mojaveri * Department of Computer Science, Erik Jonsson School of Engineering and Computer Science, The University of Texas at Dallas, P.O.B. 830688, MS-EC31, Richardson, TX 75080, USA * Correspondence: [email protected] Received: 23 June 2019; Accepted: 30 July 2019; Published: 2 August 2019 Abstract: In this survey paper, we review various concepts of graph density, as well as associated theorems and algorithms. Our goal is motivated by the fact that, in many applications, it is a key algorithmic task to extract a densest subgraph from an input graph, according to some appropriate definition of graph density. While this problem has been the subject of active research for over half of a century, with many proposed variants and solutions, new results still continuously emerge in the literature. This shows both the importance and the richness of the subject. We also identify some interesting open problems in the field. Keywords: dense subgraph; algorithms; graph density; clusters in graphs; big data 1. Introduction and Motivation In the era of big data, graph-based representations became highly popular to model various real-world systems, as well as diverse types of knowledge and data, due to the simplicity and visually appealing nature of graph models. The specific task of extracting a dense subgraph from a large graph has received a lot of attention, since it has direct applications in many fields. While the era of big data is still relatively young, the study and application of dense subgraphs started much earlier. -
Lipics-ITCS-2021-44.Pdf (0.6
Total Functions in the Polynomial Hierarchy Robert Kleinberg Cornell University, Ithaca, NY, USA Oliver Korten Columbia University, New York, NY, USA Daniel Mitropolsky Columbia University, New York, NY, USA Christos Papadimitriou Columbia University, New York, NY, USA Abstract We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for “polynomial empty pigeonhole principle”), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory’s quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes α-PEPP, which are inside FPNP|poly. Higher in the hierarchy, we identify the constructive version of the Sauer-Shelah lemma and the appropriate generalization of PPP that contains it, as well as the problem of finding a king in a tournament (a vertex k such that all other vertices are defeated by k, or by somebody k defeated). 2012 ACM Subject Classification Theory of computation → Complexity classes Keywords and phrases total complexity, polynomial hierarchy, pigeonhole principle Digital Object Identifier 10.4230/LIPIcs.ITCS.2021.44 Funding Daniel Mitropolsky: This research was supported in part by a grant from the Columbia-IBM center for Blockchain and Data Transparency, and by JPMorgan Chase & Co. This research was supported in part by NSF Awards CCF1763970 andCCF1910700, and by a research contract with Softbank Christos Papadimitriou: This research was supported in part by NSF Awards CCF1763970 and- CCF1910700, and by a research contract with Softbank. -
On Promise Problems
On Promise Problems in memory of Shimon Even Oded Goldreich Department of Computer Science Weizmann Institute of Science Rehovot Israel odedgoldreichweizmannac il First version January Current version July Abstract The notion of promise problems was introduced and initially studied by Even Selman and Yacobi Inform and Control Vol pages In this article we survey some of the applications that this notion has found in the twenty years that elapsed These include the notion of unique solutions the formulation of gap problems as capturing various approximation tasks the identication of complete problems esp ecially for the class SZK the indication of separations b etween certain computational resources and the enabling of presentations that b etter distill the essence of various pro ofs Keywords Complexity Theory NP reductions Unique Solutions Approximate Counting Com plexity of Approximation Prop erty Testing BPP Statistical ZeroKnowledge Complete Problems circuit complexity derandomization PCP machines that take advice innitely often classes This survey was written towards publication in a forthcoming b o ok in memory of Shimon Even Contents Introduction What are promise problems Some denitions Some indisp ensable uses of promise problems Relation to Shimon Even a p ersonal comment Organization Unique Solutions and Approximate Counting of Solutions The complexity -
P Versus NP Frank Vega
P versus NP Frank Vega To cite this version: Frank Vega. P versus NP. 2020. hal-02199310v14 HAL Id: hal-02199310 https://hal.archives-ouvertes.fr/hal-02199310v14 Preprint submitted on 28 Dec 2020 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. P versus NP Frank Vega CopSonic, 1471 Route de Saint-Nauphary 82000 Montauban, France [email protected] Abstract P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP ? The precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is P-Sel. P-Sel is the class of decision problems for which there is a polynomial time algorithm (called a selector) with the following property: Whenever it’s given two instances, a “yes” and a “no” instance, the algorithm can always decide which is the “yes” instance. It is known that if NP is contained in P-Sel, then P = NP .