...this paper we first see how NP problems can be categorized and look at some examples. We then analyze the subset sum problem and prove its NP completeness. We look at various algorithms used to solve the problem and the efficiency of each algorithm. Hence we prove that subset sum problem is NP complete. Keywords: NP complete, NP hard, complexity, subset sum, exponential I. Introduction In the study of computation complexity of problems, we find out the resources required during computation like time, memory and space required. We check if the problem can be solved in polynomial time by some algorithm. An algorithm is said to be solved in polynomial time if its worst case efficiency belongs to O (p (n)) where p (n) is the polynomial of the problems input size. Problems that can be solved in polynomial time are called tractable and those that cannot be solved in polynomial time are intractable. The formal definition of NP problem is as follows: A problem is said to be Non-deterministically Polynomial (NP) if we can find a nondeterministic Turing machine that can solve the problem in a polynomial number of nondeterministic moves. Equivalent definition of NP problem: A problem is said to be NP if 1. its solution comes from a finite set of possibilities, and 2. It takes polynomial time to verify the correctness of a possible solution. NP-complete problems NP-complete problems are a subset of class NP. An problem p is NP-complete, if 1. p ∈ NP (you can solve it in polynomial...
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...ACHIEVEMENT REQUIREMENTS GSW 1110 Section 146L Fall 2015 |Instructor: |Joseph Celizic | |E-mail: |cjoseph@bgsu.edu | |Office: |421 East Hall | |Office Hours: |Tuesday & Thursday: 4:00 – 5:30 | | |(and by appointment) | |Mailbox: |210 East Hall (my mailbox is above my name) | |Learning Commons: |140 Jerome Library | |Learning Commons Phone: |372-2823 (call ahead to make an appointment) | REQUIRED COURSE TEXTS AND MATERIALS • Kirszner & Mandell’s The Brief Wadsworth Handbook (BGSU Special Edition). 7th edition. Laurie Kirszner and Stephen Mandell. Wadsworth, Cengage Learning, 2013 • A laptop with a word processing program (Microsoft Word or Open Office) that you must bring to every class, fully...
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...Graphing Applications and the TSP Name: Institution: Graphing Applications and the TSP Question 1 A Hamiltonian cycle is a closed loop within a graph that visits all its vertices exactly once. An example of the Hamiltonian cycle is the Travelling salesman problem. The solution to the minimum length of a Hamiltonian cycle is an NP complete problem that cannot be computed in polynomial time. This means that the minimum possible path cannot be computed by a deterministic machine. When completing a Hamiltonian cycle, one has to make sure that there are only two edges getting in and out of a vertex. In addition one has to ensure that there are no sub-cycles in the cycle. In a Hamiltonian cycle with n vertices, the number of different cycles that can be completed is (n-1)! /2 in a complete undirected graph and (n-1)! In a complete directed graph (Narasimhan, 2009). Question 2 A Euler cycle is a path that passes through all the edges of a graph exactly once. It usually starts and ends at the same vertex. For one to construct a Euler cycle all the vertices in the graph must have an even degree. Therefore, one can conclude that any graph with all vertices of an even degree connected is a Euler cycle. Unlike the Hamiltonian cycle, the Euler cycle can be computed in polynomial time. The Euler cycle can be constructed using the Fleury’s or Hierholzer’s algorithm. Question 3 The minimum length of a Hamiltonian cycle is the shortest path that can...
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...GEOGRAPHY OF THE MIDDLE EAST & NORTH AFRICA Geog 209 - Fall 2014 T-Th 12:00-1:20 Plus Discussion Section and evening films McKenzie 240A Prof. Shaul Cohen Condon 107G Tel. 346-4500 Office Hours Tuesday 12:00-1:00 OBA scohen@uoregon.edu GTFs Ashley Wall Jennings Office Hours M 1:00-2:00 ajenning@uoregon.edu Christine Carolan ccarolan@uoregon.edu Purpose: This course explores the geography of the Middle East with an emphasis on politics, culture, and regional cohesion. Through a variety of sources including modern literature, film, images of landscape, traditional academic texts and the daily news, we will pursue an understanding of those elements that characterize the region, as well as those features that are distinct and mark different peoples and places. We will examine local, sub-national, national, and international issues relating to identity and status, history, environment, economy and other topics, in an attempt to create a portrait of daily life in the many venues of the region, whether they be urban or rural, coastal or desert, North African or Asian, Muslim, Christian, Jewish, Arab, Israeli, Turkish, Iranian, and so on. Our goal will be to use the information available to us to discern patterns in the region that allow us to grasp its richness and complexity, to gain a sense of its past, contextualize current changes, and to anticipate future directions. Resources: For this course we will work from a number of selected writings, and...
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...Algorithms Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Singapore Tokyo To my parents Preface Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872–1970) Most natural optimization problems, including those arising in important application areas, are NP-hard. Therefore, under the widely believed conjecture that P = NP, their exact solution is prohibitively time consuming. Charting the landscape of approximability of these problems, via polynomial time algorithms, therefore becomes a compelling subject of scientific inquiry in computer science and mathematics. This book presents the theory of approximation algorithms as it stands today. It is reasonable to expect the picture to change with time. The book is divided into three parts. In Part I we cover a combinatorial algorithms for a number of important problems, using a wide variety of algorithm design techniques. The latter may give Part I a non-cohesive appearance. However, this is to be expected – nature is very rich, and we cannot expect a few tricks to help solve the diverse collection of NP-hard problems. Indeed, in this part, we have purposely refrained from tightly categorizing algorithmic techniques so as not to trivialize matters. Instead, we have attempted to capture, as accurately as possible, the individual character of each problem, and point out connections between problems and algorithms for solving them. In Part II, we...
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...Academic essay on Annie Proulx's "Job Story" Choices are something we all make. Not necessarily important choices, but there will always be a time to make them. It's not always good choices, but they have to be made. There will always be consequences, whether it's bad or good. Throughout the story, Leeland Lee has to make a lot of choices. Where to live, where to work and when to work. All the different choices he made, put him in the position he is now. Leeland Lee is an awkward-looking young boy. His face is heavily boned, which he has gotten from his mom, his neck is quite thick and he has red-gold hair. His eyes are as pouchy as a middle-aged alcoholic. His nose is broad and lays close to his face. Lori Bovee is Leeland Lee's wife. She has an undistinguished oval face, and hair of medium length. Leeland Lee is the protagonist of the story, because he is the main character. I would say Leeland is a flat and static character as he is an endless optimist. He doesn't give up when it comes to finding a new job, and despite his wife dying he still gets a job at Unique Eats. The reason he is a static character is because he doesn't change at all. After getting several different jobs he doesn't change anything, after his mom and wife dies he doesn't change one single thing except the fact he isn't listening to the radio anymore, but since that have been an important factor of the story all along, it can also show a lot about how he has changed. The story starts November...
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...Essay on “Job History” written by Annie Proulx In the short story “Job History” written by Annie Proulx, we follow Leeland Lee from the time of his birth, until he is about fifty years old. In the short story we follow Leeland through his harsh life, with ups and downs, in the form of thoughts, feelings, incidents, etc. Leeland is born in a ranch in Wyoming, and lives there with his wife Lori. Leeland does not look particularly good, in fact he is a very unattractive man; (page 91, line 12)“Leeland’s face shows heavy bones from his mother’s side. His neck is thick and his red-gold hair plastered down in bangs. Even as a child his eyes are as pouchy as those of a middle-age alcoholic, the brows rod-straight above wandering out-of-line eyes. His nose lies broad and close to his face, his mouth seems to have been cut with a single chisel blow into easy flesh” And in the top of that, we see how Leeland through his life, tries to find a successful career, but fails consistently. He moves various times from place to place, too seek occupation and good business. But it is hard when you’re a high school dropout, without a career. Leeland have to changes his job constantly, because of his lack of luck, and since he can’t get along white his bosses. He is never able to stay at one job or place for long, which lead to problems in the family. He has a hard time supporting his wife, and their children financially. Throughout the story the author, Annie Proulx manages...
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...An essay is a piece of writing which is often written from an author's personal[->0] point of view[->1]. Essays can consist of a number of elements, including: literary criticism[->2], political manifestos[->3], learned arguments[->4], observations of daily life, recollections, and reflections of the author. The definition of an essay is vague, overlapping with those of an article[->5] and a short story[->6]. Almost all modern essays are written in prose[->7], but works in verse[->8] have been dubbed essays (e.g. Alexander Pope[->9]'s An Essay on Criticism[->10] and An Essay on Man[->11]). While brevity usually defines an essay, voluminous works like John Locke[->12]'s An Essay Concerning Human Understanding[->13] and Thomas Malthus[->14]'s An Essay on the Principle of Population[->15] are counterexamples. In some countries (e.g., the United States and Canada), essays have become a major part of formal education[->16]. Secondary students are taught structured essay formats to improve their writing skills, and admission essays[->17] are often used by universities[->18] in selecting applicants and, in the humanities and social sciences, as a way of assessing the performance of students during final exams. The concept of an "essay" has been extended to other mediums beyond writing. A film essay is a movie that often incorporates documentary film making styles and which focuses more on the evolution of a theme or an idea. A photographic essay[->19] is an attempt to cover a topic...
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...We all know love. We have all loved in some kind of way. We love our parents, significant others and even our friends. But we can also love other things like animals or material things. But what is the difference between loving and liking? And is it better not to love and feel pain or to love and be hurt in the progress? Jonathan Franzen seeks to answer these questions in his essay “Liking Is for Cowards. Go for What Hurts”. The essay “Liking Is for Cowards. Go for What Hurts” is, as mentioned, written by Jonathan Franzen and published in The New York Times, May 28, 2011. Jonathan Franzen is born in 1959, and he is an acclaimed American novelist and essayist. The essay is based on the commencement speech he delivered at Kenyon College in Ohio, USA. “Our technology has become extremely adept in creating products that correspond to our fantasy ideal of an erotic relationship, in which the beloved object asks for nothing and gives everything, instantly. (…)” As Franzen claims in his essay, many people can feel like they love their technological object. It gives them a satisfaction, which human interaction maybe wouldn’t. Franzen however thinks, that people in general don’t love material things: they like them. There is a major difference between loving and liking – even though it might appear small. “Liking, in general, is commercial culture’s substitute for loving.” Products are made to be likeable, but if that concept in transferred to a person, you would instantly see...
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...Reaction – “Salvation” The nonfiction short essay “Salvation” written by Langston Hughes in 1940, presents a theme on the literal and often manipulated perception of children. Hughes narrates the essay as he recounts his disappointing attempt at salvation. Hughes aunt told him that when she was saved by Jesus she saw a light, and felt something happen within herself. As children will do, Hughes took her story literally and was heartbroken as he sat in front of the church and watched other children “saved” while he was not. He believed that Jesus must not want him because he did not see or feel anything. In the end, Hughes is forced to lie about accepting Jesus and in turn rejects the Christian faith all together. I related to Hughes story on many accounts. I am a mother of three young children who perceive everything in life literally, and as a young girl I was raised in a very religious environment. I could visualize and almost feel Hughes devastation as he sat at the front of the church crushed by the thoughts of God not wanting him. “Still I kept waiting to see Jesus” (Barnet, Cain, & Burto, 2011, pp. 351). One of the churches that my family attended for a short time during my childhood practiced speaking in tongs. I specifically remember feeling just like Hughes during a service when other children were speaking in unnatural languages perceived to be sent from God himself. I could not understand why I was not chosen to talk for God and intern was hurt and...
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...write an essay on drugs for this topic. Drugs are very harmful and keep the capability of dragging an individual towards death and destruction. People all over the world want to eradicate the adverse situation of drug addiction from this world and this is the reason why they are busy in writing essays on drugs. Essays on drugs are of many types such as war on drugs essay, essay on performance enhancing drugs in sports, essay on drug abuse, essay on illegal drugs, essay on drug addiction, essay on drug use, essay on drugs and alcohol and essay on drug testing, etc. The essays on drugs should be initiated by bringing in the information related to the topic of the essay on drug. You should know what drugs are. In an essay on drugs, you will have to write about drugs, their affects and the reasons due to which people use them, you have to include the information about why the drugs are so famous and how harmful are they. A persuasive essay on drugs will be one, which will be according to the topic of the essay on drugs. It should have a full-fledged introduction, which should introduce the topic completely. The introduction should also have a thesis statement that should be the main idea of the essay on drugs. A thesis statement should be based on the essay question to which your essay on drugs is an answer. A thesis statement of an essay on drugs can be one sentence or more than one sentence but it is suggested that it should be only one sentence. Essays on drug abuse or essays on drug...
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...Dog Food Fried, Flipped and Devoured Eating dog food may not be seen as the easiest thing to stomach because the smell, texture and by-products found within the mixture are thought of as awful and revolting. In Ann Hodgman’s “No Wonder They Call Me a Bitch”, although the title may suggest otherwise, different types of dog food are discussed. In this essay, Hodgman aims to inform her readers about the inequality in different types of dog food, as well as to entertain and amuse. Hodgman is writing to a broad audience, one of dog lovers and owners, as well as those that may be curious as to what is really in dog food, or more generally the packaging and processing of mass-produced food of any kind. Language, description and humor are techniques Hodgman skillfully uses to create a strong, convincing essay. Within this essay, Hodgman continually uses diction and syntax that a diverse audience can understand and relate to. In the beginning, the reader feels repulsed by what is written. However, through a light tone of voice and sentence structure, the levity of the experiences and feelings occurring throughout the experiment is conveyed. The personal anecdotes and thoughts make the tone feel less formal and much more story-like. In this way, Hodgman produces a piece that connects the information and readers Shortly after capturing interest through stating questions, Hodgman begins using large amounts of description. Each type of dog food is explained and illustrated so well that...
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...Business Essay Format and its Structure What Makes Business Essay Format? Those students, who study different business courses, for example management, marketing, have to write many different business essays. This type of essay as well as the others (argumentative essay, classification essay, cause and effect essay, persuasive essay, comparative essay, etc) has peculiar features that should be observed carefully in order to produce an essay in conformity with all instructions of research paper writing. One of these features to be observed at writing is business essay format as there is no doubt that format of the essay plays the same important role as the content of business essay. When student has made comprehensive research of the object and has found interesting ideas and sources to support hisher position, it does not mean that heshe has coped with a writing task. The writer should make the next step that is to organize hisher research paper writing in a proper way. It means that attention should be paid to the business essay format. In general, there are different formats of research paper – ChicagoTurabian, Harvard, APA, AMA, MLA format paper; which of them suits you the most depends on the business essay topic, subject, requirements set by the professor. When the student has chosen the format of the business essay it is very important to follow it and not use any elements of the other essay format. In case when you do not get any instruction as for the business...
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...is comprised of two parts: Part A has six questions; you are to answer four (4) questions only. Each question is worth 8 marks (total of 32 marks). Each answer should be approximately half to one page long. Part B has two essay questions; you are to answer one (1) question only. This question is worth 18 marks. Each answer should be two to three pages long. It will be a closed book examination- NO MATERIAL (other than Foreign Language Dictionaries) shall be allowed inside the Exam Venue. You may wish to take a calculator into the exam if you wish. This Exam provides you with an opportunity to demonstrate knowledge application. Although diagrams, analytical modelling and calculations may help to support your answer, this will not be a key focus of the questions. PART A For Part A – Short Answer Questions - while all topics are examinable you should focus on the following themes/chapters: * Operations Management, Performance and Strategy * The Nature of Planning and Control * Capacity Planning * ERP * People, Jobs and Organisation * Lean Synchronisation * Operations Improvement Each answer should be approximately half to one page long. Using relevant examples will help support your answer. PART B For Part B – Essay Question This question will enable you to use the themes listed in Part A and additional material that you have learnt in the course more broadly to write a comprehensive answer to the question. Each answer should be approximately...
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...Mennesker i verden. How can I write a critical essay? Actually, writing such an article is very simple if you know the main characteristics of an essay. For sure, you have learned how to write the introduction, the body and the conclusion parts. Now, it is time to get familiar with the critical essay which considerably differs from other types of writing that are likely to be assigned during the studying process. Actually, its main goal is to provide an analysis of the article that you wish to work with in the future. Let us summarize the main steps in writing a critical essay and learn a tip or two to perfect your writing skills. 1. Study your topic interest. This gives you an edge to prepare in writing an essay because you can learn more about the article that you wish to “criticize”. In any case, you are still going to read an article before writing a critique about it so you might as well understand clearly what the subject is all about. 2. Look for information. Writing an essay will always involve the search for information. At the very least, looking for other sources of knowledge that are to affirm a notion in an essay is always necessary. 3. Create a thesis statement. Once you have finished with the article in reference, you can create a thesis statement that will serve as the guiding medium for you to write your critical essay. The main philosophy behind it is to make a thesis statement that will present a notion of either you agree with the author’s discussions...
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