diff --git a/Submissions/Weiyu_Wang_002746483/Assignment_4/Weiyu_Wang_Assignment_4.ipynb b/Submissions/Weiyu_Wang_002746483/Assignment_4/Weiyu_Wang_Assignment_4.ipynb new file mode 100644 index 0000000..329401a --- /dev/null +++ b/Submissions/Weiyu_Wang_002746483/Assignment_4/Weiyu_Wang_Assignment_4.ipynb @@ -0,0 +1,295 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "d0e9932e", + "metadata": {}, + "source": [ + "Question 1: \n", + "\n", + "A.Given an undirected graph G=(V,E) and a number k, determine if it is possible to color the vertices of the graph with at most k colors such that no two adjacent vertices share the same color. However, there's a twist: some vertices are pre-colored and cannot be changed.\n", + "\n", + "Is this problem NP-Complete?\n", + "\n", + "Answer: Yes, this problem is NP-Complete. It is a variant of the classic Graph Coloring Problem, which is known to be NP-Complete. The addition of pre-colored vertices does not simplify the problem, as it can be seen as a constraint on the original problem. To prove NP-Completeness, one can reduce the classical Graph Coloring Problem to this modified version in polynomial time.\n", + "\n", + "B.In a directed graph G=(V,E), is there a Hamiltonian cycle (a cycle that visits each vertex exactly once and returns to the start) such that no path between any two consecutive vertices in the cycle is longer than L (a given limit)?\n", + "\n", + "Is this problem NP-Complete?\n", + "\n", + "Answer: Yes, this problem is NP-Complete. The Hamiltonian Cycle problem itself is a well-known NP-Complete problem. The additional constraint of limiting the path length between two consecutive vertices does not make the problem any easier. In fact, it adds an extra layer of complexity. This can be shown by reducing the Hamiltonian Cycle problem to this limited path Hamiltonian cycle problem.\n", + "\n", + "C.Given a set of integers and a target sum S, determine if there exists a subset of these integers that sums up to S, with the constraint that no two elements in the subset can be consecutive numbers in the given set.\n", + "\n", + "Is this problem NP-Complete?\n", + "\n", + "Answer: This problem is NP-Complete. It is a variation of the classic Subset Sum Problem, which is known to be NP-Complete. The additional constraint of not allowing consecutive numbers in the subset adds a layer of complexity but does not simplify the problem. To prove its NP-Completeness, one can use a reduction from the classic Subset Sum Problem, considering instances where no elements are consecutive." + ] + }, + { + "cell_type": "markdown", + "id": "8c176e77", + "metadata": {}, + "source": [ + "Question 2:\n", + "\n", + "You are leading a large-scale software development project and need to form a team with a diverse set of skills. There are p distinct skills required for the project (e.g., front-end development, back-end development, database management, UI/UX design, security expertise, etc.). You have received applications from q potential team members. Each team member possesses a unique combination of these skills. The question is: For a given number ≤r≤q, is it possible to select at most r team members such that all p skills required for the project are covered? We'll call this the Optimal Project Team problem.\n", + "\n", + "Task: Show that the Optimal Project Team problem is NP-complete.\n", + "\n", + "Answer：The Optimal Project Team problem is NP-complete, demonstrated by showing it's in NP and reducing the NP complete Set Cover problem to it. Verifying a solution whether a selection of team members covers all required skills is doable in polynomial time, confirming the problem's membership in NP. For the reduction, each element in Set Cover corresponds to a skill, and each set to a potential team member with specific skills. This transformation, achievable in polynomial time, aligns with the question of covering all elements (skills) with a subset of sets (team members). Hence, the problem is as complex as the hardest problems in NP, indicating no known polynomial-time solution exists for all cases unless P equals NP.\n" + ] + }, + { + "cell_type": "markdown", + "id": "7312e055", + "metadata": {}, + "source": [ + "Question 3:\n", + " \n", + "You are organizing an international conference and have received applications from numerous experts to be keynote speakers. The conference has several topics (T1, T2, ..., Tn) to be covered. Each expert is skilled in one or more of these topics. However, due to time constraints, you can only select a maximum of s speakers. Your goal is to select these speakers in a way that all topics are covered. Each expert has a list of topics they can cover, and some topics might be covered by multiple experts.\n", + "\n", + "Problem Statement: Design an algorithm for the Efficient Conference Speaker Selection (ECSS) problem, where you need to select at most s speakers to cover all n topics.\n", + "\n", + "a. Show that the ECSS problem is NP-Complete.\n", + "\n", + "b. Is it computationally easier to determine if it's possible to cover all topics with s speakers, compared to actually selecting the s speakers?\n", + "\n", + "Answer：\n", + "a. Given a selection of s speakers, we can quickly verify if all n topics are covered by checking each speaker's list of topics. This process is efficient and can be done in polynomial time, thus the problem is in NP.The ECSS problem closely resembles the Set Cover problem, which is known to be NP-Complete. In Set Cover, we are given a universe of elements (topics in ECSS) and a collection of sets (experts and their topics in ECSS), with the goal to cover the universe with the fewest sets. The ECSS problem can be reformulated as a Set Cover problem, where each topic is an element in the universe, and each speaker represents a set containing the topics they can cover. Thus, finding the minimum number of speakers to cover all topics in ECSS is equivalent to solving the Set Cover problem, confirming the NP-Completeness of ECSS.\n", + "\n", + "b. Determining whether it is possible to cover all n topics with s speakers is a decision problem. This problem, though still challenging, is generally easier than actually finding the specific speakers. It involves checking the feasibility of the existence of such a combination of speakers, which can be done more straightforwardly.Selecting the actual s speakers to cover all topics is an optimization problem and is typically more complex than just establishing the possibility. It involves not only deciding if a solution exists but also identifying the best combination of speakers, which adds an additional layer of computational complexity." + ] + }, + { + "cell_type": "markdown", + "id": "14755aba", + "metadata": {}, + "source": [ + "Question 4:\n", + "\n", + "You are part of a band with m−1 other musicians, and you all share a rehearsal space in the city. Over the next \n", + "m weeks, each of you is supposed to lead a rehearsal session exactly once, ensuring that someone leads every week. However, everyone has scheduling conflicts on some weeks (like gigs, studio sessions, personal commitments, etc.), making it difficult to decide who should lead each week. Let's label the musicians, M∈{m1,…,mm}, the weeks, W∈{w1,…,wm}, and for musician mi, there's a set of weeks Si⊂{w1,…,wm} when they are not available to lead. A musician cannot leave Si empty. If a musician isn’t scheduled to lead in any of the \n", + "m weeks, the band must pay $100 to hire an external instructor for that week.\n", + "\n", + "Tasks:\n", + "\n", + "A. Formulate this problem as a maximum flow problem that schedules the maximum number of matches between musicians and weeks.\n", + "\n", + "B. Can all m musicians always be matched with one of the m weeks? Provide a proof or counterexample.\n", + "\n", + "Answer:\n", + "a. Construct a bipartite graph with musicians on one side, weeks on the other, and directed edges from each musician to the weeks they are available. Additionally, add a source node connected to all musicians and a sink node connected from all weeks.\n", + "Each edge from the source to a musician and from a musician to a week has a capacity of 1. This represents the musician's ability to lead one session. Similarly, edges from weeks to the sink also have a capacity of 1, indicating only one session can be held each week.\n", + "The goal is to find the maximum flow in this network, which corresponds to the maximum number of sessions that can be led by the available musicians.\n", + "\n", + "b.This scenario can be analyzed using Hall's Marriage Theorem, which states that in a bipartite graph, a perfect matching exists if and only if for every subset of the nodes on one side of the bipartition, the number of neighbors in the subset is at least as large as the subset itself.\n", + "In the context of the Busy Musicians' Schedule, a perfect matching (each musician matched to a week) is possible if and only if for every subset of musicians, the collective set of weeks they are available is at least as large as the number of musicians in the subset.\n", + "It is not guaranteed that all m musicians can always be matched with one of the m weeks. A counterexample would be if there is a subset of musicians whose combined availability does not cover the number of weeks equal to their number, violating Hall's condition." + ] + }, + { + "cell_type": "markdown", + "id": "f4f22f13", + "metadata": {}, + "source": [ + "Question 5:\n", + "\n", + "Given a directed graph G=(V,E) with n vertices and m edges, each vertex vi\n", + "has an associated value w(vi) which represents its weight. Additionally, there are k special vertices in G marked as key nodes. The task is to find the longest path in terms of the number of vertices visited, which starts and ends at any of the key nodes. This path must maximize the total weight of the visited vertices. However, there's a constraint: the path cannot visit any vertex more than once, except for the starting and ending key node which can be the same.\n", + "\n", + "Input:\n", + "G=(V,E) a directed graph\n", + "W={w(v1),w(v2)...,w(vn)} a set of vertex weights\n", + "K={k1,k2,...,kk} a set of key nodes\n", + "Output:\n", + "A path P which maximizes the total weight of visited vertices, starting and ending at key nodes, and visiting no vertex more than once (except for the start/end key node).\n", + "Example:\n", + "Consider a graph with 10 vertices, where each vertex has a weight between 1 to 10, and there are 3 key nodes. The edges are directed and form a complex network. Find the maximum coverage path.\n", + "\n", + "Answer:\n", + "This problem is NP-complete because given a path, it's easy to verify in polynomial time if it starts and ends at key nodes, does not visit any vertex more than once, and calculates the total weight. This problem can be seen as an extension of the Hamiltonian path problem (which is NP-complete), with additional constraints and the maximization of vertex weights.\n", + "\n", + "This problem is typically solved using backtracking or dynamic programming, but due to its NP-complete nature, these solutions are not efficient for large graphs.\n", + "\n", + "Enumerate All Paths Starting and Ending at Key Nodes: This step is computationally intensive and is the main reason why the problem is NP-hard.\n", + "Maximize the Total Weight: For each path, calculate the total weight and keep track of the path that yields the maximum weight.\n", + "Avoid Revisiting Vertices: Implement checks to ensure that no vertex is visited more than once (except for the starting/ending key node)." + ] + }, + { + "cell_type": "markdown", + "id": "56ed298d", + "metadata": {}, + "source": [ + "Question 6:\n", + "\n", + "The Boolean Satisfiability Problem (SAT) is one of the most famous problems in computer science. It involves determining whether there exists an assignment of truth values to variables that makes a given Boolean formula true. A Boolean formula is typically expressed in Conjunctive Normal Form (CNF), where a formula is a conjunction (AND) of clauses, and each clause is a disjunction (OR) of literals (variables or their negations).\n", + "\n", + "Your Task:\n", + "Prove that SAT is NP-complete. This involves two parts:\n", + "\n", + "Show that SAT is in NP: Demonstrate that if you are given a truth assignment for the variables, you can verify whether this assignment satisfies the formula in polynomial time.\n", + "\n", + "Prove NP-completeness: Use the concept of reduction to show that any problem in NP can be translated into an instance of SAT in polynomial time. For this, you can refer to the Cook-Levin Theorem, which establishes SAT as NP-complete by showing how any computation of a nondeterministic Turing machine can be represented as a Boolean formula.\n", + "\n", + "Specifics to Address:\n", + "Explanation of NP: Define the class NP and explain what it means for a problem to be in NP.\n", + "\n", + "Verification Process for SAT: Describe the procedure to verify whether a given assignment of truth values satisfies a Boolean formula.\n", + "\n", + "Cook-Levin Theorem: Elaborate on the Cook-Levin Theorem and explain how it can be used to prove that SAT is NP-complete.\n", + "\n", + "Reduction Example: Provide a simplified example of how a problem in NP can be reduced to SAT. You may choose a well-known NP problem like the Hamiltonian Cycle Problem for this illustration.\n", + "\n", + "This question requires a solid understanding of computational complexity theory, particularly the classes P, NP, and the concept of NP-completeness. It challenges you to articulate these concepts clearly and to demonstrate the process of proving the NP-completeness of a foundational problem in the field.\n", + "\n", + "Answer:\n", + "\n", + "A problem is in NP (Nondeterministic Polynomial time) if a given solution to the problem can be verified in polynomial time. SAT Problem Description: In SAT, we are given a Boolean formula (often in Conjunctive Normal Form, CNF), and the task is to determine whether there exists a truth assignment to the variables that makes the formula true.\n", + "\n", + "Verification Process:\n", + "\n", + "Input: A Boolean formula and a truth assignment for its variables.\n", + "Verification Procedure: For each clause in the formula, check if at least one of its literals is true under the given assignment. This step is linear in the length of each clause.\n", + "Overall Complexity: Since we check each clause independently and each check is polynomial in the length of the clause, the overall verification process is polynomial in the size of the formula.\n", + "\n", + "A problem is NP-complete if it is in NP, and every problem in NP can be reduced to it in polynomial time. The Cook-Levin Theorem is the cornerstone for proving the NP-completeness of SAT. It states that any problem in NP can be reduced to SAT in polynomial time.\n", + "\n", + "Process of Reduction:\n", + "\n", + "Turing Machine Model: Start with a nondeterministic Turing machine (NTM) that decides some language in NP.\n", + "Encoding the Computation: Encode the entire computation of the NTM on a given input string into a Boolean formula. This encoding involves creating variables to represent the state of the Turing machine, the symbols on the tape, and the position of the tape head at each step.\n", + "Formula Construction: Construct clauses in the Boolean formula to represent the transitions of the Turing machine, its initial configuration, the accept state, and the valid configurations at each step.\n", + "Satisfiability: The constructed formula is satisfiable if and only if there exists a sequence of transitions (choices made by the NTM) that leads to an accepting state." + ] + }, + { + "cell_type": "markdown", + "id": "754b28bd", + "metadata": {}, + "source": [ + "Question 7:\n", + "\n", + "You are given a problem called the Customized Delivery Route Problem (CDRP), which is suspected to be NP-hard. Your task is to demonstrate why CDRP is NP-hard by using a reduction from a known NP-complete problem.\n", + "\n", + "Customized Delivery Route Problem (CDRP)\n", + "In CDRP, you are given a list of cities, each with a set of delivery requests with specific time windows. Each request in a city must be fulfilled by visiting the city within the specified time window. The challenge is to determine whether there is a route that starts and ends at a designated home city and fulfills all delivery requests across the various cities within their respective time windows.\n", + "\n", + "Input Format\n", + "A list of cities, including the designated home city.\n", + "For each city, a set of delivery requests. Each request specifies a time window during which the delivery must be made.\n", + "The travel time between each pair of cities.\n", + "Output Format\n", + "A binary decision:\n", + "\n", + "\"YES\" if there exists a route starting and ending at the home city that fulfills all delivery requests within their time windows.\n", + "\"NO\" if no such route exists.\n", + "Specifics to Address in Your Proof\n", + "Choose an NP-complete Problem for Reduction: Select a well-known NP-complete problem, such as the Traveling Salesman Problem (TSP) or 3-SAT, and describe it briefly.\n", + "\n", + "Explain the Reduction Process:\n", + "\n", + "Show how an instance of your chosen NP-complete problem can be transformed into an instance of CDRP.\n", + "Ensure that the transformation process is done in polynomial time.\n", + "Demonstrate Equivalence:\n", + "\n", + "Explain why a solution to the NP-complete problem translates to a solution for CDRP and vice versa.\n", + "Conclude NP-hardness:\n", + "\n", + "Conclude that since CDRP can be reduced from an NP-complete problem, it is at least as hard as the hardest problems in NP, thereby proving it is NP-hard.\n", + "\n", + "Answer:\n", + " \n", + "We know the Traveling Salesman Problem is NP-complete. It's about finding the shortest path that visits each city once and returns to the start. To show CDRP is NP-hard, we transform TSP into CDRP. \n", + "\n", + "Cities and Distances: Use the same set of cities as in TSP. The travel time between each pair of cities in CDRP will be the same as the distances between cities in TSP.\n", + "Delivery Requests and Time Windows: For each city in CDRP, introduce a delivery request with a time window that is large enough to accommodate any tour that would be considered in the TSP. This time window should be consistent across all cities.\n", + "Home City: Designate any city as the home city in CDRP, similar to the origin city in TSP.\n", + "This transformation is clearly polynomial in time, as it involves only copying the city list and setting up broad time windows for delivery requests.\n", + "\n", + "If there exists a tour in TSP that visits all cities and returns to the origin, this tour can be directly used in CDRP to fulfill all delivery requests within their time windows, as the time windows are set broadly. Conversely, if a route in CDRP fulfills all delivery requests and returns to the home city, this route is a valid tour in TSP, visiting each city exactly once.\n", + "\n", + "Since we can transform TSP into CDRP in polynomial time and solutions in TSP are equivalent to solutions in CDRP, we conclude that CDRP is at least as hard as TSP.\n", + "As TSP is NP-complete, CDRP, by this reduction, is shown to be NP-hard. This means that CDRP is at least as hard as the hardest problems in NP, and therefore, it is NP-hard." + ] + }, + { + "cell_type": "markdown", + "id": "a82a83bf", + "metadata": {}, + "source": [ + "Question 8:\n", + "\n", + "You are given a necklace made of beads, each bead colored either red, blue, or green. The challenge is to determine if there exists a way to cut the necklace into three parts such that each part contains an equal number of beads, and each part has beads of all three colors.\n", + "\n", + "Input Format:\n", + "\n", + "A string representing the necklace, with characters 'R', 'B', and 'G' representing red, blue, and green beads, respectively.\n", + "Output Format:\n", + "\n", + "\"YES\" if it's possible to cut the necklace as described.\n", + "\"NO\" if no such division is possible.\n", + "Example:\n", + "\n", + "Input: \"RGBRGBRGB\"\n", + "Output: \"YES\"\n", + "\n", + "Answer:\n", + "\n", + "Given a necklace represented as a string of 'R', 'B', and 'G' characters, we need to determine if it's possible to cut this string into three equal parts, each containing all three colors.At first glance, this problem seems to involve checking multiple combinations to find a valid division, if one exists. The challenge is that as the length of the necklace increases, the number of possible divisions grows, potentially exponentially.TCNDP resembles certain partition problems, where a set (or string in this case) must be divided into subsets (or substrings) that meet specific criteria. However, unlike some classic partition problems, TCNDP requires each part to have an equal number of beads and all three colors, which adds complexity.A crucial step in proving NP-hardness would be to show that a known NP-hard problem can be reduced to TCNDP in polynomial time. For instance, if we could transform an instance of the 3-Partition Problem into an instance of TCNDP such that solving TCNDP also solves the 3-Partition Problem, we could argue for NP-hardness. However, this transformation is not straightforward due to the additional color requirement in TCNDP.\n", + "\n", + "Verification in Polynomial Time: Given a solution (a specific way to cut the necklace), it is relatively straightforward to check if each part contains all three colors and has an equal number of beads. This check can be done in polynomial time, which is a characteristic of NP problems.While TCNDP is clearly a complex problem, especially for long necklaces, it's not immediately clear if it is NP-hard. The main challenge in proving NP-hardness is the lack of an obvious polynomial-time reduction from a known NP-hard problem.Without a clear path to prove NP-hardness, TCNDP remains a potentially complex problem that might require heuristic or approximation methods for practical solutions, especially for large instances." + ] + }, + { + "cell_type": "markdown", + "id": "a861dd8f", + "metadata": {}, + "source": [ + "Summary:\n", + "\n", + "Assistance from ChatGPT\n", + "ChatGPT excelled in providing clear, concise explanations of complex topics like NP, NP-complete, and NP-hard problems. This was invaluable for building a foundational understanding of the subject matter.The tool adeptly demonstrated various instances of these problems, often with practical examples. This not only solidified theoretical knowledge but also offered a practical perspective. ChatGPT was particularly helpful in drawing comparisons between different problems, illustrating the subtle nuances that distinguish NP, NP-complete, and NP-hard problems from each other.\n", + "\n", + "Challenges in Problem Design\n", + "Maintaining Computational Complexity: One of the challenges was ensuring that the problems discussed retained their inherent computational complexity. It was crucial to avoid oversimplifying them to the point where they lost their characteristic NP, NP-complete, or NP-hard nature.\n", + "\n", + "Balancing Theory and Application: Striking a balance between theoretical accuracy and practical application was challenging. While ChatGPT provided great theoretical insights, translating these into real-world applications required careful consideration.\n", + "\n", + "Avoiding Ambiguity: Ensuring that problems were presented unambiguously was another challenge. In the realm of algorithms, especially of this complexity, clarity is key to understanding and solving problems.\n", + "\n", + "Designing problems in the realm of algorithms taught the importance of understanding the layers of complexity. Each class of problems (NP, NP-complete, NP-hard) has its own set of rules and characteristics.Practical examples are as important as theoretical knowledge. They provide concrete understanding and application contexts for these abstract concepts.Learning about these problems highlighted the need for adaptability in problem design. Real-world scenarios rarely follow textbook examples, so being able to adjust and apply these concepts flexibly is crucial." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0ac6c34c", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.13" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/Submissions/Weiyu_Wang_002746483/Assignment_4/readme.md b/Submissions/Weiyu_Wang_002746483/Assignment_4/readme.md new file mode 100644 index 0000000..d15e5b4 --- /dev/null +++ b/Submissions/Weiyu_Wang_002746483/Assignment_4/readme.md @@ -0,0 +1,28 @@ +First Name: "Weiyu" +Last Name : "Wang" +NU_ID : "002746483" + +Assignment-3 : "Summary of assignment 3" + +Completing algorithmic assignments is a challenging but rewarding experience. Using ChatGPT as a tool has greatly helped me understand complex concepts and solve complex problems. +Through interacting with ChatGPT, I gained valuable insights into algorithms, especially in the areas of Bellman-Ford, Ford-Fulkerson, Preflow-Push, polynomial-time algorithms, and master theorems. + +One of the main challenges I face is ensuring that the problems I solve maintain the spirit of the examples provided. +It was crucial not only to replicate the solution I found, but also to gain a deep understanding of the fundamentals. +ChatGPT played a key role in clarifying doubts and explaining complex steps. +This was especially helpful when I studied Ford-Fulkerson algorithms, where iterative processes require a clear understanding of the remaining capacity and enhancement path. +With detailed explanations, ChatGPT helped me grasp the nuances of the algorithm and apply it effectively to solve network traffic problems. + +In addition, ChatGPT played an important role in guiding me to understand the complexity of the Master theorem. +Understanding the time complexity of an algorithm is critical, and ChatGPT's explanation uncovers the complexity of divide-and-conquer algorithms. +With its help, I was able to confidently analyze the time complexity of recursive algorithms and infer their efficiency. + +This assignment has given me valuable experience in problem design in the field of algorithms. +I learned that mastering the basic concepts is just as important as solving the problem itself. +Algorithms are not just formulas to remember, they are formulas to remember. They are strategies to solve problems. +Through the challenges and ChatGPT's guidance, I gained a deeper understanding of the algorithm's system design. +I learned to approach problems with structured thinking, breaking them down into smaller, manageable components, and applying appropriate algorithms to solve them efficiently. + +Overall, ChatGPT proved to be a valuable resource for enhancing my understanding of algorithms. +It not only helped me overcome challenges, but also provided me with the knowledge and confidence to solve complex problems. +This experience reinforced the importance of clear conceptual understanding and thoughtful problem design, laying a solid foundation for my future efforts in the field of algorithms. \ No newline at end of file