# Programming Assignment 3: NP-completeness solution

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## Description

Introduction Welcome to your third programming assignment of the Advanced Algorithms and Complexity class! In this programming assignment, you will be practicing reducing hard real-world problems to SAT problem, which can in turn often be solved efficiently in practice using specialized programs called SAT-solvers.
Learning Outcomes Upon completing this programming assignment you will be able to: 1. Reduce real-world problems to instances of classical NP-complete problems. 2. Design and implement efficient algorithms to reduce the following computational problems to SAT: (a) assigning frequencies to the cells of a GSM network; (b) determine whether it is possible to leave no signs of a party in the apartment; (c) determine whether there is a way to allocate advertising budget given a set of constraints.
Passing Criteria: 2 out of 3 Passing thisprogramming assignmentrequires passingat least2out of3code problemsfrom thisassignment. In turn, passing a code problem requires implementing a solution that passes all the tests for this problem in the grader and does so under the time and memory limits specified in the problem statement.
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Contents 1 Problem: Assign Frequencies to the Cells of a GSM Network 3
2 Problem: Cleaning the Apartment 6
4 General Instructions and Recommendations on Solving Algorithmic Problems 13 4.1 Reading the Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4 Compiling Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.6 Submitting Your Program to the Grading System . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.7 Debugging and Stress Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Frequently Asked Questions 16 5.1 I submit the program, but nothing happens. Why? . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 I submit the solution only for one problem, but all the problems in the assignment are graded. Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.3 What are the possible grading outcomes, and how to read them? . . . . . . . . . . . . . . . . 16 5.4 How to understand why my program fails and to fix it? . . . . . . . . . . . . . . . . . . . . . 17 5.5 Why do you hide the test on which my program fails? . . . . . . . . . . . . . . . . . . . . . . 17 5.6 My solution does not pass the tests? May I post it in the forum and ask for a help? . . . . . . 18 5.7 My implementation always fails in the grader, though I already tested and stress tested it a lot. Would not it be better if you give me a solution to this problem or at least the test cases that you use? I will then be able to fix my code and will learn how to avoid making mistakes. Otherwise, I do not feel that I learn anything from solving this problem. I am just stuck. . . . 18
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1 Problem: Assign Frequencies to the Cells of a GSM Network Problem Introduction
In this problem, you will learn to reduce the real-world problem about assigning frequencies to the transmitting towers of the cells in a GSM network to a problem of proper coloring a graph into 3 colors. Then you will design and implement an algorithm to reduce this problem to an instance of SAT.
Problem Description Task. GSM network is a type of infrastructure used for communication via mobile phones. It includes transmitting towers scattered around the area which operate in different frequencies. Typically there is one tower in the center of each hexagon called “cell” on the grid above — hence the name “cell phone”. A cell phone looks for towers in the neighborhood and decides which one to use based on strength of signal and other properties. For a phone to distinguish among a few closest towers, the frequencies of the neighboring towers must be different. You are working on a plan of GSM network for mobile, and you have a restriction that you’ve only got 3 different frequencies from the government which you can use in your towers. You know which pairs of the towers are neighbors, and for all such pairs the towers in the pair must use different frequencies. You need to determine whether it is possible to assign frequencies to towers and satisfy these restrictions. This is equivalent to a classical graph coloring problem: in other words, you are given a graph, and you need to color its vertices into 3 different colors, so that any two vertices connected by an edge need to be of different colors. Colors correspond to frequencies, vertices correspond to cells, and edges connect neighboring cells. Graph coloring is an NP-complete problem, so we don’t currently know an efficient solution to it, and you need to reduce it to an instance of SAT problem which, although it is NP-complete, can often be solved efficiently in practice using special programs called SAT-solvers. Input Format. The first line of the input contains integers n and m — the number of vertices and edges in the graph. The vertices are numbered from 1 through n. Each of the next m lines contains two integers u and v — the numbers of vertices connected by an edge. It is guaranteed that a vertex cannot be connected to itself by an edge. Constraints. 2 ≤ n ≤ 500; 1 ≤ m ≤ 1000; 1 ≤ u,v ≤ n; u ̸= v. Output Format. You need to output a boolean formula in the conjunctive normal form (CNF) in a specific format. If it is possible to color the vertices of the input graph in 3 colors such that any two vertices connected by an edge are of different colors, the formula must be satisfiable. Otherwise, the formula must be unsatisfiable. The number of variables in the formula must be at least 1 and at most 3000. The number of clauses must be at least 1 and at most 5000. On the first line, output integers C and V — the number of clauses in the formula and the number of variables respectively. On each of the next C lines, output a description of a single clause. Each clause hasaform(x4 OR x1 OR x8).Foraclausewith k terms(intheexample,k = 3forx4,x1 andx8),output first those k terms and then number 0 in the end (in the example, output “4 −1 8 0”). Output each term as integer number. Output variables x1,x2,…,xV as numbers 1,2,…,V respectively. Output
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negations of variables x1,x2,…,xV as numbers −1,−2,…,−V respectively. Each number other than the last one in each line must be a non-zero integer between −V and V where V is the total number of variables specified in the first line of the output. Ensure that 1 ≤ C ≤ 5000 and 1 ≤ V ≤ 3000. See the examples below for further clarification of the output format. If there are many different formulas that satisfy the requirements above, you can output any one of them. Note that your formula will be checked internally by the grader using a SAT-solver. Although SAT-solvers often solve instances of SAT of the given size very fast, it cannot be guaranteed. If you submit a formula which cannot be resolved by the SAT-solver we use under a reasonable time limit, the grader will timeout, and the problem won’t pass. We guarantee that there are solutions of this problem that output formulas which are resolved almost instantly by the SAT-solver used. However, don’t try to intentionally break the system by submitting very complex SAT instances, because the problem still won’t pass, and you will violate Coursera Honor Code by doing that. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 1 1 1.5 5 1.5 2 5 5 3
Memory Limit. 512MB. Sample 1. Input: 3 3 1 2 2 3 1 3 Output: 1 1 1 -1 0 Explanation:
1 2
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The input graph has just 3 vertices, so of course they all can be colored in different colors using only 3 colors. That’s why we need to output a satisfiable formula. The formula in the output uses just 1 variable x1 and 1 clause, and the only clause is (x1 OR x1) which is, of course, satisfiable: for any value of x1, the boolean value of the formula is true. Note that you could output another satisfiable formula, like x1 or (x1 OR x2 OR x3) AND (x1 OR x2), or one of many others.
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Sample 2. Input: 4 6 1 2 1 3 1 4 2 3 2 4 3 4 Output: 2 1 1 0 -1 0
Explanation:
1 2
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The input graph has 4 vertices, and each pair of them is connected by an edge. In a proper coloring, all these vertices must be of different colors, but we have only 3 different colors, so it is impossible. Thus, we need to output an unsatisfiable formula. The formula in the output has 2 clauses with one variable, itis (x1)AND(x1).Notethatyoucouldoutputanotherformula,like (x1 OR x2) AND (x1) AND (x2), or one of many other unsatisfiable formulas.
Starter Files The starter solutions for this problem read the data from the input, pass it to a procedure that outputs a fixed satisfiable formula. You need to change the main procedure to implement some reduction of the graph coloring problem to SAT problem if you’re using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: gsm_network
What To Do Create a separate variable xij for each vertex i of the initial graph and each possible color j, 1 ≤ j ≤ 3 which means “vertex i has color j”. Think how to write down conditions like “each vertex has to be colored by some color” and “vertices connected by an edge must have different colors” with clauses of CNF using these variables.
Need Help? Ask a question or see the questions asked by other learners at this forum thread.
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2 Problem: Cleaning the Apartment Problem Introduction
In this problem, you will learn to determine whether it is possible to clean an apartment after a party without leaving any traces of the party. You will learn how to reduce it to the classic Hamiltonian Path problem, and then you will design and implement an efficient algorithm to reduce it to SAT.
Problem Description Task. You’ve just had a huge party in your parents’ house, and they are returning tomorrow. You need to not only clean the apartment, but leave no trace of the party. To do that, you need to clean all the rooms in some order. After finishing a thorough cleaning of some room, you cannot return to it anymore: you are afraid you’ll ruin everything accidentally and will have to start over. So, you need to move from room to room, visit each room exactly once and clean it. You can only move from a room to the neighboring rooms. You want to determine whether this is possible at all. This can be reduced to a classic Hamiltonian Path problem: given a graph, determine whether there is a route visiting each vertex exactly once. Rooms are vertices of the graph, and neighboring rooms are connected by edges. Hamiltonian Path problem is NP-complete, so we don’t know an efficient algorithm to solve it. You need to reduce it to SAT, so that it can be solved efficiently by a SAT-solver. Input Format. The first line contains two integers n and m — the number of rooms and the number of corridors connecting the rooms respectively. Each of the next m lines contains two integers u and v describing the corridor going from room u to room v. The corridors are two-way, that is, you can go both from u to v and from v to u. No two corridors have a common part, that is, every corridor only allows you to go from one room to one other room. Of course, no corridor connects a room to itself. Note that a corridor from u to v can be listed several times, and there can be listed both a corridor from u to v and a corridor from v to u. Constraints. 1 ≤ n ≤ 30; 0 ≤ m ≤ n(n−1) 2 ; 1 ≤ u,v ≤ n. Output Format. You need to output a boolean formula in the CNF form in a specific format. If it is possible to go through all the rooms and visit each one exactly once to clean it, the formula must be satisfiable. Otherwise, the formula must be unsatisfiable. The sum of the numbers of variables used in each clause of the formula must not exceed 120 000. On the first line, output integers C and V — the number of clauses in the formula and the number of variables respectively. On each of the next C lines, output a description of a single clause. Each clause hasaform(x4 OR x1 OR x8).Foraclausewith k terms(intheexample,k = 3forx4,x1 andx8),output first those k terms and then number 0 in the end (in the example, output “4 −1 8 0”). Output each term as integer number. Output variables x1,x2,…,xV as numbers 1,2,…,V respectively. Output negations of variables x1,x2,…,xV as numbers −1,−2,…,−V respectively. Each number other than the last one in each line must be a non-zero integer between −V and V where V is the total number of variables specified in the first line of the output. Ensure that the total number of non-zero integers in the C lines describing the clauses is at most 120 000. See the examples below for further clarification of the output format. If there are many different formulas that satisfy the requirements above, you can output any one of them.
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Note that your formula will be checked internally by the grader using a SAT-solver. Although SAT-solvers often solve instances of SAT of the given size very fast, it cannot be guaranteed. If you submit a formula which cannot be resolved by the SAT-solver we use under a reasonable time limit, the grader will timeout, and the problem won’t pass. We guarantee that there are solutions of this problem that output formulas which are resolved almost instantly by the SAT-solver used. However, don’t try to intentionally break the system by submitting very complex SAT instances, because the problem still won’t pass, and you will violate Coursera Honor Code by doing that. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 2 2 3 10 3 4 10 10 6
Memory Limit. 512MB. Sample 1. Input: 5 4 1 2 2 3 3 5 4 5 Output: 1 1 1 -1 0
Explanation:
1 2 3 5 4
There is a Hamiltonian path 1−2−3−5−4, so we need to output a satisfiable formula. The formula in the output uses just 1 variable x1 and 1 clause, and the only clause is (x1 OR x1) which is, of course, satisfiable: for any value of x1, the boolean value of the formula is true. Note that you could output another satisfiable formula, like x1 or (x1 OR x2 OR x3) AND (x1 OR x2), or one of many others. Sample 2. Input: 4 3 1 2 1 3 1 4 Output: 2 1 1 0 -1 0
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Explanation:
1 2
3 4
There is no way to visit each room exactly once: either we don’t visit one of the rooms 2, 3 or 4, or we visit room 1 at least twice. Thus, we need to output an unsatisfiable formula. The formula in the output has 2 clauses with one variable, it is (x1)AND(x1). Note that you could output another formula, like (x1 OR x2) AND (x1) AND (x2), or one of many other unsatisfiable formulas.
Starter Files Thestartersolutionsforthisproblemreadthedatafromtheinput,passittoaprocedurethatoutputsafixed satisfiable formula. You need to change the main procedure to implement some reduction of the Hamiltonian path problem to SAT if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: cleaning_apartment
What To Do Create a separate variable xij for each vertex i and each position in the Hamiltonian path j. xij is true if vertex i is at the position j of Hamiltonian path. Think how to express all the restrictions in terms of these variables: all vertices must be on the path, each vertex must be visited exactly once, there is only one vertex on each position in the path, two successive vertices must be connected by an edge.
Need Help? Ask a question or see the questions asked by other learners at this forum thread.
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3 Advanced Problem: Advertisement Budget Allocation Westronglyrecommendyoustartsolvingadvancedproblemsonlywhenyouaredonewiththebasicproblems (for some advanced problems, algorithms are not covered in the video lectures and require additional ideas to be solved; for some other advanced problems, algorithms are covered in the lectures, but implementing them is a more challenging task than for other problems).
Problem Introduction
In the previous programming assignment, you worked for an online advertisement system. In this programming assignment, you’ll work for a big company that uses advertising to promote itself. You will need to determine whether it is possible to allocate advertising budget and satisfy all the constraints. You will learn how to reduce this problem to a particular type of Integer Linear Programming problem. Then you will designandimplementanefficientalgorithmtoreducethistypeofInteger Linear Programming to SAT.
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Constraints. 1 ≤ n,m ≤ 500; −100 ≤ Aij ≤ 100; −1 000 000 ≤ bi ≤ 1 000 000. Output Format. You need to output a boolean formula in the CNF form in a specific format. If it is possible to accept some of the proposals and decline all the others while satisfying all the constraints, the formula must be satisfiable. Otherwise, the formula must be unsatisfiable. The number of variables in the formula must not exceed 3000, and the number of clauses must not exceed 5000. On the first line, output integers C and V — the number of clauses in the formula and the number of variables respectively. On each of the next C lines, output a description of a single clause. Each clause hasaform(x4 OR x1 OR x8).Foraclausewith k terms(intheexample,k = 3forx4,x1 andx8),output first those k terms and then number 0 in the end (in the example, output “4 −1 8 0”). Output each term as integer number. Output variables x1,x2,…,xV as numbers 1,2,…,V respectively. Output negations of variables x1,x2,…,xV as numbers −1,−2,…,−V respectively. Each number other than the last one in each line must be a non-zero integer between −V and V where V is the total number of variables specified in the first line of the output. Ensure that 1 ≤ C ≤ 5000 and 1 ≤ V ≤ 3000. See the examples below for further clarification of the output format. If there are many different formulas that satisfy the requirements above, you can output any one of them. Note that your formula will be checked internally by the grader using a SAT-solver. Although SAT-solvers often solve instances of SAT of the given size very fast, it cannot be guaranteed. If you submit a formula which cannot be resolved by the SAT-solver we use under a reasonable time limit, the grader will timeout, and the problem won’t pass. We guarantee that there are solutions of this problem that output formulas which are resolved almost instantly by the SAT-solver used. However, don’t try to intentionally break the system by submitting very complex SAT instances, because the problem still won’t pass, and you will violate Coursera Honor Code by doing that. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 1 1 1.5 5 1.5 2 5 5 3 Memory Limit. 512MB. Sample 1. Input: 2 3 5 2 3 -1 -1 -1 6 -2 Output: 1 1 1 -1 0
Explanation: Here we have two inequalities: 5·x1 + 2·x2 + 3·x3 ≤ 6 and (−1)·x1 + (−1)·x2 + (−1)·x3 ≤−2. The binary vector x1 = 0,x2 = 1,x3 = 1 satisfies all the inequalities, so we need to output a satisfiable formula. The formula in the output uses just one variable x1 and one clause, and the only clause is (x1 OR x1) which is, of course, satisfiable: for any value of x1, the boolean value of the formula is true. Note that you could output another satisfiable formula, like x1 or (x1 OR x2 OR x3) AND (x1 OR x2), or one of many others. You do not have to make sure that the formula you output is satisfied only by the binary vectors which satisfy the given system of inequalities, but the intended solution ensures that.
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Sample 2. Input: 3 3 1 0 0 0 1 0 0 0 1 1 1 1 Output: 1 1 1 -1 0
Explanation: There are three inequalities of the form xi ≤ 1. Of course, any binary vector satisfies all those inequalities, so we need to output a satisfiable formula. The formula in the output is the same as in the previous example. Sample 3. Input: 2 1 1 -1 0 -1 Output: 2 1 1 0 -1 0
Explanation: There are two inequalities x1 ≤ 0 and −x1 ≤ −1 ⇒ x1 ≥ 1, but x1 cannot be less than 0 and more than 1 simultaneously, so we need to output an unsatisfiable formula. The formula in the output has 2 clauses with one variable, it is (x1)AND(x1). Note that you could output another formula, like (x1 OR x2) AND (x1) AND (x2), or one of many other unsatisfiable formulas. Sample 4. Input: 2 3 1 1 0 0 -1 -1 0 -2 Output: 2 1 1 0 -1 0
Explanation: There are two inequalities: x1 + x2 ≤ 0 and (−1)·x2 + (−1)·x3 ≤−2 ⇒ x2 + x3 ≥ 2. From the first one, it follows for a binary vector x that x1 = x2 = 0. From the second one, it follows for a binary vector x that x2 = x3 = 1. But x2 cannot be 0 and 1 simultaneously, so there is no binary vector x satisfying all the inequalities, and so we need to output an unsatisfiable formula. The formula in the output is the same as in the previous example.
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Starter Files The starter solutions for this problem read the data from the input, pass it to procedure that outputs a fixed satisfiable formula. You need to change the main procedure to implement some reduction of Integer Linear Programming problem to SAT if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: budget_allocation
What To Do In this case, you can use the binary vector x that you are looking for itself as the basis for the CNF formula. Think how to use the fact that there can be at most 3 non-zero coefficients in each inequality. Try thinking about assignments of all the corresponding variables that invalidate a particular inequality — how many different such assignments are there at most? You know that none of those assignments should hold if the binary vector x satisfies all the inequalities. Think how to write this condition compactly in the CNF form, then join those clauses for each inequality.
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4 General Instructions and Recommendations on Solving Algorithmic Problems Your main goal in an algorithmic problem is to implement a program that solves a given computational problem in just few seconds even on massive datasets. Your program should read a dataset from the standard input and write an answer to the standard output. Below we provide general instructions and recommendations on solving such problems. Before reading them, go through readings and screencasts in the first module that show a step by step process of solving two algorithmic problems: link.
4.1 Reading the Problem Statement You start by reading the problem statement that contains the description of a particular computational task as well as time and memory limits your solution should fit in, and one or two sample tests. In some problems your goal is just to implement carefully an algorithm covered in the lectures, while in some other problems you first need to come up with an algorithm yourself.
4.2 Designing an Algorithm If your goal is to design an algorithm yourself, one of the things it is important to realize is the expected running time of your algorithm. Usually, you can guess it from the problem statement (specifically, from the subsection called constraints) as follows. Modern computers perform roughly 108–109 operations per second. So, if the maximum size of a dataset in the problem description is n = 105, then most probably an algorithm with quadratic running time is not going to fit into time limit (since for n = 105, n2 = 1010) while a solution with running time O(nlogn) will fit. However, an O(n2) solution will fit if n is up to 103 = 1000, and if n is at most 100, even O(n3) solutions will fit. In some cases, the problem is so hard that we do not know a polynomial solution. But for n up to 18, a solution with O(2nn2) running time will probably fit into the time limit. To design an algorithm with the expected running time, you will of course need to use the ideas covered in the lectures. Also, make sure to carefully go through sample tests in the problem description.
4.3 Implementing Your Algorithm When you have an algorithm in mind, you start implementing it. Currently, you can use the following programming languages to implement a solution to a problem: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, Scala. For all problems, we will be providing starter solutions for C++, Java, and Python3. If you are going to use one of these programming languages, use these starter files. For other programming languages, you need to implement a solution from scratch.
4.4 Compiling Your Program For solving programming assignments, you can use any of the following programming languages: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, and Scala. However, we will only be providing starter solution files for C++, Java, and Python3. The programming language of your submission is detected automatically, based on the extension of your submission. We have reference solutions in C++, Java and Python3 which solve the problem correctly under the given restrictions, and in most cases spend at most 1/3 of the time limit and at most 1/2 of the memory limit. You can also use other languages, and we’ve estimated the time limit multipliers for them, however, we have no guarantee that a correct solution for a particular problem running under the given time and memory constraints exists in any of those other languages. Your solution will be compiled as follows. We recommend that when testing your solution locally, you use the same compiler flags for compiling. This will increase the chances that your program behaves in the
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same way on your machine and on the testing machine (note that a buggy program may behave differently when compiled by different compilers, or even by the same compiler with different flags). ∙ C (gcc 5.2.1). File extensions: .c. Flags: gcc -pipe -O2 -std=c11