Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 15, Problem 3P
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A graduate student is working on a problem X. After working on it for several days she is unable to find a polynomial-time solution to the problem. Therefore, she attempts to prove that he problem is NP-complete. To prove that X is NP-complete she first designs a decision version of the problem. She then proves that the decision version is in NP. Next, she chooses SUBSET-SUM, a well-known NP-complete problem and reduces her problem to SUBSET-SUM (i.e., she proves X £p SUBSET-SUM). Is her approach correct? Explain your answer.
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Prove that the following problem is NP-hard and determine whether it is also NP-complete or not – if the problem is NP-complete, you also need to give a poly-time verifier and argue the correctness of your verifier.
Given a problem X and Y, if X reduces to Y in polynomial time, and Y is known to be NP-Complete, what can be said about X?
Chapter 15 Solutions
Introduction to Algorithms
Ch. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5ECh. 15.2 - Prob. 1ECh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5E
Ch. 15.2 - Prob. 6ECh. 15.3 - Prob. 1ECh. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.4 - Prob. 1ECh. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15 - Prob. 1PCh. 15 - Prob. 2PCh. 15 - Prob. 3PCh. 15 - Prob. 4PCh. 15 - Prob. 5PCh. 15 - Prob. 6PCh. 15 - Prob. 7PCh. 15 - Prob. 8PCh. 15 - Prob. 9PCh. 15 - Prob. 10PCh. 15 - Prob. 11PCh. 15 - Prob. 12P
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- Which of the following is NOT enough to show that a problem P is NP-Hard? Reduce the traveling salesperson problem to P in polynomial time Reduce every NP problem to P in polynomial time reducing the C-SAT problem to p in polynomial time Reduce the 01-knapsack problem to P in polynomial timearrow_forwardUSING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.arrow_forwardThe sum-of-subsets problem is the following: Given a sequence a1 , a2 ,..., an of integers, and an integer M, is there a subset J of {1,2,...,n} such that i∈J ai = M? Show that this problem is NP-complete by constructing a reduction from the exact cover problem.arrow_forward
- Show that the following problem belongs to NP class: we are given a set S of integer numbers and an integer number t. Does S have a subset such that sum of its elements is t? Note: Data Structures and Algorithm problemarrow_forwardThe Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4arrow_forwardWe know that when we have a graph with negative edge costs, Dijkstra’s algorithm is not guaranteed to work. (a) Does Dijkstra’s algorithm ever work when some of the edge costs are negative? Explain why or why not. (b) Find an algorithm that will always find a shortest path between two nodes, under the assumption that at most one edge in the input has a negative weight. Your algorithm should run in time O(m log n), where m is the number of edges and n is the number of nodes. That is, the runnning time should be at most a constant factor slower than Dijkstra’s algorithm. To be clear, your algorithm takes as input (i) a directed graph, G, given in adjacency list form. (ii) a weight function f, which, given two adjacent nodes, v,w, returns the weight of the edge between them. For non-adjacent nodes v,w, you may assume f(v,w) returns +1. (iii) a pair of nodes, s, t. If the input contains a negative cycle, you should find one and output it. Otherwise, if the graph contains at least one…arrow_forward
- Imagine that you have a problem P that you know is N P-complete. For this problem you have two algorithms to solve it. For each algorithm, some problem instances of P run in polynomial time and others run in exponential time (there are lots of heuristic-based algorithms for real N P-complete problems with this behavior). You can’t tell beforehand for any given problem instance whether it will run in polynomial or exponential time on either algorithm. However, you do know that for every problem instance, at least one of the two algorithms will solve it in polynomial time. (a) What should you do? (b) What is the running time of your solution? 564 Chap. 17 Limits to Computation (c) What does it say about the question of P = N P if the conditions described in this problem existed?arrow_forwardPlease state if the following questions are TRUE or FALSE ? -Suppose problem A reduces to problem B and there is an exponential time algorithm for B. Then there is an exponential time algorithm for A -It has been (mathematically) proven that there is no polynomial time algorithm for any NP-Complete problem. -If we have a polynomial time algorithm for 3SAT, then P = NP. -If problem A reduces to problem B and problem C reduces to problem B, then problem A (always) reduces to problem C. -The total weight of the minimum spanning tree is always the same as total weight of a traveling salesman tour.arrow_forwardTheorem to prove: Any NP-complete issue that can be solved in polynomial time is P D NP. In other words, if any NP problem is not solvable in polynomial time, then no NP-complete problem is solvable in polynomial time.arrow_forward
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