Show that the independent set problem is NP-complete through the following two steps:
1. Show that the problem is in NP.
2. Show that 3SAT is poly-time reducible to the problem.
Show that the independent set problem is NP-complete through the following two steps: 1. Show that...
25. (1 point) Suppose A is some language in the class NP and B is NP-complete. Which of the following could be false? A. A is polynomial time reducible to B. B. Given a decider for B which runs in polynomial time, it is possible to decide A in polynomial time. C. Given a decider for A which runs in polynomial time, it is possible to decide B in polynomial time. D. Given a decider for B which runs in...
Prove that the following problem is NP-complete: given an undirected graph G = (V, E) and an integer k, return a clique of size k as well as an independent set of size k, provided both exist.
Algorithms Given the following 3SAT formula, convert the problem to Independent Set and determine from there if the formula is satisfiable: $ = (x1 V x2 V x3) ^ (X1 V X2 V x3) ^ (X1 V x2 V x3)
3. (3 pts) Two well-known NP-complete problems are 3-SAT and TSP, the traveling salesman problem. The 2-SAT problem is a SAT variant in which each clause contains at most two literals. 2-SAT is known to have a polynomial-time algorithm. Is each of the following statements true or false? Justify your answer. a. 3-SAT sp TSP. b. If P NP, then 3-SAT Sp 2-SAT. C. If P NP, then no NP-complete problem can be solved in polynomial time.
Consider the following set of deependent and independent variables. Complete parts a through c below. 11 14 14 21 24 26 31 Y X1 1 8 10 15 21 4 4. 7 X2 16 9 13 11 2 8 8 4 CO b. Test the significance of each independent variable using a 0.10. Calculate the appropriate test statistic The test statistic (Round to two decimal places as needed.) What is the critical t-score? Use the t-test to determine the significance...
Show that the decision version of the knapsack problem is NP-complete. (Hint: In your reduction, make use of the partition problem: given n positive integers, partition them into two disjoint subsets with the same sum of their elements. The partition problem is NP-complete.)
Is the following problem NP-Complete? The Rice bowl problem is to pick the ingredients for your bowl. You are given a set of ingredients I1 to In. Each ingredient Ii comes with a quality qi and a quantity si . You are also given a bowl size S and a quality goal Q. Can you select a subset of the ingredients that both fit in the bowl (the sum of their si is at most S) and have enough quality...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete. Note:...
Show that PARTITION is NP-complete by reduction from SUBSET-SUM. Given a set of integers, we say that can be partitioned if it can be split into two sets U and V so that considering all u EU and all v € V, Eu = Ev. Let PARTITION = { <S> S can be partitioned ). Show that PARTITION IS NP-complete by reduction from SUBSET-SUM.
5. The Hitting Set Problem (HS) is the following decision problem. Input. A finite set S, a collection (s1, s2,... , sn) of subsets of S, and a positive integer K. Question. Does there exist a subset t of S such that (a) t has exactly K members and (b) for i 1,..., n, sint6For example, suppose S # {1, 2, 3, 4, 5, 6, 7. the collection of subsets is (2.45), (34).(1,35) and K - 2. Then the answer...