Given a 3-CNF formula φ with n variables and m clauses, where m is even. We wish to determine whether there exists a truth assignment to the variables of φ such that exactly half the clauses evaluate to 0 and exactly half the clauses evaluate to 1. Prove that this problem is NP-complete.
Given a 3-CNF formula φ with n variables and m clauses, where m is even. We...
7. (10 pts) STINGY SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that STINGY SAT is NP-complete. 7. (10 pts) STINGY SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are...
3, (30 points) Given a directed graph G - N. E), each edge eEhas weight We, 3, (30 points) Given a directed graph G (V, E), each edgee which can be positive or negative. The zero weight cycle problem is that whether exists a simple cycle (each vertex passes at most once) to make the sum of the weights of each edge in G is exactly equal to 0. Prove that the problem is NP complete. 3, (30 points) Given...
4. The NOT-ALL-EQUAL 3SAT problem is defined as follows: Given a 3-CNF formula F, is there a truth assignment for the variables such that each clause has at least one true literal and at least one false literal? The NOT-ALL-EQUAL 3SAT problem is NP-complete. This question is about trying to reduce the NOT-ALL-EQUAL 3SAT problem to the MAX-CUT problem defined below to show the latter to be NP-complete. A cut in an undirected graph G=(V.E) is a partitioning of the...
10 points (bonus) A propositional formula on n variables, P(ri,2,... ,Tn) is satisfiable if there exists an assignment of truth values (true or false) to its variables such that it evaluates to true. (a) Give an algorithm (pseudocode) that, given a formula P determines if it is satisfiable or not. Analyze your algorithm. b) Suppose that we are given a free" algorithm A that, given P and a partial assignment of truth values (that is, some variables are set to...
NP-completeness. We are given an undirected graph where each edge has a positive weight. Given (k, alpha), the problem asks whether there is a subgraph with k nodes such that the total weight of the edges in the subgraph is at least alpha. Prove this problem is NP-Complete.
MEMB343 MECHANICAL VIBRATIONS ASSIGNMENT l. For the system shown in Figure 1, where mi=5 kg, m,-10 kg, ki=1000 N/m, k2-500 N/m, k, 2000 N/m, fi-100sin(15t) N and f-0, use modal analysis to determine the amplitudes of masses m, and m2. The equations of motion are given as sin(15t), wth natura frequencies 5 01[i, 0 10 500-500x, 500 2500jx, x,[100 ω,-14.14 rad's and a, = 18.71 rad/s, and mode shapes, Φ',, and Φ' k, Im Figure 1 MEMB343 MECHANICAL VIBRATIONS ASSIGNMENT...
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si 3. (20 pts) Let ụ be a finite set, and let...
3. (a) Let Knbe the complete bipartite graph with n vertices in each part of its bipartition, where n 21. Determine the number of perfect matchings of Kn (b) A matching M in a graph Gis ca a mazimal matching if there exists no matching M' of G such that M is a proper subset of M' Prove that, for any graph G and any edges e,f of G which are not incident with a common vertex, there exists a...
MATLAB 5. Given the recurrence formula In+1 = g(n), where g(x) = 1.2 + €0.52 repeat the evaluation of the recurrence formula until the criteria n+1-en < 10-6 is satisfied. Start the procedure with n = 0 and .co = 1. Solve the problem with a while- loop. (Hint: You may wish to corroborate your results by conducting this procedure in an Excel Spreadsheet.) (a) Store the value of n which satisfies the given criteria in a scalar variable. Apply...
Problem. In class we saw that the even moments of the standard Gaussian Xx(0,1) are given by: EX2 = (2k – 1)!! = (2k – 1)(2k – 3)...31 (2k)! 2kk! Meanwhile the odd moments vanish. The goal of this exercise is to prove the CLT by the method of moments. Suppose X1, X2, ..., X, are independent identically distirbuted random variables with EX; = 0 (mean zero), EX = 1 (unit variance) and bounded higher moments EX < M. Let's...