Student Suite Cd-rom For Winston's Operations Research: Applications And Algorithms
4th Edition
ISBN: 9780534423551
Author: Wayne L. Winston
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 6.7, Problem 7P
Program Plan Intro
Dual of Linear
The dual of the given LP is another LP derived in the following schematic way from the original (the primal) LP.
- Every variable in the primal Linear Programming becomes a constraint in the dual LP.
- Every limit in the primal Linear Programming becomes a variable in the dual LP.
- In the dual LP, the objective direction is inversed. That is, the maximum in the primal becomes minimum in the dual and vice-versa.
Weak Duality:
- The weak duality is an optimization concept that states that the gap in duality is always greater than or equal to 0.
- This means the solution to the primary(minimization) problem is always greater than or equal to the solution to a dual problem associated with it.
Explanation of Solution
b.
Validity of the form of Lemma 3:
For showing validity of form of Lemma 3, it need to show that the dual is feasible and this will imply that the primal is bounded.
Assuming that the dual is to be feasible, let
Then Weak Duality implies that
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
(a) 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-hard.
(b) The Double SAT problem asks whether a given satisfiability problem has at least two
different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}}
is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}}
has exactly two solutions. Show that Double-SAT is NP-hard.
Correct answer will be upvoted else downvoted.
Annie's nursery is a limitless 2D plane. She has n potatoes to plant, and the I-th potato should be planted at (xi,yi). Beginning at the point (0,0), Annie starts strolling, in one stage she can travel one unit right or up (expanding her x or y facilitate by 1 individually). Anytime (X,Y) during her walk she can establish a few potatoes at discretionary focuses utilizing her potato weapon, burning-through max(|X−x|,|Y−y|) units of energy to establish a potato at (x,y). Track down the base absolute energy needed to establish each potato.
Note that Annie might establish quite a few potatoes from any point.
Input :The primary line contains the integer n (1≤n≤800000).
The following n lines contain two integers xi and yi (0≤xi,yi≤109), addressing the area of the I-th potato. It is conceivable that a few potatoes ought to be planted in a similar area.
Output :Print the base absolute energy to establish all potatoes.
1. Using the pumping lemma and contradiction, prove that L = (ab² n>=0} is not regular.
2. Using the pumping lemma and contradiction, prove that L = {a"bm n> m} is not regular.
3. Using the pumping lemma and contradiction, prove that L = (aºb n
Chapter 6 Solutions
Student Suite Cd-rom For Winston's Operations Research: Applications And Algorithms
Ch. 6.1 - Prob. 1PCh. 6.1 - Prob. 2PCh. 6.1 - Prob. 3PCh. 6.1 - Prob. 4PCh. 6.1 - Prob. 5PCh. 6.2 - Prob. 1PCh. 6.2 - Prob. 2PCh. 6.3 - Prob. 1PCh. 6.3 - Prob. 2PCh. 6.3 - Prob. 3P
Ch. 6.3 - Prob. 4PCh. 6.3 - Prob. 5PCh. 6.3 - Prob. 6PCh. 6.3 - Prob. 7PCh. 6.3 - Prob. 8PCh. 6.3 - Prob. 9PCh. 6.4 - Prob. 1PCh. 6.4 - Prob. 2PCh. 6.4 - Prob. 3PCh. 6.4 - Prob. 4PCh. 6.4 - Prob. 5PCh. 6.4 - Prob. 6PCh. 6.4 - Prob. 7PCh. 6.4 - Prob. 8PCh. 6.4 - Prob. 9PCh. 6.4 - Prob. 10PCh. 6.4 - Prob. 11PCh. 6.4 - Prob. 12PCh. 6.4 - Prob. 13PCh. 6.5 - Prob. 1PCh. 6.5 -
Find the duals of the following LPs:
Ch. 6.5 - Prob. 3PCh. 6.5 - Prob. 4PCh. 6.5 - Prob. 5PCh. 6.5 - Prob. 6PCh. 6.6 - Prob. 1PCh. 6.6 - Prob. 2PCh. 6.7 - Prob. 1PCh. 6.7 - Prob. 2PCh. 6.7 - Prob. 3PCh. 6.7 - Prob. 4PCh. 6.7 - Prob. 5PCh. 6.7 - Prob. 6PCh. 6.7 - Prob. 7PCh. 6.7 - Prob. 8PCh. 6.7 - Prob. 9PCh. 6.8 - Prob. 1PCh. 6.8 - Prob. 2PCh. 6.8 - Prob. 3PCh. 6.8 - Prob. 4PCh. 6.8 - Prob. 5PCh. 6.8 - Prob. 6PCh. 6.8 - Prob. 8PCh. 6.8 - Prob. 9PCh. 6.8 - Prob. 10PCh. 6.8 - Prob. 11PCh. 6.9 - Prob. 1PCh. 6.9 - Prob. 2PCh. 6.9 - Prob. 3PCh. 6.10 - Prob. 1PCh. 6.10 - Prob. 2PCh. 6.10 - Prob. 3PCh. 6.11 - Prob. 1PCh. 6.11 - Prob. 3PCh. 6.11 - Prob. 4PCh. 6.12 - Prob. 5PCh. 6.12 - Prob. 6PCh. 6.12 - Prob. 7PCh. 6 - Prob. 1RPCh. 6 - Prob. 2RPCh. 6 - Prob. 3RPCh. 6 - Prob. 4RPCh. 6 - Prob. 5RPCh. 6 - Prob. 6RPCh. 6 - Prob. 7RPCh. 6 - Prob. 8RPCh. 6 - Prob. 9RPCh. 6 - Prob. 10RPCh. 6 - Prob. 11RPCh. 6 - Prob. 13RPCh. 6 - Prob. 14RPCh. 6 - Prob. 15RPCh. 6 - Prob. 17RPCh. 6 - Prob. 18RPCh. 6 - Prob. 19RPCh. 6 - Prob. 20RPCh. 6 - Prob. 21RPCh. 6 - Prob. 22RPCh. 6 - Prob. 25RPCh. 6 - Prob. 29RPCh. 6 - Prob. 33RPCh. 6 - Prob. 34RPCh. 6 - Prob. 35RPCh. 6 - Prob. 36RPCh. 6 - Prob. 37RP
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- Question 2: Consider the 0/1 knapsack problem. Given Nobjects where each object is specified by a weight and a profit, you are to put the objects in a bag of capacity C such that the sum of weights of the items in the bag does not exceed Cand the profits of the items is maximized. Note that you cannot use an item type more than once. a. Using dynamic programming, write an algorithm that finds the maximum total value according to the above constraints. b. What is the complexity of your algorithm? c. Show the dynamic programming table for the following data: W= { 2 ,7 , 1} , P={ 3 ,15 , 2 } and C=8.arrow_forwardTheorem 1.2.3 Suppose that both (LP) and (LD) are feasible. Then there exists a primal-dual feasible pair (x*,y*, s*) such that (x*)Ts* = = 0 and x*+s* > 0. A solution (x*, s*) with this property is called strictly complementary.arrow_forwardShow 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_forward
- c) Knowing that we have the equality: 12Ar+3= B2hez determine the value of x and encircle it.arrow_forwardNP-Complete Reductions: Assume we have the 3-SAT problem (x |l y || z ) && ( ¬x |I y II ¬z) && (¬x || ¬y || z). a) Convert this to a maximal clique problem. b) One satisfying assignment for the Boolean logic problem is x=1, y=0, z=0. Show the clique in your solution to part (a) that corresponds to this solution of the logic problem. c) What minimal vertex cover problem does this reduce to? What solution to that problem results from the specified solution in part (b)?arrow_forwardWe are given a 3-CNF formula with n variables and m clauses, where m is even, in the half 3-CNF satisfiability issue. We want to know if there is a true variable assignment where precisely half of the clauses evaluate to 0 and exactly half of the clauses evaluate to 1. prove the NP-completeness of the partial 3-CNF satisfiability issue.arrow_forward
- 4. Reducibility, NP-completeness 4a. Recall that linear ax + b = 0 is reducible to the quadratic ax²+bx+c=0. Show that the quadratic ax2+bx+c=0 is reducible to quartic ax*+bx+cx+dx+e=0 (Lec14.1). Is quartic reducible to quadratic? 4b. Consider the 4-way Partition Problem: Given a set S of positive integers, determine whether S can be partitioned into four subsets S1, S2, S3, and S4 such that the sum of the numbers in S, is equal to S2, S3, and S4. Demonstrate that the problem is NP-complete based on NP-completeness of the 2-way Partition Problem. 4c. Prove that P != NP.arrow_forward3. Using the pumping lemma and contradiction, prove that L = {ab narrow_forward2. Can someone help with this problem 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.arrow_forwardThe missionaries and cannibals problem is usually stated as follows. Three missionariesand three cannibals are on one side of a river, along with a boat that can hold one ortwo people. Find a way to get everyone to the other side without ever leaving a group of missionariesin one place outnumbered by the cannibals in that place. This problem is famous inAI because it was the subject of the first paper that approached problem formulation from ananalytical viewpoint (Amarel, 1968).b. Implement and solve the problem optimally using an appropriate search algorithm. (Mention the name of the search algorithm, and write the complete answers, Draw the answer using the searching algorithms with complete and all paths and branches.)arrow_forwardPlease prove the following without using outside lemmas, "Let a, b ∈ Z. Prove that if ab is odd, then a2 + b2 is even."arrow_forwardSolve this one using DLV. A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet two inhabitants: Sue and Zippy. Sally claims, “I and Zippy are not the same.” Zippy says, “Of I and Sally, exactly one is a knight.” Using DLV, determine who is a knight and who is a knave.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole
Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole