/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q3E STINGY SAT is the following prob... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q3E (page 278)

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 isNP -complete.

Short Answer

Expert verified

STINGY SAT is the required assignment, it has been proved that STINGY SAT is NP complete.

Step by step solution

01

Statement of  is -complete

There are steps to show the given hassle as NP (Non-deterministic polynomial) Completeness.

Take a look at whether the STINGY SAT is in 鈥淣P鈥�

To test whether the STINGY SAT is in 鈥淣P鈥�, first test whether or not the answer incorporates an enjoyable challenge via evaluating the system.

Moreover, check that fewer than 鈥渙kay鈥� literals are assigned with 鈥渁ctual鈥� cost by means of examining the literals as soon as.

Reduce a recognized NP-whole hassle to STINGY SAT

To prove the NP-completeness, allow us to reduce SAT to STINGY SAT.

It's far supplied that in a SAT method if there is 鈥�f鈥� with 鈥�k鈥� variables, then it virtually chooses the for instance of STINGY SAT.

Now, its miles had to show that: 鈥�f鈥� is a 鈥渟ure-example鈥� of SAT if and best if is a 鈥測es-example鈥� of STINGY SAT.

02

Prove

Proof:

Anticipate that 鈥�f鈥� is a 鈥渟ure-example鈥� of SAT. There are absolutely 鈥�k鈥� variables. Consequently, no more than 鈥�k鈥� variables are 鈥済enuine鈥�.

Therefore, any gratifying venture of instance 鈥�f鈥� for SAT will also be the enjoyable venture of example for STINGY SAT.

Subsequently, is a 鈥測es-instance鈥� of STINGY SAT.

Count on that is a 鈥渟ure-example鈥� of STINGY SAT, then any fulfilling assignment of that example will also be gratifying undertaking of instance 鈥�f鈥� for SAT.

Therefore, 鈥�f鈥� is a 鈥測es-instance鈥� of SAT.

Because SAT is NP-Complete, the STINGY SAT is NP-Complete.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove that the following problem is NP-complete: given an undirected graph

G=V,Eand an integer k, return a clique of size kas well as an independent set of size k, provided both exist.

In the HITTING SET problem, we are given a family of sets {S1,S2K,Sn}and budget , and we wish to find a set H of size 鈮�b which intersects every Si, if such an exists. In other words, we want H鈭�Si鈮�蠒for all i.Show that HITTING SET is NP-complete.

Question: In an undirected graph G=(V,E), we say D鈯�Vis a dominating set if every v鈭�V is either in D or adjacent to at least one member of D. In the DOMINATING SET problem, the input is a graph and a budget , and the aim is to find a dominating set in the graph of size at most , if one exists. Prove that this problem is NP-complete.

Determine which of the following problems are NP-complete and which are solvable in polynomial time. In each problem you are given an undirected graph G=(V,E), along with:

(a)A set of nodesL鈯�V , and you must find a spanning tree such that its set of leaves includes the set L.

(b)A set of nodes L鈯�V, and you must find a spanning tree such that its set of leaves is precisely the set L.

(c)A set of nodesL鈯�V , and you must find a spanning tree such that its set of leaves is included in the set L.

(d)An integer k, and you must find a spanning tree withk or fewer leaves.

(e)An integer k, and you must find a spanning tree withk or more leaves.

(f)An integer k, and you must find a spanning tree with exactlyk leaves.

Proving NP-completeness by generalization. For each of the problems below, prove that it is NP-complete by showing that it is a generalization of some NP-complete problem we have seen in this chapter.

  1. SUBGRAPH ISOMORPHISM: Given as input two undirected graphsG and H, determine whetherG is a subgraph of H (that is, whether by deleting certain vertices and edges ofH we obtain a graph that is, up to renaming of vertices, identical toG ), and if so, return the corresponding mapping ofV(G) intoV(H) .
  2. LONGEST PATH: Given a graph role="math" localid="1658141805147" Gand an integerg find inG a simple path of lengthg .
  3. MAX SAT: Given a CNF formula and an integer g, find a truth assignment that satisfies at least gclauses.
  4. DENSE SUBGRAPH: Given a graph and two integersa and b, find a set of a vertices ofG such that there are at leastb edges between them.
  5. SPARSE SUBGRAPH: Given a graph and two integersa andb , find a set of a vertices ofG such that there are at most bedges between them.
  6. SET COVER. (This problem generalizes two knownNP-complete problems.)
  7. RELIABLE NETWORK: We are given twon脳n matrices, a distance matrixdij and a connectivity requirement matrixrij , as well as a budgetb ; we must find a graph G=({1,2,.....,n},E)such that (1) the total cost of all edges isb or less and (2) between any two distinct verticesi andj there arerij vertex-disjoint paths.
See all solutions

Recommended explanations on Computer Science Textbooks

Data Structures

Read Explanation

Problem Solving Techniques

Read Explanation

Issues in Computer Science

Read Explanation

Computer Systems

Read Explanation

Functional Programming

Read Explanation

Databases

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.