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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.

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Most popular questions from this chapter

On page 266we saw that 3SATremainsNP-complete even when restricted to formulas in which each literal appears at most twice.

(a)Show that if each literal appears at mostonce,then the problem is solvable in polynomial time.

(b)Show that INDEPENDENT SET remains NP-complete even in the special case when all the nodes in the graph have degree at most 4.

Optimization versus search.Recall the traveling salesman problem:

TSP

Input: A matrix of distances; a budget b

Output: A tour which passes through all the cities and has length鈮�b, if such a tour exists.

The optimization version of this problem asks directly for the shortest tour.

TSP-OPT

Input:A matrix of distances

Output:The shortest tour which passes through all the cities.

Show that if TSP can be solved in polynomial time, then so can TSP-OPT.

Consider the CLIQUE problem restricted to graphs in which every vertex has degree at most v. Call this problem CLIQUE-3 .

(a) Prove that CLIQUE-3 is in NP .

(b) What is wrong with the following proof of NP-completeness for CLIQUE-3 ? We know that the CLIQUE problem in general graphs is NP-complete, so it is enough to present a reduction from CLIQUE-3 to CLIQUE . Given a graph G with vertices of degree 鈮�3, and a parameter g, the reduction leaves the graph and the parameter unchanged: clearly the output of the reduction is a possible input for the CLIQUE problem. Furthermore, the answer to both problems is identical. This proves the correctness of the reduction and, therefore, the NP-completeness of CLIQUE-3 .

(c) It is true that the VERTEX COVER problem remains NP-complete even when restricted to graphs in which every vertex has degree at most 3 . Call this problem VC-3 . What is wrong with the following proof of NP-completeness for CLIQUE ? We present a reduction from VC-3 to CLIQUE-3 . Given a graph G=(V,E) with node degrees bounded by 3 , and a parameter b , we create an instance of CLIQUE-3 by leaving the graph unchanged and switching the parameter to |V|-b. Now, a subset C鈯�Vis a vertex cover in G if and only if the complementary set V-C is a clique in G. Therefore G has a vertex cover of size鈮�bif and only if it has a clique of size 鈮�|V|-b. This proves the correctness of the reduction and, consequently, the NP-completeness of CLIQUE-3 .

(4)Describe an O(V)algorithm for CLIQUE-3 .

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.

Akiteis a graph on an even number of vertices, say 2n, in which of the vertices form a clique and the remaining vertices are connected in a 鈥渢ail鈥� that consists of a path joined to one of the vertices of the clique. Given a graph and a goal , the KITE problem asks for a subgraph which is a kite and which contains 2g nodes. Prove that KITE is NP-complete.
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