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# Clique NP complete

The Clique Decision Problem is NP and NP-Hard. Therefore, the Clique decision problem is NP-Complete. Attention reader! Don't stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready Theorem 1. CLIQUE is NP-complete. Proof. 1. To show CLIQUE is in NP, our veri er takes a graph G(V;E), k, and a set Sand checks if jSj k then checks whether (u;v) 2Efor every u;v2S. Thus the veri cation is done in O(n2) time. 2. Next we need to show that CLIQUE is NP-hard; that is we need to show that CLIQUE is at least as hard any other.

### Proof that Clique Decision problem is NP-Complete

In this article, we will prove that the Clique Detection Problem is NP-Complete by the help of Independent Set problem, which is NP-Complete. Refer to Proof that Clique Decision problem is NP-Complete, for the proof with the help of Boolean Satisfiability Problem.. Clique Problem is in NP If any problem is in NP, then, given a 'certificate', which is a solution to the problem and an. Why is Clique NP-complete while k-Clique is in P for all k? 0. If the Clique-k Problem is in P, why not Clique as well? 0. Time Complexity of k-clique problem with fixed k. Related. 21. HALF CLIQUE - NP Complete Problem. 11. NP-complete proof from Dasgupta problem on Kite. 27 Show that HALF-CLIQUE is NP-complete using a reduction from CLIQUE. Solution: Clearly HALF-CLIQUE NP, since given a certificate of a subset of vertices we can verify whether they form a clique in time polynomial in V and E. To show that HALF-CLIQUE is NP-Complete, we give a reduction from CLIQUE to MIN - CLIQUE, as follows

1. Why is Clique NP-complete while k-Clique is in P for all k? Ask Question Asked 4 years, 6 months ago. Active 4 years, 6 months ago. Viewed 3k times 4 \$\begingroup\$ I just stumbled upon this question here Why is the clique problem NP-complete? and I am confused by the given answer. The question.
2. Just as CLIQUE is NP-complete, the complement of CLIQUE is co-NP-complete. (More generally, the complement of any NP-complete problem is co-NP-complete). There's a theorem that if any co-NP-complete problem is in NP, then co-NP = NP,which would be a huge theoretical breakthrough
3. The question is to show that the recognition version of Clique is in NP. I have started with a graph G=(V,E) and integer k such that G does have a clique C of cardinality k. How to proceed further..

Overview. NP-complete problems are in NP, the set of all decision problems whose solutions can be verified in polynomial time; NP may be equivalently defined as the set of decision problems that can be solved in polynomial time on a non-deterministic Turing machine.A problem p in NP is NP-complete if every other problem in NP can be transformed (or reduced) into p in polynomial time Prove that it is NP-Complete to determine given input G and k whether G has both a clique of size k and an independent set of size k. Note that this is 1 problem, not 2; the answer is yes if and only if G has both of these subsets.. We were given this problem in my algorithms course and a large group of students could not figure it out

### Proof that Clique Decision problem is NP-Complete Set 2

1. e whether this language belongs to NP complete or not,.
2. Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in monadic second-order logic with second-order quantification on vertex sets, which includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width
3. Disclaimer: I did not produce this video, just found it (can't recall from where) and thought that it might be useful to others
4. NP-Hard Graph Problem - Clique Decision Problem CDP is proved as NP-HardPATREON : https://www.patreon.com/bePatron?u=20475192Courses on Udemy=====..
5. Problem 1. [Clique] Solution To prove that Half-Clique is NP-complete we have to prove that 1) Half-Clique 2NP 2) Half-Clique is NP-hard 1) To prove that Half-Clique 2NP we consider an instance of the problem (G;jVj=2) and a subset HC V. To prove that that HCis an actual solution to the problem we have to verify that the vertices in HCare a.

In this article, we have explored the idea of NP Complete Complexity intuitively along with problems like Clique problem, Travelling Salesman Problem and more. In simple terms, a problem is NP Complete if a non-deterministic algorithm that be designed for the problem to solve it in polynomial time O(N^K) and it is the closest thing in NP to P Theorem: CLIQUE is NP-complete. CLIQUE 2NP: Given an instance (G;k) for CLIQUE, we guess the k vertices that will form the clique. (These vertices form the certi cate.) We can easily verify in polynomial time that all pairs of vertices in the set are adjacent (e.g., by inspection of O(k2) entries Lecture 21 2 Fall 201 The clique decision problem is NP-complete (one of Karp's 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques Clique is NP-complete SAT can be reduced to clique by the following construction. Suppose we have a formula F with m clauses. 1) Vertices are going to be of the form <xa,i>where xa is a literal that occurs in clause Ci 2) Edges are going to be of the form {<xa,i>,<xb,j>} for all xa =¯xb and i = j

### Why is the clique problem NP-complete? - Computer Science

• To prove that CLIQUE is NP-complete, we need to reduce SAT to CLIQUE. It's a bit easier to reduce 3-SAT to CLIQUE (although we could do a direct reduction from SAT). Therefore I'm going to do this in two steps SAT -> 3-SAT -> CLIQUE Generally spea..
• In it, they reduce 3SAT to Clique, proving Clique is NP-Complete, and then reduce Clique to VC. This works in the exact same way as the reduction from VC to Clique that I'll be doing here next. Leave a comment. Posted in Core Problems. Tagged 3-sat, Clique, core problems, Difficulty 7, reductions, Vertex Cover
• Cliques¶. Find and manipulate cliques of graphs. Note that finding the largest clique of a graph has been shown to be an NP-complete problem; the algorithms here could take a long time to run
• Solutions of exercise set 5 Exercise 1 Prove that CLIQUE is NP-complete. CLIQUE: given a graph G and a number k, decide whether G has a clique with k nodes (any two nodes in a clique are adjacent)
• Corollary 2 : The CLIQUE problem is NP-complete. A clique in a simple undirected graph G= (V;E) is a subset C V of vertices, each pair of which is connected by an edge in E. In other words, a clique is a complete subgraph of G. The size of a clique is the number of vertices it contains; that is, jCj. The CLIQUE problem is to nd a clique of.
• The Maximal Clique Problem. Although many theoretical papers have been published on DNA computing since Adleman's first crude demonstration in 1994, it would be over two years, in 1997, until another NP complete problem would be solved

### complexity theory - Why is Clique NP-complete while k

• e whether a clique of size k exists in a graph. Here, we want to prove that CLIQUE is also NP Complete, by reduction from VERTEX-COVER. Before we do the reduction, we need to rst prove that CLIQUE 2NP. Recall that all problems in NP must be veri ble in polynomial time
• 20.3 CLIQUE We will now use the fact that 3-SAT is NP-complete to prove that a natural graph problem called the Max-Clique problem is NP-complete. Deﬁnition 20.2 Max-Clique: Given a graph G, ﬁnd the largest clique (set of nodes such that all pairs in the set are neighbors). Decision problem: Given G and integer k, does G contain
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• We also show that, given a graph G and an integer k, deciding whether the clique-width of G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique.
• The Clique problem is NP-Complete We will show that the Clique problem is NP-complete. It is easy to see that the algorithm above is a polynomial-time algorithm. Putting those facts together, if the algorithm above works, then P = NP
• NP-Complete problems are in both NP and NP-Hard, and therefore their solutions can be checked in polynomial time, but they cannot be solved in polynomial time. The clique cover problem is an NP-Complete problem and can include finding the maximum clique, the maximum weight clique, or the maximal clique. Cliques: {A,B,C}, {B,D}
• 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. Reference - DPV 8.14. Solution: We will try to solve the problem by proving CLIQUE-IS problem as NP Complete in following steps: CLIQUE-IS problem is NP Problem

### Is the complement of the language CLIQUE - Stack Overflo

Abstract. Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in monadic second-order logic with second-order quantification on vertex sets, which includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width The proof that the Clique problem is NP-complete depends on a construction given in Theorem 34.11 (p. 1087), which reduces 3SAT to Clique. Apply this construction to the 3SAT instance Clique problem np complete Complete - i nettapotek - Gratis frakt og rask leverans . Nå kan du bestille Complete fra Vitusapotek på nett. Se vårt utvalg The clique decision problem is NP-complete.It was one of Richard Karp's original 21 problems shown NP-complete in his 1972 paper Reducibility Among Combinatorial Problems Theorem Clique problem is NP complete 9 Clique Problem Proof Step 1 Prove that from CS 010 at Kendriya Vidyalaya Khanapar

### np complete - Max Clique - NP Problem - Mathematics Stack

• Slide 20 of 2
• HALF CLIQUE - NP Complete Problem posted by , on 4:46:00 AM, No Comments. Problem Detail: Let me start off by noting this is a homework problem, please provide only advice and related observations, NO DIRECT ANSWERS please. With that said, here is the problem I am looking at: Let HALF.
• NP-complete Reductions 1. DOUBLEProve that 3SAT P-SAT, i.e., show DOUBLE SAT is NP complete by reduction from 3SAT. The 3-SAT problem consists of a conjunction of clauses over n Boolean variables, where each clause is a disjunction of 3 literals, e.g., (
• In computer science, the clique problem is the computational problem of finding a maximum clique, or all cliques, in a given graph. It is NP-complete, one of Karp's 21 NP-complete problems ().It is also fixed-parameter intractable, and hard to approximate.Nevertheless, many algorithms for computing cliques have been developed, either running in exponential time (such as the Bron-Kerbosch.

### NP-completeness - Wikipedi

• There is clearly a reduction from CLIQUE to k-Color because they're both NP-Complete. In fact, I can construct one by composing a reduction from CLIQUE to 3-SAT with a reduction from 3-SAT to k-Color. What I'm wondering is whether there is a reasonable direct reduction between these problems
• NP-complete problems have no known p-time solution, considered intractable. Tractability Difference between tractability and intractability can be slight If G has clique of size k, contains exactly one vertex per clause satisfied by assigning 1 to corresponding literals
• CLIQUE is NP-complete (4) Proof (cont.): We construct a graph G as follows: 1. For each literal x j,q, we create a distinct vertex in G representing it 2. G contains all edges, except those (i) joining two vertices in same clause, (ii) joining two vertices whose literal
• Complexity Theory 67 Clique 2 CLIQUE is in NP by the algorithm which guesses a clique and then veri es it. CLIQUE is NP-complete, since IND P CLIQUE by the reduction that maps the pair (G;K) to (G ;K), where G i
• PDF of Eric's handwritten notes are here. In this last lecture we took for granted that SAT is NP-Complete, and then we used this fact to show that 3-SAT is NP-Complete. Now we'll build on the NP-completeness of 3-SAT to prove that a few graph problems, namely, Independent Set, Clique, and Vertex Cover, are NP-Complete

Clique is in NP clique is NP-complete. Graph clique solver. Independent Sets Independent set problem: given a graph and an integer k, is there a pairwise non-adjacent node subset of size k? Theorem: The independent set problem is NP-complete. Proof: Reduction from graph clique Some NP-Complete Problems 10.1 Statements of the Problems In this chapter we will show that certain classical algo-rithmic problems are NP-complete. This chapter is heavily inspired by Lewis and Papadim-itriou's excellent treatment [?]. In order to study the complexity of these problems in terms of resource (time or space) bounded Turing ma NP completeness 1. Design and Analysis of Algorithms NP-COMPLETENESS 2. Instructor Prof. Amrinder Arora amrinder@gwu.edu Please copy TA on emails Please feel free to call as well Available for study sessions Science and Engineering Hall GWU Algorithms NP-Completeness 2 LOGISTIC CLIQUE = f(G;k) : graph G has a clique of size kg We will show that CLIQUE is an NP complete problem. To do so, we start with the easier direction: Lemma 0.5. CLIQUE 2NP: Proof. The veri er can take an indicator vector for the set of vertices in the clique, y. Checking whether the size of y is at least k can be done in O(n) time, and checkin

### algorithm - Prove NP-Completeness clique + independent set

1 De nition of NP-Complete Languages 2 naesat is NP-complete 3 0-1 integer programming is NP-complete 4 independent set is NP-complete 5 clique is NP-complete 6 Reductions and Computational Feasibility John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 2, 2009 2 / 1 Some NP-Complete Problems Most natural problems in NP are either in P or NP-complete. Six basic genres and paradigmatic examples of NP-complete problems. Packing problems: SET-PACKING, INDEPENDENT-SET. Covering problems: SET-COVER, VERTEX-COVER. Sequencing problems: HAMILTONIAN-CYCLE, TSP. Partitioning problems: 3-COLOR, CLIQUE

For example, the maximum clique problem, which is NP-hard, can actually be solved eﬃciently assuming a random input because the maximum clique in a randomly chosen graph the ﬁrst problems that was shown to be NP-complete, and for Maximum Independent Set, another NP-complete problem. Lecture 20:. An Annotated List of Selected NP-complete Problems. The standard textbook on NP-completeness is: . Michael Garey and David Johnson: Computers and Intractability - A Guide to the Theory of NP-completeness; Freeman, 1979.. David Johnson also runs a column in the journal Journal of Algorithms (in the HCL; there is an on-line bibliography of all issues) . On the Web the following sites may be of.

np complete problems in graph theory 1. indian institute of information technology, design & manufacturing, kancheepuram study of np-complete problems: poly-time reductions and computing opt solutions submitted by: k seshagiri rao id: 2010uit190 information technology mnit jaipur guide: dr. n sadagopan assistant professor computer engg. dept. iiit d&m kancheepuram signature (dr. n sadagopan. S forms a clique in G, and we have a clique of size k. Since VC is in NP, and we can reduce an NP-complete problem to it, VC must be NP-complete. Problem #3 Denote the Dominating Set problem by DS. Given a potential dominating set S, we can check in polynomial time that every vertex is either in S,or adjacent to a vertex in S. Thus, DS is in NP

clique problems 389 1. INTRODUCTION In this paper,we study biclique and multipartite clique problems. Given a bipartite graph B=V1 ∪ V2E ,abiclique C= U1 ∪ U2 is a subset of the node set,such thatU1 ⊆ V1, U2 ⊆ V2,and for everyu∈ U1, v∈ U2 the edge uv∈ E G=(V,E) and a positive integer k, is there a clique cover of size k or less for G, that is, a partition of V into k disjoint subsets V1,V2,...,Vk such that, for 1 i k, the subgraph induced by Vi is a complete graph? Theorem 2.1. The clique cover problem on circle graphs is NP-complete. Proof. It is not hard to see that the problem is in NP in a clique and are connected to no other vertices in G. This new graph trivially has a clique of size k now. Now run A on this augmented graph. If it answers YES, then that must mean that G had an independent set of size k. If it answers NO, then there cannot be an independent set of size k. This shows that this problem is NP-complete. 5. 9. ### CLIQUE to UNARY-CLIQUE reduction NP complet

• NP-Complete problems are problems that live in both the NP and NP-Hard classes. This means that NP-Complete problems can be verified in polynomial time and that any NP problem can be reduced to.
• tion in the binary search is actually a k-clique problem, which is NP-complete, we utilize uniform sampling to extracts a seed set S s.t., ﬁnding a ! t-clique in Gis equivalent to ﬁnding a ! t-clique in Swith probability guarantees ( 1 n c). For each seed in S, we introduce algorithms scSeed and tciSeed to iteratively shrink its subgraphs
• A clique in an undirect graph G=(V,E) is a subset U of V such that every pair of vertices in U is joined by an edge. E.g., mutual friends on facebook, genes that vary together An optimization problem: How large is the largest clique in G A search problem: Find the/a largest clique in G A search problem: Given G and integer k, find a k-clique in
• 09:46am April 16, 2003 † We can reduce a clique problem to an independent set problem. Suppose we have a clique problem for general graph. We already know it is NP-Complete. We can create an independent set problem for G0, which is the complement of G in the clique problem
• 在计算复杂度理论中，分团问题（clique problem）是图论中的一个NP完全（NP-complete）问题。..

G&J don't even bother to prove Clique is NP-Complete, just stating that it (and Independent Set) is a different version of Vertex Cover. But the problem comes up often enough that it's worth seeing in its own right. The problem: Clique (I've also seen Max Clique or Clique Decision Problem (CDP) The maximal clique problem has been solved by means of molecular biology techniques. A pool of DNA molecules corresponding to the total ensemble of six-vertex cliques was built, followed by a series of selection processes. The algorithm is highly parallel and has satisfactory fidelity. This work represents further evidence for the ability of DNA computing to solve NP-complete search problems Homophones: click, klick; Rhymes: -ɪk Noun []. clique (plural cliques) . A small, exclusive group of individuals, usually according to lifestyle or social status; a cabal. This school used to be really friendly, but now everyone keeps to their own cliques.. 1931, Dorothy L. Sayers, The Five Red Herrings There had been talk of some disagreement about a picture, but in Sir Maxwell's experience. Finding the largest clique in a graph is NP-complete problem, so most of these algorithms have an exponential running time; for more information, see the Wikipedia article on the clique problem _.. Clique ϵ NP; 1) Clique. Definition: - In Clique, every vertex is directly connected to another vertex, and the number of vertices in the Clique represents the Size of Clique. CLIQUE COVER: - Given a graph G and an integer k, can we find k subsets of verticesV 1, V 2...V K, such that UiVi = V, and that each Vi is a clique of G

other exercises to be NP-complete yourself. On question 1 you can earn 2 points. For each of the questions 2 { 8 you Argue that Independent Set is polynomial-time reducible to Clique. (e) Argue that Clique is NP-complete. 2. NP-completeness of Vertex Cover A set of vertices W V is a vertex cover in a graph G = (V;E), if for eac Download PDF Abstract: Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for. Lecture 23: NP-complete Problems Marius Minea marius@cs.umass.edu University of Massachusetts Amherst 22 March 2019 Review: Polynomial-time Reductions can have k-CLIQUE only by choosing one node per clause equivalent to satisfying each clause I no connection between any xi and

### Clique-Width is NP-Complete SIAM Journal on Discrete

work in creating the clauses, and end with a formula with at most 8 m clauses, and 3 m + n variables. Both of these are polynomial, thus the input formula to Exact4SAT will be polynomial in the original input. Because all steps in the reduction take polynomial time, this implies that a polytime algorithm for Exact 4SAT would result in a polytime algorithm for 3-SAT 3DM is also NP-complete, via a reduction from 3SAT. We build gadgets for the variables and clauses. The variable gadget for variable. x. i. is displayed in the picture. Only red or blue triangles can be chosen, where the red triangles correspond to the true literal, while the blue triangles correspond to the false literal CLIQUE therefore CLIQUE is NP-Complete General form: Reduce from known NP-Complete (e.g., SAT) to a new problem. Question: What happens if you reduce to SAT rather than from it? Theorem: Vertex Cover To show that all NP-Complete problems are polynomial-time reducible to it

### NP Completeness for The Clique Problem - YouTub

Example of NP-Complete problem. NP problem: - Suppose a DECISION-BASED problem is provided in which a set of inputs/high inputs you can get high output. Criteria to come either in NP-hard or NP-complete The CLIQUE problem is: given an undirected graph G and an integer g, find a set of ≥ g vertices such that all possible edges between them are present, or report that none exists. Prove that CLIQUE is NP-complete. Show that CLIQUE ∈ NP. Describe a polynomial-time reduction from INDEPENDENT-SET to CLIQUE

Clique in an undirected graph is a subgraph that is complete. Particularly, if there is a subset of k vertices that are connected to each other, we say that graph contains a k-clique. To fin Say that G is a clique graph if there exists a graph H such that G=K(H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. We prove that the clique graph recognition problem is NP-complete Approximating clique is almost NP-complete Abstract: The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist, then EXPTIME=NEXPTIME and NP=P Second, clique-trees of height 2 are the first known example of a class of graphs where µ-coloring is polynomial-time solvable and precoloring extension is NP-complete, thus being at the same. deﬁne polynomial time computable function f: Σ* →Σ* ∀ w ∈ Σ* w ∈ 3SAT 㱻 f(w) ∈ CLIQUE F: On input w • If w is not a valid encoding of a 3CNF formula • then output ; ;{},{} <, 1 < (this is obviously not in CLIQUE) • If w = ;Φ < is a valid encoding of a 3CNF formula • then construct a graph G as speciﬁed and output ;G, k <where k = # clause

Question: We Have Seen That The Clique Problem Is NP-complete, But We Have Also Seen That Sometimes A Restricted Version Of An NP-complete Problem Is Solvable In Polynomial Time. Consider The Restricted Version Of The Clique Problem: Does Graph G Contain A Clique Whose Size Is Exactly 10% Of The Number Of Nodes In The Graph The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist, then EXPTIME=NEXPTIME and NP=P

In the CLIQUE problem, we are given a graph G and a number c 2 IN, and the goal is to decide whether G contains a clique of size at least c. It is not di-cult to show that CLIQUE is NP-complete. In this section we will prove that also approximating the clique number of a graph within a reasonably good bound is NP-hard. For this we need the. How hard is it to check for a clique of size at least k= 4? We just showed that CLIQUE is NP-complete! No luck! Even with small subgraphs (4 vertices :( ). Without further ado, let's prove our theorem: Theorem 1. 3-COLOURING is NP-complete. Where: 3-COLOURING: Given a graph G(V;E), return 1 if and only if there is a proper colouring of Gusing a NP-Complete Algorithms. The next set is very similar to the previous set. Taking a look at the diagram, all of these all belong to , but are among the hardest in the set. Right now, there are more than 3000 of these problems, and the theoretical computer science community populates the list quickly

Max-Clique problem is a non-deterministic algorithm. In this algorithm, first we try to determine a set of k distinct vertices and then we try to test whether these vertices form a complete graph. There is no polynomial time deterministic algorithm to solve this problem. This problem is NP-Complete. Example. Take a look at the following graph NP Hard and NP-Complete Classes. A problem is in the class NPC if it is in NP and is as hard as any problem in NP. A problem is NP-hard if all problems in NP are polynomial time reducible to it, even though it may not be in NP itself. If a polynomial time algorithm exists for any of these problems, all problems in NP would be polynomial time. if G has an independent set of size at least K, then Gc has a clique of size at least K, and if Gc has a clique of size at least K, then G has an independent set of size at least K> (d) Argue that Independent Set is polynomial-time Karp reducible to Clique. (e) Argue that Clique is NP-complete. CLIQUE ¥ Instance: Ð a graph, e.g. Ð a number k (e.g. 4) ¥ Question: Is there a clique of size k, i.e., a set of k vertices such that there is an edge between each pair? ¥ CLIQUE ! NP. ¥ VC P CLIQUE. More NP-complete/ NP-hard Problems ¥ Ha mil t onia n Cir cuit (and hence T r avel l ing Sa l esma n P r obl em) (see Sipser for related.   Clique Partitioning (Cont'd) • Formulation of storage allocation as a clique partitioning problem: - Each value needed to be stored is mapped to a vertex. - Two vertices are connected iff the life-time of the two val-ues do not intersect. ☞The clique partitioning problem is NP-complete. ☞Efﬁcient heuristics have been developed; e.g. The most-degree-central clique problem is introduced and studied. • The decision version of this problem is proven to be NP-complete. • An exact algorithm is proposed. • Experiments with real-life and random networks are performed CLIQUE is NP-complete B Since there is a polynomial time reduction from 3SAT into CLIQUE and CLIQUE 2NP, we can conclude that CLIQUE is NP-complete B Polynomial time reducibility allows us to link these two very different problems B Similar links may be made among other problems 8 / 13 Proof: An input to the Clique problem is a pair (G;k), where G is a graph and k is a number,and we want to decide if 9 a complete subgraph (a subset of nodes such that each pair of nodes have an edge between them) of G with k nodes. We can prove that Clique is an NP-complete problem by proving that 3-SAT ` Clique. The idea is the following: Given an input (l11 _l12 _l13)^¢¢¢^(lm1 _lm2 _lm3.

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