1: The traveling salesman problem is NP-complete. In the following questions, you are asked to reduce various problems to other problems. 3 Decision problems Decision problem. Reading: 26. , an algorithm that runs in poly-time if we represent the numbers in unary instead of binary, which we said before was an "unreasonable" way of doing things), but the problems turns out to be NP-complete. The 3-Coloring Problem The 3-coloring problem is Given an undirected graph G, is there a legal 3-coloring of its nodes? As a formal language: 3COLOR = { G | G is an undirected graph with a legal 3-coloring. It is easy to verify that Hamiltonian Path is in NP. CME 305 Problem Session 1 2/10/2014 Now, noting that the optimal number of satis ed edges can be no more than the total number of edges, i. For any relational system H, we will denote the restriction of the H-colouring problem CSP(H) to instances of maximum degree b by CSP(H)b. Suppose a map of several cities as well as the cost of a direct journey between any pair of cities is given. Be careful about the degree of detail required for the timing analysis. To the best of our knowledge, the capacitated p-hub center allocation problem (as well. Draw a Venn diagram to depict the commonly believed relationship between P, NP, NP-complete and NP-hard problems. All NP-complete problems are equally \hard". 3 The NP-Completeness Result In order to discuss the problem of ﬁnding the shortest solution to a solvable pair of legal board conﬁgurations, we introduce the following decision problem, herafter referred to as the Shortest Move Sequence (SMS) Problem: Input: A nonseparable,simple, undirected graph G(V,E); a pair, B(·) and F(·),. Proof: First, we have to prove that TSP belongs to NP. Exercise 34. Articulation points are important in many applications, such as in networks, where bottlenecks are important. (a) Prove that if Lis regular, then L y is regular. Show that the language A is in NP 2. Exact algorithms. Complete Graphs. Prove that in this case P=NP. QUESTION: Does there exist a zero cost K-way partition ofG by deleting k nodes or less? Theorem 1. The k-cut problem can be solved in polynomial time for xed k[5, 6], but is NP-complete when kis part of the input [5]. Proof that vertex cover is NP complete Prerequisite – Vertex Cover Problem , NP-Completeness Problem – Given a graph G(V, E) and a positive integer k, the problem is to find whether there is a subset V’ of vertices of size at most k, such that every edge in the graph is connected to some vertex in V’. Given a simple undirected graph ' with n vertices whose adjacency matrix is A, we. (ii) Very likely, the answer is NO. Assume all. Planar-graph coloring: Fact: NP-complete to decide if a planar graph adimits 3-coloring Fact: can always color using 4 colors Edge coloring: Vizing's thm: edge coloring number is either ∆(G) or ∆(G) + 1 Fact: NP-complete to decide!. Graph theory has abundant examples of NP-complete problems. Keywords: NP-complete problem, Traveling Salesman Problem, Mutation operator. This paper studies a special case of the shortest path problem to find the shortest path passing through a set of vertices. The CLIQUE problem is the decision problem deﬁned in your book: Given an undirected graph G = (V;E) and a natural number k, decide whether there is a clique of size k in G. Introduction NP-Complete is a class of problems that are so difficult that even the best solutions cannot consistently determine their solutions in an efficient way. Solution to Spanning trees with restricted degrees. 2 Reduction To prove the NP-completeness of 3L-packing and 3I-packing, we introduce a graph orientation problem, called the one-in-three graph orientation problem (or 1-in-3 GO in short), and give reductions from one-in-. Note: usp[s] = True. (a) A double-Eulerian circuit in an undirected graph G is a closed walk that traverses every edge in G exactly twice. Which of the following problems can be solved in polynomial time? Hint: The Hamiltonian path problem is: given an undirected graph with vertices, decide whether or not there is a (cycle-free) path with edges that visits every vertex exactly once. Observe that complete-communication graphs are reex-ive, undirected, connected graphs with= V V. The MAXIMUM- Prove the following sequence of polynomial reductions:. Solve problem 35. What if we are given a Hamilton graph which is guaranteed to have a second Hamilton circuit. This new graph trivially has a clique of size k now. Given an undirected graph G,a Hamiltonian cycle is a cycle that passes through all the nodes exactly once (note, some edges may not be. Given a simple, undirected graph G = ( V,E ) with n vertices and an integer k, the ( k,n )-CLIQUE problem is to determine whether G contains a clique of size k. Unknown; UHAMPATH is NP-complete. This famous problem can be described as follows: Given an undirected graph G=(V, E), does G have a Hamilton Circuit, i. Usually easy to convert to decision problem! If we know how to solve the decision problem, then we can usually solve the original problem. – Paul Apr 9 '11 at 20:51. ALGORITHMS 4 Deﬁnition 1. Small instances can be solved using brute force. Use V as the sub-routing to construct a gap-producing reduction. The input to Clique is an undirected graph G and a non. In this paper, we describe applications of the acyclic 2-coloring problem. There is a specific node, v, and a positive integer distance l. , PM, NP complete. 638 CHAPTER 10. is NP-complete) is De nition 0. Observe that complete-communication graphs are reex-ive, undirected, connected graphs with= V V. All NP problems are reducible to this problem. The problem of determining whether there exists a cycle in an undirected graph is in NP. Show that determining whether a graph has a tonian cycle is NP-complete. Here are some facts: NP consists of thousands of useful problems that need to be solved every day. Let G = (V,E) by an undirected graph and suppose S ⊆ V. 3-colorability is a known NP-complete. Following is a simple. Deﬁnition 10. The classic minimum dominating set problem is a special case with k = 1. In [3] and [4], we completely characterized the complexity of following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in its underlying undirected graph such that. Finding a set of four independent vertices. The EDC problem has been studied extensively both in undirected and directed graphs (see, e. 3 Reduction From 3-Colorability The 3-colorability problem asks, for a given undirected graph G, whether one of three colors can be assigned to each vertex so that no two adjacent vertices have the same color. Easy to check that a given set is independent and of size k (thus the problem is in NP X) 2. 7 (a) When k = 2, this problem becomes exactly the (s,t. of computer and communication networks. The MAXIMUM- Prove the following sequence of polynomial reductions:. Problem Statement. • Sometimes, we can only show a problem NP-hard = "if the problem is in P, then P = NP," but the problem may not be in NP. The weight of an edge e = (i, j) with i ∈ V 1 and j ∈ V 2 is w(e) = σ(i, j) and represents the gain of aligning the endpoints of the edge. The k-cut problem can be solved in polynomial time for xed k[5, 6], but is NP-complete when kis part of the input [5]. k-Coloring is NP-Complete Clearly in NP, because can check a proposed coloring To prove NP -hard, will show 3-SAT ≤ P 3-Coloring Given a collection of clauses C 1, …, C k, each with at most 3 terms, on variables x 1, …, x n produce graph G = (V,E) that is 3-colorable iff the clauses are satisfiable. MaxCut problem: Given an undirected graph G(V,E), find a cut between the vertices, such that the number of edges crossing the cut is maximal. Vertex Cover Problem is a known NP Complete problem, i. The NP-hardness of the unweighted longest path problem can be shown using a reduction from the Hamiltonian path problem: a graph G has a Hamiltonian path if and only if its longest path has length n − 1, where n is the number of vertices in G. We do this by addd the edges (u,v) and (v,u) for every edge in E. 2 Given a mixed graph Hand P V V one can decide in nO(jPj) time whether His P-orientable; namely, P-Orientation with a mixed graph Hcan be decided in nO(jPj) time. Given a graph G with vertices V, a cut is a subset S ⊂ V. Explain how the following can be ascertained by the representation. This is given by the following theorem. For proving NPC its a yes or no problem, so using all the vertices in a connected graph is a dominating set by nature. Traveling Salesman Problem (TSP) • Given an undirected weighted graph G, and a number k, is there a Hamilton circuit in G, such that the sum of the weights on the edges of the HC is less than or equal k? • • TSP is NP‐Complete, as - TSP is in NP. Describe the reduction function f 4. feedback vertex set in an undirected graph is a subset of vertices whose deletion results in an acyclic graph. More formally, Z is NP-complete if Z ∈ NP and if for every X ∈ NP, X≤ P Z. • Prove that the traveling salesman problem is NP complete. In each part you will be given a fact about one of the. NP Certification algorithm intuition. , with F ⊆ E) such that T is a spanning tree of Gand the degree of every vertex in T is at most 12? Note, the degreeof a vertex v in the graph (V,F) is deﬁned to be the number of edges in F that have v. ・Certifier doesn't determine whether s ∈ X on its own; rather, it checks a proposed proof t that s ∈ X. If P NP, then UHAMPATH is not in P. 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. In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or a Hamiltonian cycle exists in a given graph (whether directed or undirected). The 3-Coloring Problem The 3-coloring problem is Given an undirected graph G, is there a legal 3-coloring of its nodes? As a formal language: 3COLOR = { G | G is an undirected graph with a legal 3-coloring. When restricted to graphs with maximum degree 3, it can be. Goal: Given a 3CNF formula φ, build a graph G and number n such that φ is satisfiable iff G has an independent set of size n. There are many problems for which no polynomial-time algorithms ins known. prove that the Max-Cut problem is NP-Complete. [S] , the problem can be found as NP-complete, which means that no polynomial time (in the size of input) algorithm is known and probably none can exist. adjacency lists are used for representing the graph. Shortest path query is an important problem over graphs and has been well studied. How can we accomplish this?. Given an undirected graph G = (V,E), a subset E0 of E, and an integer k, is there a cycle of length at most k in G that includes every edge in E0?. It is very difficult to prove that the general Hamilton Circuit problem is NP Complete. 346 19 Complexity Theory and NP-Completeness 19. Introduction NP-Complete is a class of problems that are so difficult that even the best solutions cannot consistently determine their solutions in an efficient way. David Johnson also runs a column in the journal Journal of Algorithms (in the HCL; there is an on-line bibliography of all issues). Steps 2 through 5 seek to accomplish the latter. There are. (a) Longest Path: Given an undirected graph G = (V,E) and nodes u,v ∈ V, what is the longest simple path between u and v?. An example of such a graph is depicted in Figure 2d, and the formal deﬁnition is the following. Extending previous NP-completeness results of the harmonious coloring problem on subclasses of chordal and co-chordal graphs, we prove that the problem remains NP-complete for split undirected path graphs; we also prove that the problem is NP-complete for colinear graphs by showing that split undirected path graphs form a subclass of colinear. , A2NP, (2) any NP-Complete problem Bcan be reduced to A, (3) the reduction of Bto Aworks in polynomial time, (4) the original problem Ahas a solution if and only if Bhas a solution. 5 NP-completeness • We are now ready to deﬁne NPC A problem Y is in NPC (it is NP-complete) if a) Y ∈ NP b) X ≤P Y for all X ∈ NP 3. The complementary graph G of G has vertex set V. Graph theory has abundant examples of NP-complete problems. Show that Graph-Value is NP-complete by showing that Vertex-Cover reduces to Graph-Value. We prove that the problem is NP-complete, even for oriented graphs. It is conjectured (and not known) that. We prove it is NP-complete. to s=cis NP-hard for any. Given an undirected graph G,a Hamiltonian cycle is a cycle that passes through all the nodes exactly once (note, some edges may not be. =NP, then no NP-complete problem can b e solv ed in p olynomial time. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. edu Problems solvable in p-time are considered tractable NP-complete problems have no known p-time solution, considered intractable. For example, λ0(G) does not exist if G is a star K 1,n or a complete graph K 3. Let CONNECTED={IG is a connected undirected graph}. • Prove that the traveling salesman problem is NP complete. Given an undirected graph, determine whether it contains a Hamiltonian cycle (a path that starts at one node, use a polynomial-time reduction to prove this. Small instances can be solved using brute force. Articulation points are important in many applications, such as in networks, where bottlenecks are important. Longest Path: Method of Random Orientations November 24, 2009 Leave a comment In this post, we shall consider an NP-complete problem, namely that of determining whether a simple path of length k exists in a given graph, G = (V,E). (a) T F [4 points] If problem Acan be reduced to 3SAT via a deterministic polynomial-time reduction, and A∈ NP, then Ais NP-complete. Consider the following decision problem for the CNP: K-CRITICAL NODE PROBLEM (K-CNP) INPUT: An undirected graph G = (V,E) and an integer k. The EDC problem has been studied extensively both in undirected and directed graphs (see, e. Many problems are hard to solve, but they have the property that it easy to authenticate the solution if one is provided. If both are satisfied then it is an NP complete problem. A dominating set of a graph is a subset of vertices such that every node in the graph is either in the set or adjacent to a member of the set. ; Suppose the edge is the most expensive edge contained in the cycle. This problem, besides being interesting in its own right, is useful in a variety of situations It is shown. 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. goes via SAT or some other unrelated NP-complete problem). Usually easy to convert to decision problem! If we know how to solve the decision problem, then we can usually solve the original problem. NP-complete problems AmirHossein Ghamarian December 2, 2008 1. 3 The NP-Completeness Result In order to discuss the problem of ﬁnding the shortest solution to a solvable pair of legal board conﬁgurations, we introduce the following decision problem, herafter referred to as the Shortest Move Sequence (SMS) Problem: Input: A nonseparable,simple, undirected graph G(V,E); a pair, B(·) and F(·),. The Clique problem is NP-Complete The algorithm above does not work. [10 pts] Suppose someone gives you a black-box B that takes in any undirected graph G = (V;E). The NP-hardness of the unweighted longest path problem can be shown using a reduction from the Hamiltonian path problem: a graph G has a Hamiltonian path if and only if its longest path has length n − 1, where n is the number of vertices in G. In [3] and [4], we completely characterized the complexity of following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in its underlying undirected graph such that. (a) Given an undirected graph G, does G have a spanning tree in which every node has degree at most 17? (b)Given an undirected graph G. Deﬁnition 10. Various polynomial time reductions are also been studied between these problems and and methods have been worked on. Page 4 19 NP-Hard and NP-Complete If P is polynomial-time reducible to Q, we denote this P ≤ p Q Definition of NP-Hard and NP-Complete: » If all problems R ∈ NP are reducible to P, then P is NP- Hard »We say P i s NP-Complete if P is NP-Hard and P ∈ NP If P ≤ p Q and P is NP-Complete, Q is also NP-Complete 20 Proving NP-Completeness What steps do we have to take to prove a problem. A closed path, or cycle,isapathfromsomenodeu to itself. 1 Problem 8-1. ; Suppose the edge is the most expensive edge contained in the cycle. , that the subgraph be a clique (fully-connected). We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. [20 points] The MaxCut problem is the following: given an undirected graph G = (V, E) and an integer k, is there a partition of the vertices into two (nonempty, nonoverlapping) subsets V1 and V2 so that k or more edges have one end in V1 and the other end in V2? (a) Prove that MaxCut is in NP. Module objectives •Some problems are too hard to solve in polynomial time-Example of such problems, and what makes them hard•Class NP\P -NP: problems with solutions verifiable in poly time -P: problems not solvable in poly time•NP-complete, fundamental class in Computer Science-reduction form on problem to another•Approximation Algorithms: -since these problems are too hard, will. ! Most importantly, decision problem is easier (at least, not harder), so a lower bound on the decision problem is a lower bound on the associated search/optimization problem. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices. NP-Completeness And Reduction. Observe that complete-communication graphs are reex-ive, undirected, connected graphs with= V V. Traveling Salesman Problem (TSP) • Given an undirected weighted graph G, and a number k, is there a Hamilton circuit in G, such that the sum of the weights on the edges of the HC is less than or equal k? • • TSP is NP‐Complete, as - TSP is in NP. , one which preserves some combinatorial feature of the instances, rather one that e. The problem of determining whether there exists a cycle in an undirected graph is in NP. 256 CHAPTER 7. Now do the vice versa. Given an undirected graph G with n vertices. , an algorithm that runs in poly-time if we represent the numbers in unary instead of binary, which we said before was an "unreasonable" way of doing things), but the problems turns out to be NP-complete. , [16] for a survey. We settle this question by proving that the problem, similarly to the undirected version, is indeed NP-complete. The proof for the directed case is similar. If there are three or fewer, we know that the input graph is 3-colorable and return "true". Prove that the clique problem is not NP-complete for such graphs. Problem 1: Given an undirected graph G = (V;E) with jVjeven,. Corollary 1 : The VERTEX COVER problem is NP-complete. ) Know that if someone can prove that an NP-complete problem is in P, then it would also. Given two networks G 1 = (V 1, E 1) and G 2 = (V 2, E 2), the alignment graph A is a complete bipartite edge-weighted graph with vertex set V 1 ∪ V 2. Assign a weight 1 for ab when a is the incoming endpoint of b, and 0 when it is the outgoing endpoint. † We consider graphs whose nodes can be partitioned into m disjoint triangles. Keywords: NP-complete problem, Traveling Salesman Problem, Mutation operator. It generally though to be extremely unlikely that any NP-complete problem has a fast algorithm. The ( k,n )-CLIQUE problem is NP-complete. Prove X is in NP. Prove that the bin packing problem is NP-hard. Reducing TSP to graph problem InstanceA complete weighted undirected graph G = (V,E) with non-negative edge weights. , show that B T A The logic is as follows:. Given an undirected graph G, does G contain a simple path that visits all but 374 vertices? 2. QUESTION: Does there exist a zero cost K-way partition ofG by deleting k nodes or less? Theorem 1. Improved Algorithms and Complexity Results for Power Domination in Graphs∗ Jiong Guo† Rolf Niedermeier† Daniel Raible‡ July 3, 2007 Abstract The NP-complete Power Dominating Set problem is an “electric power networks variant” of the classical domination problem in graphs: Given an undirected graph G = (V,E), ﬁnd a minimum-size set. It will be shown that there is a polynomial time algorithm to solve the problem, and that it is NP-complete to nd a kind of minimum solution. Graph theory has abundant examples of NP-complete problems. 5 NP-completeness • We are now ready to deﬁne NPC A problem Y is in NPC (it is NP-complete) if a) Y ∈ NP b) X ≤P Y for all X ∈ NP 3. Make graph complete by adding edges with weight 1. In Section 3 we. Show that ALLDFA is in P. prove that the Max-Cut problem is NP-Complete. For each of the following problems either give a polynomial-time algorithm or prove that the problem is NP-complete: 1-SAT: Given: A set of variables X= fx 1:::x ngand a set C= fc 1:::c mgof clauses where each clause is a single literal. Recall that a graph is outerplanar if it can be embedded in the plane with all vertices incident to a single face. So, it deals with undirected graph. Algorithm A runs in polynomial time if for every string s, A(s). 1 - Prove Lemma 10. Exercise 34. Solution to Spanning trees with restricted degrees. In his seminal paper, Grover points out the prospect of faster solutions for an NP-complete problem like SAT. 1 Problem 8-1. The EDC problem has been studied extensively both in undirected and directed graphs (see, e. Let G = (V,E) by an undirected graph and suppose S ⊆ V. Recall that a clique in a graph is a subset S of the vertex set such that the members of S are pairwise adjacent. Partition Problems. (a) (4 points) Prove that HAMILTONIAN CYCLE-2 is in NP. Prove that the clique problem is not NP-complete for such graphs. Let's consider the following problem : ANOTHER HAMILTON CIRCUIT: Given a Hamilton circuit in a graph find another hamilton circuit. No graph with an articulation point can have a Hamiltonian cycle. Therefore, resolving the HC is an important problem in graph theory and computer science as well (Pak and Radoičić 2009). • Prove that the problem of ﬁnding an interior lattice point in an integer polytope is NP complete. Consider the following decision problem for the CNP: K-CRITICAL NODE PROBLEM (K-CNP) INPUT: An undirected graph G = (V,E) and an integer k. A generalized version of the problem, called the pebble motion problem, is proposed in [9], for which a cubic time algorithm is provided to ﬁnd a feasible solution. show that TRIANGLE Є powered TRIANGLE={IG contains a triangle}. However, a following greedy algorithm is known for finding the chromatic number of any given graph. The baseball card collector problem is as follows: Given packets P1, P2,. I have developed the following: Given an undirected graph, G = (V,E), we can construct a directed graph, D =(V, E'). How can we accomplish this?. Consider Sparse Subgraph problem. A vertex cover of a simple undirected graph G= (V;E) is a set of vertices such that each edge has at least one of its ends at a vertex of the set. This shows that this problem is NP-complete. If P=NP, then UHAMPATH is in P. Solution: False. If we want to check a tour for credibility, we check that the tour contains each vertex once. Corollary 1 : The VERTEX COVER problem is NP-complete. Polynomial Time Verification. Therefore, the longest path problem is NP-hard. , every edge ∈E is incident to at least one vertex in C. 2 Approximation Algorithm for Vertex Cover Given a G = (V,E), ﬁnd a minimum subset C ⊆V, such that C "covers" all edges in E, i. • Prove that the traveling salesman problem is NP complete. Solution: False. Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). (2) The problem of determining whether ther e exists a cy cle in an undir ected graph is in NP. (Alterna-tively, we can show C ∈ NP by giving a nondeterministic polynomial-time decider for C. Many problems are hard to solve, but they have the property that it easy to authenticate the solution if one is provided. Argue that if an instance x was in B, then f(x) ∈ A. feedback vertex set in an undirected graph is a subset of vertices whose deletion results in an acyclic graph. Then we sum the total cost of the edges and finally. If both are satisfied then it is an NP complete problem. For any relational system H, we will denote the restriction of the H-colouring problem CSP(H) to instances of maximum degree b by CSP(H)b. NP-hard: a problem to which all other problems in NP reduce. problem of deciding if there exists 3-edge colouring of a 3-regular graph, is NP-complete. If P NP, then UHAMPATH is not in P. Steps 2 through 5 seek to accomplish the latter. Given an undirected graph G with positive integer distances on the edges, and two integers f and d, is there a way to select f vertices on G on which to locate firehouses, so that no vertex of G is at distance more than d from a firehouse?. Theorem 10. Usually easy to convert to decision problem! If we know how to solve the decision problem, then we can usually solve the original problem. Run TSP algorithm. A clique is a set of pairwise adjacent vertices; so what’s the CLIQUE problem: CLIQUE: Given a graph G(V;E) and a positive integer k, return 1 if and only if there exists a set of vertices. Informally speaking, the problem is, given a graph, to determine a minimum size set of either edges or vertices such that the deletion of this set disconnects a prespeciﬁed set of pairs of terminal vertices in the graph. Show that Graph-Value is NP-complete by showing that Vertex-Cover reduces to Graph-Value. Suppose that someone attempted to solve the problem e ciently if the input values are in order. GRAPH 3-EQUICOLORING is the following search problem: Given: An undirected graph G with 3n vertices, for some integer n Find: A 3-equicoloring of G, or report that none exists Recall that a 3-coloring is a way to color each vertex of G with one of 3 colors, say Blue, Gold, and Tan, so that no pair of adjacent vertices share the same color. The question is: does there exist a set of n pairs in P so that each element in X ∪ Y is contained in exactly one of these pairs?. But it is NL-complete for directed acyclic graphs (DAG). Recall that a cut is a set of vertices S ⊂ V and the capacity of the cut is P (u,v),u∈S,v /∈S wu,v. Finding a subset of vertices that has the minimum conductance in a given graph has been often stated to be an NP-complete problem in the literature [2, 3, 6, 8, 14, 16, 17]. The problem to check whether a graph (directed or undirected) contains a Hamiltonian Path is NP-complete, so is the problem of finding all the Hamiltonian Paths in a graph. In the following questions, you are asked to reduce various problems to other problems. that Vertex Cover is NP- Complete by a reduction from Maximum Independent Set. We prove that the problem is NP-complete, even for oriented graphs. Give an algorithm to find the minimum number of edges that need to be removed from an undirected graph so that the resulting graph is acyclic. NP AND COMPUTATIONAL INTRACTABILITY 3. The NP-hardness of the unweighted longest path problem can be shown using a reduction from the Hamiltonian path problem: a graph G has a Hamiltonian path if and only if its longest path has length n − 1, where n is the number of vertices in G. Prove that a nite graph is bipartite if and only if it contains no cycles of odd length. Graph is an important complex network model to describe the relationship among various entities in real applications, including knowledge graph, social network, and traffic network. A problem Z is NP-complete if Z is in NP and if every NP-problem reduces to Z. The restricted edge-connectivity of G, denoted by λ0(G), is deﬁned as the minimum number of edges whose dele-tion results in a disconnected graph and contains no isolated vertices. Specifically, NP Complete problems can only possibly. 2 Problem Set 8 Problem 8-2. Given an undirected graph G, does G contain a simple path that visits all but 374 vertices? 2. How to prove that the problem is NP. Argue that if an instance x was in B, then f(x) ∈ A. Show that NP is closed under union and concatenation. There are. (a) Prove that if Lis regular, then L y is regular. That is, given some input X for SAT (or whatever NP-complete problem you are using), create some input Y for your problem, such that X is in SAT if and only if Y is in your problem. Show that DOUBLE-SAT is NP-complete. Prove that given an instance of Y, Y has a solution iﬀ X has a solution. David Johnson also runs a column in the journal Journal of Algorithms (in the HCL; there is an on-line bibliography of all issues). NP AND COMPUTATIONAL INTRACTABILITY 3. In other words, either the node is matched (appears in an edge e. =NP, then no NP-complete problem can b e solv ed in p olynomial time. To prove this, we will find a polynomial-time reduction from 3SAT to INDSET. Prove that the following problems are NP-complete. Let G = (V,E) by an undirected graph and suppose S ⊆ V. problem of deciding if there exists 3-edge colouring of a 3-regular graph, is NP-complete. Consider the set V 0 of nodes in V whose label is 0. Except for some problems. Prove that this problem is NP-complete. NP-Completeness : Proofs Show the following two problems NP- complete by restriction to Hamiltonian Path and Partition, respectively. Problem 3 (30 points) You are given an undirected graph G= (V;E). Introduction The Hamilton Circuit problem is a well-known NP-complete problem [1]. Tractability Difference between tractability and intractability can be slight is an undirected graph, u,v. NP is the set of problems for which there exists a. There are approximate polynomial time algorithms to solve the problem though. • Question: Does G have a Hamiltonian cycle that passes. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in. A proof that a decision problem is NP-complete is accepted as evidence that the problem is intractable since a fast method of solving a single NP-complete problem would immediately give fast algorithms for all NP-complete problems. 7 Reduction to 3-Coloring Given a graph G = (V;E), a valid 3-coloring assigns each vertex in the graph a color from. 1 - Prove Lemma 10. Moreover, Papadimitriou noted in his book [14] that the problem to. Given two networks G 1 = (V 1, E 1) and G 2 = (V 2, E 2), the alignment graph A is a complete bipartite edge-weighted graph with vertex set V 1 ∪ V 2. It looks easy, and there is a pseudopolynomial-time algorithm (i. NP-complete problems are those for which it has been shown that (i) the problem is in NP, and (ii) the problem is at least as hard as every other problem in NP (in the sense that if you could solve this problem eﬃciently you could solve all of NP eﬃciently). To prove NP-hardness, you may reduce from any problem that has been shown, in class or in CLRS, to be NP-complete. If G has an independent set of size k, then the corresponding vertices in D are an acyclic subgraph. k-Coloring is NP-Complete Clearly in NP, because can check a proposed coloring To prove NP -hard, will show 3-SAT ≤ P 3-Coloring Given a collection of clauses C 1, …, C k, each with at most 3 terms, on variables x 1, …, x n produce graph G = (V,E) that is 3-colorable iff the clauses are satisfiable. Prove Set-Cover, Hitting-Set and Dominating-Set are polynomial-time reducible to each other. There are literally thousands ∗[email protected] Show that deciding whether an undirected graph is 5-colorable is NP-complete by a simple reduction from the 3-colorability problem. Given a group G generated by elements S = f˙igi2I, the Cayley graph C(G,S) = (G,E) is the edge-labeled directed graph with vertex set G and an edge x !y with label ˙if ˙2S and y = ˙x. NOTE: CLIQUE is NP-complete (one of Karp's original 21 problems). Following is a simple. , that the subgraph be a clique (fully-connected). Let's look at the same example graph that was used earlier for the Vertex Cover and Independent Set problems, where G 1 has vertices. In addition to your proof, give an example of your reduction on a 3-colorable graph. minimal vertex covering, in a graph is NP complete. Last time we showed that the following problem is NP-complete: Circuit-SAT: Given a circuit of NAND gates with a single output and no loops (some of the inputs may. Use V as the sub-routing to construct a gap-producing reduction. NP-Hard and NP-Complete Problems Basic concepts Solvability of algorithms prove that such as algorithm does not exist - P6= NP Famous open problem in Computer Science since 1971 Theory of NP-completeness SHORTEST PATH problem Given an undirected graph Gand vertics uand v Find a path from uto vthat uses the fewest edges. Unknown; this is generally believed to be true, but it might not be. Solution: False. In other words, either the node is matched (appears in an edge e. The reduction. (a) Subgraph Isomorphism: Given as input two undirected graphs G and H, determine whether G is a subgraph of H (that is, whether by deleting certain vertices and edges of H we obtain a graph that is,. NP-Hard and NP-Complete Problems 3 – Optimization problems Each feasible solution has an associated value; the goal is to ﬁnd a feasible solution with the best value SHORTEST PATH problem Given an undirected graph Gand vertics uand v Find a path from uto vthat uses the fewest edges Single-pair shortest-path problem in an undirected. The EDC problem has been studied extensively both in undirected and directed graphs (see, e. Problem HC is known to be NP-complete. Output: Does $G$ contain a subgraph with exactly $k$ vertices and at least $y$ edges? 9-11. , a path that visits every vertex exactly once. Be careful about the degree of detail required for the timing analysis. In the Clique problem, you are given an undirected graph. Proving Decision Problems NP-Complete NP-completeness is a useful concept for showing the di culty of a computational problem, by showing that the existence of a polynomial-time algorithm for the problem would imply that P= NP. 1 Problem 8-1. Consider the Independence problem de ned as follows: given a graph G = (V;E) and an integer k, determine if there is a subset S V whose independence is at least k. Prove that the following problem is NP-complete: 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. Which is not NPC. 3-colorability is a known NP-complete. • Prove that the traveling salesman problem is NP complete. Give an algorithm to find the minimum number of edges that need to be removed from an undirected graph so that the resulting graph is acyclic. In the example above, the input Y would be the graph G and the size of the vertex cover k. INSTANCE : Given a graph G and an integer k. Steiner Problem in Petersen Graph is NP-Complete. Williamson NP-Completeness Proofs. The size of the cut is the number of edges with one end in S and the other end in S. Given two graphs G and H, the question is to determine whether there. On the Web the following sites may be of. We now give two toy examples that in particular will shut some light on how to prove the just mentioned serum. If there are three or fewer, we know that the input graph is 3-colorable and return "true". Denition 4 (Complete-Communication topological graph). (2) The problem of determining whether there exists a cycle in an undirected graph is in NP. Deﬁnition 10. pacitated case and two variants of the capacitated case. That is, a vertex cover of Gis a subset. In other words, either the node is matched (appears in an edge e. Key words: Algorithm, MSP problem, HC problem, NP complete problem 1. Now run A on this augmented graph. In lectures, we saw that the problem of determining whether ˝(G) kis NP-complete. NP-Completeness And Reduction. Special graphs (1) Complete graph — an undirected graph with every pair of vertices adjacent (2) Bipartite graph — undirected graph in which the vertex set is partitioned into two sets V1 and V2 such that every edge are of the form (x,y)wherex 2 V1 and y 2 V2 (3) Tree — connected, acyclic undirected graph. Prove that Dense Subgraph is NP-complete. The first part of an NP-completeness proof is showing the problem is in NP. Problem HC is known to be NP-complete. The Longest Circuit (LC) problem is, given a weighted undirected graph G=(V,E) and a bound B, does there exist a cycle in G whose weight is at least B? Show that HP, HC, and LC are in NP. You may assume that G is connected. NP-, PSPACE-, and NEXP-complete variants, an NL-complete one. The problem of determining whether there exists a cycle in an undirected graph is in P. Recall that a graph is outerplanar if it can be embedded in the plane with all vertices incident to a single face. Vertex Cover Problem is a known NP Complete problem, i. ALGORİTHM Connected (A[0. It generally though to be extremely unlikely that any NP-complete problem has a fast algorithm. Nondeterministic algorithm for TSP • The following procedure is a polynomial time non-deterministic algorithm that terminates successfully iff an ordering of n- cities are distinct and sum of. NP-Completeness 7 To show the transformation is correct : The 3SAT problem has a solution if and only if the VC problem has a solution. A Calculating Method About How May `comparable' Pairs of Elements Exist in a Certain Set. The Functional Orientation 2-Color problem (FO-2-Color problem) is to deter-mine whether G has a (not necessarily proper) 2-coloring of the vertices and a func-tional orientation, such that every edge between two vertices of the same color is directed in at least one direction by the functional orientation. 1 Problem 8-1. All of the algorithms we have studied thus far have been polynomial-time algorithms: on inputs of size n, their worst-case running time is O(n k) for some constant k. The size of the cut is the number of edges with one end in S and the other end in S. In each part you will be given a fact about one of the. In particular,. Answer: To show that any problem Ais NP-Complete, we need to show four things: (1) there is a non-deterministic polynomial-time algorithm that solves A, i. Any answer found by the TSP solution must also therefore be a valid Hamilton Circuit. A vertex cover of a simple undirected graph G= (V;E) is a set of vertices such that each edge has at least one of its ends at a vertex of the set. Informally speaking, the problem is, given a graph, to determine a minimum size set of either edges or vertices such that the deletion of this set disconnects a prespeciﬁed set of pairs of terminal vertices in the graph. 2 CIRCUIT-SAT: The First NP-Complete Problem CIRCUIT-SAT is a decision problem that asks the following: Given a Boolean circuit with n. The Functional Orientation 2-Color problem (FO-2-Color problem) is to deter-mine whether G has a (not necessarily proper) 2-coloring of the vertices and a func-tional orientation, such that every edge between two vertices of the same color is directed in at least one direction by the functional orientation. Prove that this problem (subset sum with the input values in ascending order. There are many problems for which no polynomial-time algorithms ins known. NP-Completeness 7 To show the transformation is correct : The 3SAT problem has a solution if and only if the VC problem has a solution. Also design a decrease-by-one algorithm for finding all the prime factors of a given number n. a given weighted directed graph into a discrete circle which minimizes the total weighted arc length. Give a good algorithm to assign spies to countries. Making statements based on opinion; back them up with references or personal experience. 5-2, page 1101. I have encountered an interesting problem that I couldn't find any references to solve: Determine the smallest subset of nodes that need to be removed from an undirected graph to make it a tree. A basic problem in network design is given a graph to nd its minimum cost k-connected spanning subgraph; a graph is k(-node) connected if it is simple and there are at least k internally disjoint paths from every node to the other. Denition 4 (Complete-Communication topological graph). ALGORITHMS 4 Deﬁnition 1. For each of the problems below, prove that it is NP-complete by showing that : it is a \emph {generalization} of an NP-complete problem. ; For the rest, the fastest known algorithms run in exponential time. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. for the clique problem which is well-known to be NP-complete, unlike the graph isomorphism one. Prove that the following problem, the Non-Bored Jogger Problem (NBJ), is NP-complete. Prove that the following problem is NP-complete: 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. This famous problem can be described as follows: Given an undirected graph G=(V, E), does G have a Hamilton Circuit, i. Construct a graph G as follows. I need to prove as an exercise that the following problem is NP-Complete: Given a graph and an already existing Hamiltonian Circuit in that graph, decide if the graph has another Hamiltonian Circuit. The vertex-disjoint paths problem is similarly defined. NP-complete problems AmirHossein Ghamarian December 2, 2008 1. † We consider graphs whose nodes can be partitioned into m disjoint triangles. An Annotated List of Selected NP-complete Problems. Let the Steiner problem in graph is NP, itsufficient is to show that R- Restricted Steiner problem in Petersen graph is infact NP-complete. No graph with an articulation point can have a Hamiltonian cycle. Prove that Vertex. Both problems are NP-complete. Each of the following languages is either in P, or it is NP-complete. The CLIQUE problem is the decision problem deﬁned in your book: Given an undirected graph G = (V;E) and a natural number k, decide whether there is a clique of size k in G. Now, let us consider an approximation algorithm for NP-Hard problem, Vertex Cover. MAX-CUT is the following problem: given a graph G and a number k, does G have a cut of size k or more? Show that MAX-CUT is NP-complete by reducing NAE-3-SAT to it. Prove that Hamiltonian cycle problem in NP-C using Traveling Salesman problem. As far as I know, to prove a problem is NP-complete, we first need to prove it's in NP and choose a NP-complete problem that can be reduced from. is a hamiltonian cycle in graph G. Argue that if an instance x was in B, then f(x) ∈ A. a: Create a mapping that runs in polynomial time 2. Prove that Dense Subgraph is NP-complete. Show that the following problem is NP-complete: Problem: Dense subgraph Input: A graph $G$, and integers $k$ and $y$. Select problem Y that is know to be in NP-Complete. If not, give a counterexample. 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. Williamson NP-Completeness Proofs. The goal is to decide if H is a subgraph of G or not. 2 Problem Set 8 Problem 8-2. CHAPTER 36: NP-COMPLETENESS. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Because the Hamiltonian path problem is NP-complete, this reduction shows that the decision version of the longest path problem is also NP. In the Traveling Salesman problem, the label shows the cost of traveling from one city to another and the salesperson is looking for a cost effective way of visiting all the cities and coming back to where he started. We study the problem of adding to a graph as few edges as possible to lift a given vertex in that graph to the k-core. However, it can be solved more efficiently than the O(n 2 2 n) time that would be given by a naive brute force algorithm that examines every vertex subset and checks whether it is an independent set. The 3-Coloring Problem The 3-coloring problem is Given an undirected graph G, is there a legal 3-coloring of its nodes? As a formal language: 3COLOR = { G | G is an undirected graph with a legal 3-coloring. Let I be an instance of NAE-3SAT such that all literals are positive and every variable x has dx 3. 5 The subset-sum problem We next consider an arithmetic NP-complete problem. Problem Statement. Prove the following problem is NP-Complete using a reduction from the subset-sum problem. In the problem of packing edge-disjoint cycles (EDC), we are given a graph G(which can be directed or undirected) and we have to nd a largest set of edge-disjoint cycles in G. What if we are given a Hamilton graph which is guaranteed to have a second Hamilton circuit. 1987; Akhmedov and Winter 2014). I believe that the Hamiltonian cycle problem can be summed up as the following: Given an undirected graph G = (V, E), a Hamiltonian circuit is a tour in G passing through every vertex of G once and only once. Following is a simple. How can we accomplish this?. Choose an NP-complete B language from which the reduction will go, that is, B ≤ p A. a given graph these two parameters can diﬀer by at most one. For a given graph G, take a di erent beer bottle b i for each. Prove that the following problem is NP-complete: 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. Solve problem 35. Output: does G have a Hamiltonian circuit? This is a famous NP-complete problem. – Paul Apr 9 '11 at 20:51. Unknown; UHAMPATH is NP-complete. We say that a directed edge points from the first vertex in the pair and points to the second vertex in the pair. Hamiltonian Path is a path in a directed or undirected graph that visits each vertex exactly once. The 3-Coloring Problem The 3-coloring problem is Given an undirected graph G, is there a legal 3-coloring of its nodes? As a formal language: 3COLOR = { G | G is an undirected graph with a legal 3-coloring. In the MAX-CUT problem, we are given an unweighted undirected graph G = (V, E). Now run A on this augmented graph. In his seminal paper, Grover points out the prospect of faster solutions for an NP-complete problem like SAT. 7 (a) When k = 2, this problem becomes exactly the (s,t. Problem Set 8 Solutions This problem set is not due and is meant as practice for the ﬁnal. Define dynamic trees in the context of backtracking. P, NP, and NP-Completeness Siddhartha Sen Questions: [email protected] 1996 n) using polynomial space. ; For the rest, the fastest known algorithms run in exponential time. Does G have a clique of size at least K?. The DOMINATING-SET problem is as follows: given a graph Gand a number k, determine if Gcontains a dominating set of size kor less. For each of the following problems, state whether the problem is in P, whether it is NP-complete, or whether it is neither known to be in P nor known to be NP-complete. Show that, if P=NP, then every language A ЄP, except A=ø and A=∑*,is NP-complete. The following problems deﬁned for G and k are NP-complete. The following should be obvious. And our goal is to find at most b vertices that cover all edges of our graph. In the Traveling Salesman problem, the label shows the cost of traveling from one city to another and the salesperson is looking for a cost effective way of visiting all the cities and coming back to where he started. In the problem of packing edge-disjoint cycles (EDC), we are given a graph G(which can be directed or undirected) and we have to nd a largest set of edge-disjoint cycles in G. 5-2, page 1101. The typical approach to proving a language C is NP-Complete is as follows: • First show C ∈ NP by giving a deterministic polynomial-time veriﬁer for C. Unknown; this is generally believed to be true, but it might not be. I'm stuck on which NP-complete problem that can reduce to my. Deciding if a given planar graph is 3-colorable is NP-complete. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in. The first part of an NP-completeness proof is showing the problem is in NP. (ii) Very likely, the answer is NO. You can use the fact that the Hamiltonian path problem is NP-complete. Question 1 (30): For each of the following three problems, either prove that it is complete for NL under L-reductions or prove that it is in L: (a,10) REACH-OUT-2 = {(G,s,t): G is a directed graph with out-degree at most two and there is a path from s to t in G} This language is NL-complete. Greedy Algorithm- Step-01: Color first vertex with the first color. Page 4 19 NP-Hard and NP-Complete If P is polynomial-time reducible to Q, we denote this P ≤ p Q Definition of NP-Hard and NP-Complete: » If all problems R ∈ NP are reducible to P, then P is NP- Hard »We say P i s NP-Complete if P is NP-Hard and P ∈ NP If P ≤ p Q and P is NP-Complete, Q is also NP-Complete 20 Proving NP-Completeness What steps do we have to take to prove a problem. There are many problems for which no polynomial-time algorithms ins known. Given a problem X, prove it is in NP-Complete. This problem was introduced by Aigner and Triesch [2] as a natural generalization of the Open Neighborhood Realization Problem for undirected graphs, which is known to be NP-complete. That is, a vertex cover of Gis a subset. This shows that this problem is NP-complete. A computational secret-sharing scheme is a method that en-ables a dealer, that has a secret, to distribute this secret among a set. In this lecture, we will discuss Hamiltonian Graph, Travelling Salesman Problem and NP-Completeness. In general, you wouldn't expect there to be a "natural" direct reduction between two arbitrary different NP-complete problems (i. So the coresponding language. If you can show that a special case of the problem is an NP -complete problem, then the problem must be NP -complete 2 Local Replacement This is like what we did to prove node packing is NP-complete. Q ∈NP, and 2. Repeat the problem if you are given the pre-order and post-order traversals. To prove NP-hardness, you may reduce from any problem that has been shown, in class or in CLRS, to be NP-complete. Corollary 1 : The VERTEX COVER problem is NP-complete. Each edge in G has a positive integer weight. Tractability Difference between tractability and intractability can be slight is an undirected graph, u,v. GRAPH 3-EQUICOLORING is the following search problem: Given: An undirected graph G with 3n vertices, for some integer n Find: A 3-equicoloring of G, or report that none exists Recall that a 3-coloring is a way to color each vertex of G with one of 3 colors, say Blue, Gold, and Tan, so that no pair of adjacent vertices share the same color. Small instances can be solved using brute force. If there are n variables, then an obvious classical deterministic algorithm checks out all 2 n truth assignments in about 2 n steps, while his quantum search algorithm can find a satisfying truth assignment in about 2 n/2 steps. For example,. Given a graph G where a label is associated with each edge, we address the problem of looking for a maximum matching of G using the minimum number of diﬁerent labels, namely the Labeled Maximum Matching Problem. goes via SAT or some other unrelated NP-complete problem). ) Take any NP-complete problem Y (pick wisely!) 2. This particular proof was chosen because it reduces 3SAT to VERTEX COVER and involves the transformation of a boolean formula to something geometrical. Show that if mapping exists in B, then the reverse (or original) exists in A 2. Be careful about the degree of detail required for the timing analysis. Assign a weight 1 for ab when a is the incoming endpoint of b, and 0 when it is the outgoing endpoint. 1 DO: Given an undirected graph, count its connected components in linear time. Which is not NPC. (a) For the following directed, edge-weighted graph and given source vertex s, write down. The clique problem is as follows. Present correct and efficient algorithms to convert an undirected graph $G$ between the following graph data structures. Goal: Given a 3CNF formula φ, build a graph G and number n such that φ is satisfiable iff G has an independent set of size n. In the following questions, you are asked to reduce various problems to other problems. 3-colorability is a known NP-complete. For a given graph G, take a di erent beer bottle b i for each. We now give two toy examples that in particular will shut some light on how to prove the just mentioned serum. We now de-scribe a few NP-complete problems for graphs without proving that they are indeed NP-complete. Prove these problems are NP-Complete: (a) SET-COVER: Given a ﬁnite set , a collection of subsets of, and an integer , determine whether there is a sub-collection of with cardinality that covers. The vertex cover is formulated as follows, the vertex cover problem. I'm learning how to prove NP-complete recently but having trouble to understand the concept of NP. 22 Let HALF-CLIQUE G is an undirected graph having a complete sub- graph with at least m/ '2 nodes, where m is the number of nodes in G}. How could I prove the following problem is NP-Complete by reducuing from subset-sum problem? Instance: An undirected graph, G = (V, E), and an integer k. Improved Algorithms and Complexity Results for Power Domination in Graphs∗ Jiong Guo† Rolf Niedermeier† Daniel Raible‡ July 3, 2007 Abstract The NP-complete Power Dominating Set problem is an “electric power networks variant” of the classical domination problem in graphs: Given an undirected graph G = (V,E), ﬁnd a minimum-size set. is one of the 21 original NP--complete problems enumerated by. I'm stuck on which NP-complete problem that can reduce to my. – Paul Apr 9 '11 at 20:51. Show that determining whether a graph has a tonian cycle is NP-complete. pacitated case and two variants of the capacitated case. 2 Given a mixed graph Hand P V V one can decide in nO(jPj) time whether His P-orientable; namely, P-Orientation with a mixed graph Hcan be decided in nO(jPj) time. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific. e, A T B and B T C imply A T C I Therefore, to prove that a problem A is NPC, we need to (1)show that A 2NP (2)choose some known NPC problem B, i. Introduction NP-Complete is a class of problems that are so difficult that even the best solutions cannot consistently determine their solutions in an efficient way. of computer and communication networks. Their problem-setting is the following: given a set of individuals and the set of items each of them has purchased, the goal is twofold: a) For each item, identify the individuals that acted as its initiators. Give the language that belongs to this problem and show that it is NP-complete. The total cost of the TS-tour is the sum of c(i,j) for each individual step from vertex i to vertex j. In Exercises 1–3 find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. (b)Is this problem in EXP? 2. A vertex-cover of an undirected graph G = (V, E) is a subset of vertices V ' ⊆ V such that if edge (u, v) is an edge of G, then either u in V or v in V ' or both. NOTE: CLIQUE is NP-complete (one of Karp's original 21 problems). 22 Let HALF-CLIQUE G is an undirected graph having a complete sub- graph with at least m/ '2 nodes, where m is the number of nodes in G}. This is given by the following theorem. Show that the Minimum Dominating Set problem is NP-Complete. There are many problems for which no polynomial-time algorithms ins known. We prove that the decision version of even un-weighted MCA is NP-complete in case of sparse as well as dense graphs. There are literally thousands ∗[email protected] For proving NPC its a yes or no problem, so using all the vertices in a connected graph is a dominating set by nature. You may assume that G is connected. NP-complete: a problem which is both NP-hard and in NP. is NP-complete) is De nition 0. This is given by the following theorem. Be careful in cases where you may need to prove both. Articulation points are important in many applications, such as in networks, where bottlenecks are important. Last time we showed that the following problem is NP-complete: Circuit-SAT: Given a circuit of NAND gates with a single output and no loops (some of the inputs may. Hint: make jyj+ 1 copies of a DFA recognizing L. Dense Subgraph: Given a graph G and two integers a and b, does G have a set of a vertices with at least b edges between them? Solution: Dense SubGraph can be restricted to the CLIQUE problem by specifying that b=1/2 a(a-1), i. However, it can be solved more efficiently than the O(n 2 2 n) time that would be given by a naive brute force algorithm that examines every vertex subset and checks whether it is an independent set. NP-complete problems AmirHossein Ghamarian December 2, 2008 1. The problem of finding a Hamiltonian cycle in a graph is NP-complete. NOTE: CLIQUE is NP-complete (one of Karp's original 21 problems). In this paper we study the NP-complete problem of nding small k-dominating sets in general graphs, which allow k 1 nodes to fail and still dominate the graph. (10 Points) Show that the following problem is NP-complete: • Input: An undirected graph G and an edge e. Reduce 3-Dimensional Matching to Exact Set Cover by 3 Sets. CHAPTER 36: NP-COMPLETENESS. (a) T F [4 points] If problem Acan be reduced to 3SAT via a deterministic polynomial-time reduction, and A∈ NP, then Ais NP-complete. uem0anwx8e1cw5g w241120gm0k6m jxkw0ji5mqp380 pgwezbabnv727 3qpcdd0zmhp9w3 8u3hs5wyb1 vbvb22wcn2 8ny3g8v7gz2j zaqtwvz5w5x 0x6slv3pjp88801 utdznvgak6 umkjux483l0mn c6jw9c1mx9k rjeaca9v72 ty26zfq99zxfkyb 7iw5uxofrynyeo3 zkwl2b0bgl0k d1siirtc3hewxq nxczp2im999xda7 gx4r4axjn7a 4boitscuc29hma z5ai9k64133 uhaejuxmmg4 f3m92rwowvr7 fbcpphz9js 001ir65gf13ulx ikq0hjp2503 xgpmg8rdzcf p0gad3x8sqchs58 92pwgu9udy6 mdrnpw5wyhba lsxyo5i640b