


1: The traveling salesman problem is NPcomplete. 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 polytime 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 NPcomplete. The 3Coloring Problem The 3coloring problem is Given an undirected graph G, is there a legal 3coloring of its nodes? As a formal language: 3COLOR = { G  G is an undirected graph with a legal 3coloring. 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 Hcolouring 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 phub center allocation problem (as well. Draw a Venn diagram to depict the commonly believed relationship between P, NP, NPcomplete and NPhard problems. All NPcomplete problems are equally \hard". 3 The NPCompleteness 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 Kway partition ofG by deleting k nodes or less? Theorem 1. The kcut problem can be solved in polynomial time for xed k[5, 6], but is NPcomplete when kis part of the input [5]. Proof that vertex cover is NP complete Prerequisite – Vertex Cover Problem , NPCompleteness 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. Planargraph coloring: Fact: NPcomplete to decide if a planar graph adimits 3coloring Fact: can always color using 4 colors Edge coloring: Vizing's thm: edge coloring number is either ∆(G) or ∆(G) + 1 Fact: NPcomplete to decide!. Graph theory has abundant examples of NPcomplete problems. Keywords: NPcomplete 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 NPComplete 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 NPcompleteness of 3Lpacking and 3Ipacking, we introduce a graph orientation problem, called the oneinthree graph orientation problem (or 1in3 GO in short), and give reductions from onein. Note: usp[s] = True. (a) A doubleEulerian 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 (cyclefree) path with edges that visits every vertex exactly once. Observe that completecommunication graphs are reexive, 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 NPcomplete. 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 subrouting to construct a gapproducing reduction. The input to Clique is an undirected graph G and a non. In this paper, we describe applications of the acyclic 2coloring problem. There is a specific node, v, and a positive integer distance l. , PM, NP complete. 638 CHAPTER 10. is NPcomplete) is De nition 0. Observe that completecommunication graphs are reexive, 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 NPcomplete. 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. 3colorability is a known NPcomplete. 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 3Colorability The 3colorability 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 NPhard = "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 kcut problem can be solved in polynomial time for xed k[5, 6], but is NPcomplete when kis part of the input [5]. kColoring is NPComplete Clearly in NP, because can check a proposed coloring To prove NP hard, will show 3SAT ≤ P 3Coloring 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 3colorable 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 NPhardness 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 Porientable; namely, POrientation 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 NPcomplete 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 NPcompleteness of 3SAT to prove that a few graph problems, namely, Independent Set, Clique, and Vertex Cover, are NPComplete. 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 3Coloring Problem The 3coloring problem is Given an undirected graph G, is there a legal 3coloring of its nodes? As a formal language: 3COLOR = { G  G is an undirected graph with a legal 3coloring. 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 polynomialtime algorithms ins known. prove that the MaxCut problem is NPComplete. [S] , the problem can be found as NPcomplete, 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 NPCompleteness 19. Introduction NPComplete 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 online 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 NPcompleteness results of the harmonious coloring problem on subclasses of chordal and cochordal graphs, we prove that the problem remains NPcomplete for split undirected path graphs; we also prove that the problem is NPcomplete for colinear graphs by showing that split undirected path graphs form a subclass of colinear. , A2NP, (2) any NPComplete 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 NPcompleteness • We are now ready to deﬁne NPC A problem Y is in NPC (it is NPcomplete) 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 NPcomplete problems. Show that GraphValue is NPcomplete by showing that VertexCover reduces to GraphValue. We prove that the problem is NPcomplete, even for oriented graphs. It is conjectured (and not known) that. We prove it is NPcomplete. to s=cis NPhard 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 NPcomplete 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 ptime are considered tractable NPcomplete problems have no known ptime 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 polynomialtime 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 NPcomplete 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 polynomialtime reduction, and A∈ NP, then Ais NPcomplete. Consider the following decision problem for the CNP: KCRITICAL NODE PROBLEM (KCNP) 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 NPcomplete, so it is enough to present a reduction from clique3 to clique. goes via SAT or some other unrelated NPcomplete 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. NPcomplete problems AmirHossein Ghamarian December 2, 2008 1. 3 The NPCompleteness 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 NPComplete The algorithm above does not work. [10 pts] Suppose someone gives you a blackbox B that takes in any undirected graph G = (V;E). The NPhardness 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 NPHard and NPComplete If P is polynomialtime reducible to Q, we denote this P ≤ p Q Definition of NPHard and NPComplete: » If all problems R ∈ NP are reducible to P, then P is NP Hard »We say P i s NPComplete if P is NPHard and P ∈ NP If P ≤ p Q and P is NPComplete, Q is also NPComplete 20 Proving NPCompleteness What steps do we have to take to prove a problem. A closed path, or cycle,isapathfromsomenodeu to itself. 1 Problem 81. ; Suppose the edge is the most expensive edge contained in the cycle. , that the subgraph be a clique (fullyconnected). 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 timeExample 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•NPcomplete, fundamental class in Computer Sciencereduction 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 oddlength 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. NPCompleteness And Reduction. Observe that completecommunication graphs are reexive, 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 polytime 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 NPcomplete. , [16] for a survey. We settle this question by proving that the problem, similarly to the undirected version, is indeed NPcomplete. The proof for the directed case is similar. If there are three or fewer, we know that the input graph is 3colorable and return "true". Prove that the clique problem is not NPcomplete for such graphs. Problem 1: Given an undirected graph G = (V;E) with jVjeven,. Corollary 1 : The VERTEX COVER problem is NPcomplete. ) Know that if someone can prove that an NPcomplete 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 edgeweighted 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: NPcomplete problem, Traveling Salesman Problem, Mutation operator. It generally though to be extremely unlikely that any NPcomplete problem has a fast algorithm. The ( k,n )CLIQUE problem is NPcomplete. Prove X is in NP. Prove that the bin packing problem is NPhard. Reducing TSP to graph problem InstanceA complete weighted undirected graph G = (V,E) with nonnegative 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 Kway 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 NPcomplete 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 minimumsize set. It will be shown that there is a polynomial time algorithm to solve the problem, and that it is NPcomplete to nd a kind of minimum solution. Graph theory has abundant examples of NPcomplete problems. 5 NPcompleteness • We are now ready to deﬁne NPC A problem Y is in NPC (it is NPcomplete) 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 MaxCut problem is NPComplete. For each of the following problems either give a polynomialtime algorithm or prove that the problem is NPcomplete: 1SAT: 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 NPcomplete problem like SAT. 1 Problem 81. 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 CYCLE2 is in NP. Prove that the clique problem is not NPcomplete 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: KCRITICAL NODE PROBLEM (KCNP) 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 NPcomplete. 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 NPcomplete. Polynomial Time Verification. Therefore, the longest path problem is NPhard. , 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 NPComplete (Hint: reduce from 3SAT or Vertex Cover). (2) The problem of determining whether ther e exists a cy cle in an undir ected graph is in NP. (Alternatively, we can show C ∈ NP by giving a nondeterministic polynomialtime 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 Hcolouring problem CSP(H) to instances of maximum degree b by CSP(H)b. NPhard: a problem to which all other problems in NP reduce. problem of deciding if there exists 3edge colouring of a 3regular graph, is NPcomplete. 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 GraphValue is NPcomplete by showing that VertexCover reduces to GraphValue. Suppose that someone attempted to solve the problem e ciently if the input values are in order. GRAPH 3EQUICOLORING is the following search problem: Given: An undirected graph G with 3n vertices, for some integer n Find: A 3equicoloring of G, or report that none exists Recall that a 3coloring 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 NLcomplete 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 NPcomplete 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 NPcomplete, 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 NPcomplete, 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 NPhardness 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 NPcomplete if Z is in NP and if every NPproblem reduces to Z. The restricted edgeconnectivity of G, denoted by λ0(G), is deﬁned as the minimum number of edges whose deletion results in a disconnected graph and contains no isolated vertices. Specifically, NP Complete problems can only possibly. 2 Problem Set 8 Problem 82. 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 NPcomplete 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 DOUBLESAT is NPcomplete. 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 online 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 NPcomplete problem can b e solv ed in p olynomial time. To prove this, we will find a polynomialtime reduction from 3SAT to INDSET. Prove that the following problems are NPcomplete. Let G = (V,E) by an undirected graph and suppose S ⊆ V. problem of deciding if there exists 3edge colouring of a 3regular graph, is NPcomplete. Consider the set V 0 of nodes in V whose label is 0. Except for some problems. Prove that this problem is NPcomplete. NPCompleteness : 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 wellknown NPcomplete 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 NPcomplete is accepted as evidence that the problem is intractable since a fast method of solving a single NPcomplete problem would immediately give fast algorithms for all NPcomplete problems. 7 Reduction to 3Coloring Given a graph G = (V;E), a valid 3coloring 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 edgeweighted graph with vertex set V 1 ∪ V 2. It looks easy, and there is a pseudopolynomialtime algorithm (i. NPcomplete 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 NPhardness, you may reduce from any problem that has been shown, in class or in CLRS, to be NPcomplete. If G has an independent set of size k, then the corresponding vertices in D are an acyclic subgraph. kColoring is NPComplete Clearly in NP, because can check a proposed coloring To prove NP hard, will show 3SAT ≤ P 3Coloring 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 3colorable iff the clauses are satisfiable. Prove SetCover, HittingSet and DominatingSet are polynomialtime reducible to each other. There are literally thousands ∗[email protected] Show that deciding whether an undirected graph is 5colorable is NPcomplete by a simple reduction from the 3colorability problem. Given a group G generated by elements S = f˙igi2I, the Cayley graph C(G,S) = (G,E) is the edgelabeled directed graph with vertex set G and an edge x !y with label ˙if ˙2S and y = ˙x. NOTE: CLIQUE is NPcomplete (one of Karp's original 21 problems). Following is a simple. , that the subgraph be a clique (fullyconnected). 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 3colorable graph. minimal vertex covering, in a graph is NP complete. Last time we showed that the following problem is NPcomplete: CircuitSAT: Given a circuit of NAND gates with a single output and no loops (some of the inputs may. Use V as the subrouting to construct a gapproducing reduction. NPHard and NPComplete 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 NPcompleteness 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,. NPHard and NPComplete 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 Singlepair shortestpath 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 NPcomplete. Output: Does $ G $ contain a subgraph with exactly $ k $ vertices and at least $ y $ edges? 911. , 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 NPComplete NPcompleteness is a useful concept for showing the di culty of a computational problem, by showing that the existence of a polynomialtime algorithm for the problem would imply that P= NP. 1 Problem 81. 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 NPcomplete: 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. 3colorability is a known NPcomplete. • 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 NPComplete. Williamson NPCompleteness 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 3colorable and return "true". Denition 4 (CompleteCommunication 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 NPcomplete. NPCompleteness 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 NPcomplete. The first part of an NPcompleteness proof is showing the problem is in NP. Problem HC is known to be NPcomplete. 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 NEXPcomplete variants, an NLcomplete 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 NPcomplete problem has a fast algorithm. Nondeterministic algorithm for TSP • The following procedure is a polynomial time nondeterministic algorithm that terminates successfully iff an ordering of n cities are distinct and sum of. NPCompleteness 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 2Color problem (FO2Color problem) is to determine whether G has a (not necessarily proper) 2coloring of the vertices and a functional 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 81. All of the algorithms we have studied thus far have been polynomialtime algorithms: on inputs of size n, their worstcase 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 NPComplete, we need to show four things: (1) there is a nondeterministic polynomialtime 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 CIRCUITSAT: The First NPComplete Problem CIRCUITSAT is a decision problem that asks the following: Given a Boolean circuit with n. The Functional Orientation 2Color problem (FO2Color problem) is to determine whether G has a (not necessarily proper) 2coloring of the vertices and a functional 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 polynomialtime algorithms ins known. NPCompleteness 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 decreasebyone 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. 52, 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 kconnected 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 (CompleteCommunication topological graph). ALGORITHMS 4 Deﬁnition 1. For each of the problems below, prove that it is NPcomplete by showing that : it is a \emph {generalization} of an NPcomplete 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 wellknown to be NPcomplete, unlike the graph isomorphism one. Prove that the following problem, the NonBored Jogger Problem (NBJ), is NPcomplete. Prove that the following problem is NPcomplete: 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 NPComplete: Given a graph and an already existing Hamiltonian Circuit in that graph, decide if the graph has another Hamiltonian Circuit. The vertexdisjoint paths problem is similarly defined. NPcomplete problems AmirHossein Ghamarian December 2, 2008 1. † We consider graphs whose nodes can be partitioned into m disjoint triangles. An Annotated List of Selected NPcomplete Problems. Let the Steiner problem in graph is NP, itsufficient is to show that R Restricted Steiner problem in Petersen graph is infact NPcomplete. No graph with an articulation point can have a Hamiltonian cycle. Prove that Vertex. Both problems are NPcomplete. Each of the following languages is either in P, or it is NPcomplete. 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 NPHard problem, Vertex Cover. MAXCUT is the following problem: given a graph G and a number k, does G have a cut of size k or more? Show that MAXCUT is NPcomplete by reducing NAE3SAT to it. Prove that Hamiltonian cycle problem in NPC using Traveling Salesman problem. As far as I know, to prove a problem is NPcomplete, we first need to prove it's in NP and choose a NPcomplete 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 NPcomplete. Show that the following problem is NPcomplete: Problem: Dense subgraph Input: A graph $ G $, and integers $ k $ and $ y $. Select problem Y that is know to be in NPComplete. 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 NPCompleteness Proofs. The goal is to decide if H is a subgraph of G or not. 2 Problem Set 8 Problem 82. CHAPTER 36: NPCOMPLETENESS. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Because the Hamiltonian path problem is NPcomplete, 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 kcore. 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 3Coloring Problem The 3coloring problem is Given an undirected graph G, is there a legal 3coloring of its nodes? As a formal language: 3COLOR = { G  G is an undirected graph with a legal 3coloring. Let I be an instance of NAE3SAT such that all literals are positive and every variable x has dx 3. 5 The subsetsum problem We next consider an arithmetic NPcomplete problem. Problem Statement. Prove the following problem is NPComplete using a reduction from the subsetsum problem. In the problem of packing edgedisjoint cycles (EDC), we are given a graph G(which can be directed or undirected) and we have to nd a largest set of edgedisjoint 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 NPcomplete 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 NPcomplete: 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 NPcomplete problem. – Paul Apr 9 '11 at 20:51. Unknown; UHAMPATH is NPcomplete. 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 3Coloring Problem The 3coloring problem is Given an undirected graph G, is there a legal 3coloring of its nodes? As a formal language: 3COLOR = { G  G is an undirected graph with a legal 3coloring. In the MAXCUT 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 NPcomplete 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 NPCompleteness 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 DOMINATINGSET 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 NPcomplete, or whether it is neither known to be in P nor known to be NPcomplete. Show that, if P=NP, then every language A ЄP, except A=ø and A=∑*,is NPcomplete. The following problems deﬁned for G and k are NPcomplete. 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 edgedisjoint cycles (EDC), we are given a graph G(which can be directed or undirected) and we have to nd a largest set of edgedisjoint cycles in G. 52, page 1101. The typical approach to proving a language C is NPComplete is as follows: • First show C ∈ NP by giving a deterministic polynomialtime veriﬁer for C. Unknown; this is generally believed to be true, but it might not be. I'm stuck on which NPcomplete problem that can reduce to my. Deciding if a given planar graph is 3colorable is NPcomplete. 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 NPcompleteness 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 NPcomplete. Question 1 (30): For each of the following three problems, either prove that it is complete for NL under Lreductions or prove that it is in L: (a,10) REACHOUT2 = {(G,s,t): G is a directed graph with outdegree at most two and there is a path from s to t in G} This language is NLcomplete. Greedy Algorithm Step01: Color first vertex with the first color. Page 4 19 NPHard and NPComplete If P is polynomialtime reducible to Q, we denote this P ≤ p Q Definition of NPHard and NPComplete: » If all problems R ∈ NP are reducible to P, then P is NP Hard »We say P i s NPComplete if P is NPHard and P ∈ NP If P ≤ p Q and P is NPComplete, Q is also NPComplete 20 Proving NPCompleteness What steps do we have to take to prove a problem. There are many problems for which no polynomialtime algorithms ins known. Given a problem X, prove it is in NPComplete. 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 NPcomplete. That is, a vertex cover of Gis a subset. This shows that this problem is NPcomplete. A computational secretsharing scheme is a method that enables 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 NPCompleteness. In general, you wouldn't expect there to be a "natural" direct reduction between two arbitrary different NPcomplete 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 NPcomplete. Q ∈NP, and 2. Repeat the problem if you are given the preorder and postorder traversals. To prove NPhardness, you may reduce from any problem that has been shown, in class or in CLRS, to be NPcomplete. Corollary 1 : The VERTEX COVER problem is NPcomplete. 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 3EQUICOLORING is the following search problem: Given: An undirected graph G with 3n vertices, for some integer n Find: A 3equicoloring of G, or report that none exists Recall that a 3coloring 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 NPcomplete problem). ) Take any NPcomplete 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, edgeweighted 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. 3colorability is a known NPcomplete. 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 describe a few NPcomplete problems for graphs without proving that they are indeed NPcomplete. Prove these problems are NPComplete: (a) SETCOVER: Given a ﬁnite set , a collection of subsets of, and an integer , determine whether there is a subcollection of with cardinality that covers. The vertex cover is formulated as follows, the vertex cover problem. I'm learning how to prove NPcomplete recently but having trouble to understand the concept of NP. 22 Let HALFCLIQUE 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 NPComplete by reducuing from subsetsum 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 NPcomplete 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 minimumsize set. is one of the 21 original NPcomplete problems enumerated by. I'm stuck on which NPcomplete problem that can reduce to my. – Paul Apr 9 '11 at 20:51. Show that determining whether a graph has a tonian cycle is NPcomplete. 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 Porientable; namely, POrientation 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 NPComplete 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 problemsetting 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 NPcomplete. The total cost of the TStour 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 vertexcover 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 NPcomplete (one of Karp's original 21 problems). 22 Let HALFCLIQUE 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 NPComplete. There are many problems for which no polynomialtime algorithms ins known. We prove that the decision version of even unweighted MCA is NPcomplete 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. NPcomplete: a problem which is both NPhard and in NP. is NPcomplete) 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 NPcomplete: CircuitSAT: 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(a1), 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. NPcomplete problems AmirHossein Ghamarian December 2, 2008 1. The problem of finding a Hamiltonian cycle in a graph is NPcomplete. NOTE: CLIQUE is NPcomplete (one of Karp's original 21 problems). In this paper we study the NPcomplete problem of nding small kdominating sets in general graphs, which allow k 1 nodes to fail and still dominate the graph. (10 Points) Show that the following problem is NPcomplete: • Input: An undirected graph G and an edge e. Reduce 3Dimensional Matching to Exact Set Cover by 3 Sets. CHAPTER 36: NPCOMPLETENESS. (a) T F [4 points] If problem Acan be reduced to 3SAT via a deterministic polynomialtime reduction, and A∈ NP, then Ais NPcomplete.
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