# Prove That The Following Problem Is Np Complete Given An Undirected Graph

Now given a different problem P' If we show P. Let G = (V,E)be an undirected graph with nvertices and k a pos-itive integer, as before. We prove the following: • Let L = {3,4,5,} \ L be the set of forbidden cycle lengths. This is given by the following theorem. Polynomial-time reduction: convert an instance of A to an instance of another decision problem B in polynomial-time so that answer to A is “yes” if and only if the answer to B is “yes” If you can do this for all instances of A, then it proves that B is HARDER than A w. We also show that it is unlikely. 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. For series-parallel graphs it is complete for L [16]. Complexity: It is NP-complete, it contains as a special case (D 1 =(V,t 1 s 1), D 2 =(V,t 2 s 2) and D 3 =D) the following directed 2-commodity integral flow problem that is NP-complete : Given a directed graph D and two pairs of vertices, s 1,t 1 and s 2,t 2, decide whether there exist a path from s 1 to t 1 and a path from s 2 to t 2 that are. , UHAMPATH= is an undirected graph with a Hamiltonian path from to ). , every edge ∈E is incident to at least one vertex in C. ) Give your solution in simple and elegant pseudocode. Each edge in G has a positive integer weight. The related term strongly NP-complete (or unary NP-complete) refers to those problems that remain NP-complete even if the data are encoded in unary (that is, if the data are “small” relative to the overall input size). The MAXIMUM-CLIQUE problem is an optimization problem deﬁned by: Given an undirected graph G = (V;E),. Question: Is ˝(G) k? Does it follow that this problem NP-complete? Why or why not? 3. Given a graph G = (V,E) and a parameter k, we consider the problem of ﬁnding a subset U ⊆ V of size k that maximizes the number of induced edges (DkS). Solution to Spanning trees with restricted degrees. True, False, or Unknown: The Hamiltonian path problem for undirected graphs is in P (i. We widely generalize the result of [1] as follows. 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. This particular proof was chosen because it reduces 3SAT to VERTEX COVER and involves the transformation of a boolean formula to something geometrical. Thus by the examples above, CSP(K3)b is polynomial time solvable for any b ≤ 3 and NP-complete for any b ≥ 4. (Feedback set) Given an undirected graph G = (V;E), a feedback set is a set X V with the property that G X has no cycles. NP AND COMPUTATIONAL INTRACTABILITY 3. In other words, given an undirected graph, find the largest set of vertices such that no two are connected with an edge. (3) If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve A. that Vertex Cover is NP- Complete by a reduction from Maximum Independent Set. Although no one has found polynomial-time algorithms for these problems, no one has proven that no such algorithms exist for them either! In fact, it is quite possible that all problems in. In contrast, the multiway cut problem is NP-complete for all k 3 and is also APX-hard for all k 3 [2]. Prove that LastToken is NP-complete. An undirected graph G and an integer K Question: Is there a vertex cover of size K. ) Show that there is a polynomial time method for changing any instance of problem Y into an instance of problem X. However, much of the work has focused on special graph structures, such as grids [9,18,31], and that too usually with the MaxSave objective. We ask whether there exists a subset S0 S whose elements sum to t. 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). We have displayed several Hamiltonian paths. To this question some negative results are known. The problem of finding a Hamiltonian cycle in a graph is NP-complete. Use V as the sub-routing to construct a gap-producing reduction. 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. [KT-Chapter8] Given an undirected graph G = (V;E), a feedback set is a set X V with the property that G X has no cycles. ・Algorithm A solves problem X: A(s) = yes iff s ∈ X. A computer graph is a graph in which every two distinct vertices are joined by exactly one edge. It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where ). 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. There are two things that have to be proved: (1) B is in NP and (2) HC0. 3 Decision problems Decision problem. 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]. 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. , UHAMPATH= is an undirected graph with a Hamiltonian path from to ). To do so, we give a reduction from 3-SAT (which we've shown is NP-complete) to CLIQUE. [10 pts] Suppose someone gives you a black-box B that takes in any undirected graph G = (V;E). An example of such a graph is depicted in Figure 2d, and the formal deﬁnition is the following. However, in this problem we work with an array of arbitrary numbers (in particular, unsorted values). Following is a simple. NP-complete ,and remains NP-complete for bipartite undirected graphs with maximum degree six. 2 Directed Graphs. 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. In this paper, we describe applications of the acyclic 2-coloring problem. •Instance: an undirected graph. We are again given a graph, a simple, unweighted, undirected graph, together with the budget b. This new graph trivially has a clique of size k now. True, False, or Unknown: The Hamiltonian path problem for undirected graphs is in P (i. The NP-hard 2-Club problem is, given an undirected graph G= (V;E) and a positive integer ', to decide whether there is a vertex 2-Club is NP-complete even on split graphs and, thus, also on chordal graphs [3]. Prove that given an instance of Y, Y has a solution iﬀ X has a solution. Given an undirected unweighted graph G and a parameter k, answer whether there is a cut S V(G) so that the number of edges leaving S is at least k. We prove that one variant of the capacitated case is NP-complete, where the other is polynomially solvable. The problems can all be easily veriﬁed to be in NP. Prove Set-Cover, Hitting-Set and Dominating-Set are polynomial-time reducible to each other. Prove that Dense Subgraph is NP-complete. However we shall prove the following strengthening of Theorem 3(b). 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. problem is NP-complete, e. There are many problems for which no polynomial-time algorithms ins known. (green text added 7 August 2003) This algorithm can be used to decide the NP-complete language 3-COLOR. We show that HamiltonianCycle / HamiltonianPath. Then prove the even stronger statement that any two of the following statements imply the third: Let G be an undirected graph with n nodes: 1) G is connected 2) G does not contain a cycle 3) G has n-1 edges Thanks!!!!. The following two corollaries are immediate from the above theroem. • Question: Does G have a Hamiltonian cycle that passes. Consider the set V 0 of nodes in V whose label is 0. Construction problem: Find the largest clique in the input graph G. 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. Algorithm Note, Oct 14 Wenjian Yang. For example, the NP-completeness proof due to Papadimitrou [21] for the problem of ﬁnding. Posted 5 years ago. Example: How can I prove that this is NPC problem? This is homework, I know I should show you what I have tried, but I have no idea how to do it. 1: The traveling salesman problem is NP-complete. Suppose that someone attempted to solve the problem e ciently if the input values are in order. We define a cut (S, V − S) of G as a partition of V, and the weight of a cut as the number of edges crossing the cut. So we know that the Clique problem is NP-complete and by having b=a(a-1)/2 we can restrict the Dense Subgraph problem to the Clique problem. Speci cally, choose some known NP-complete problem B, and reduce Bto A(note: not the other way around!), i. Prove that if uis a vertex of odd degree in a graph, then there exists a path from uto another. Williamson NP-Completeness Proofs. Prove that the following problem, the Non-Bored Jogger Problem (NBJ), is NP-complete. Michael Garey and David Johnson: Computers and Intractability - A Guide to the Theory of NP-completeness; Freeman, 1979. Suppose a map of several cities as well as the cost of a direct journey between any pair of cities is given. a directed graph which has exactly one edge between each pair of vertices. To prove this, we will find a polynomial-time reduction from 3SAT to INDSET. 1 Examples and Overview 19. The computational problem is called the minimum spanning tree (MST) problem. Deciding if a given planar graph is 3-colorable is NP-complete. Hamiltonian Path is a path in a directed or undirected graph that visits each vertex exactly once. Assuming that the HC problem is NP-Complete, prove that the Traveling Salesman problem is NP-Complete. Since graph isomorphism is not known to be in P nor is it known to be NP-complete, it is “natural” to define the complexity class related to graph isomorphism, GI, which is made up of problems with a polynomial time reduction to the graph isomorphism problem. Construct a Peterson graph of 10 vertices. 1 illustrates three edge-disjoint paths P 1, P 2 and P 3 in a series-parallel graph. isomorphism problem could also be applicable in resolving the much more important problems of determining a graph's clique number and finding its maximum clique, i. Given a complete graph with edge weights, the weight of a cycle cover is the sum of the weights of its edges. 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. , there is no polynomial time solution for this unless P = NP. Given an integer k > 0, determine if there is a connected subgraph of G that. , [Balister. We prove it is NP-complete. It is easy to verify that Hamiltonian Path is in NP. Problem Statement. 7 (a) When k = 2, this problem becomes exactly the (s,t. Bounded Degree Spanning Tree Instance : An undirected graph G = (V, E) and a positive integer k ≤≤≤. Example: How can I prove that this is NPC problem? This is homework, I know I should show you what I have tried, but I have no idea how to do it. Show that the following problem is NP-complete: Problem: Clique, no-clique Input: An undirected graph $ G=(V,E) $ and. We shall apply Theorem 2 to the undirected case in order to obtain Theorem 4. Output: does G have a Hamiltonian circuit? This is a famous NP-complete problem. Therefore, the longest path problem is NP-hard. HINT: rst gure out the following reduction involving a single edge: for two Boolean. • Question: Does G have a Hamiltonian cycle that passes. In this section we prove that the edge disjoint paths problem on directed and undirected rectangle graphs remains NP-complete even in the restricted case when G+H is Eulerian. Prove that a complete graph with nvertices contains n(n 1)=2 edges. More specifically, assume that there is a polynomial algorithm A that solves a problem P in NP. , UHAMPATH= is an undirected graph with a Hamiltonian path from to ). Edge Cover, given an undirected graph, asks for the smallest set of edges such that every vertex in the graph is incident to at least one of the edges. Except for some problems Input: An (undirected) graph G= (V;E), and vertices s;t2V. , a weighted directed graph. Prove that a connected undirected graph with n vertices and n-1 edges cannot have any cycles. Math Basics. Making statements based on opinion; back them up with references or personal experience. Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). We shall apply Theorem 2 to the undirected case in order to obtain Theorem 4. } This problem is known to be NP-complete by a reduction from 3SAT. Construction problem: Find the largest clique in the input graph G. Prove that a connected undirected graph with n vertices and n-1 edges cannot have any cycles. (Feedback set) Given an undirected graph G = (V;E), a feedback set is a set X V with the property that G X has no cycles. , [Balister. (a) [5 points] Let TRIPLE-SAT denote the following decision problem: given a Boolean formula. Show that the following problem is NP-complete: Problem: Dense subgraph Input: A graph $ G $, and integers $ k $ and $ y $. $\begingroup$ The easiest way to prove a problem is NP complete is usually to show that you can use it to solve a different NP-complete problem with only polynomial many questions and polynomialy many extra steps. If G has an independent set of size k, then the corresponding vertices in D are an acyclic subgraph. 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. 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. Given an undirected graph G, does G have a spanning tree in which every node has degree at most 374?. Show that this problem is NP-complete. Let the Minimum Dominating Set problem be the task of determining whether there is a dominating set of size k. Show that DOUBLE-SAT is NP-complete. Except for some problems. 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. In this section we prove that the edge disjoint paths problem on directed and undirected rectangle graphs remains NP-complete even in the restricted case when G+H is Eulerian. Asked: May 07,2020 In:Algorithms. 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. We will show that the Clique problem is NP-complete. 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). NP-complete: a problem which is both NP-hard and in NP. HAMILTONIAN CYCLE Input: Undirected graph G = (V;E). ! 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. Q ∈NP, and 2. Show that DOUBLE-SAT is NP-complete. Intuitively, a problem isin P1 if thereisan efﬁcient (practical) algorithm toﬁnd a solutiontoit. Problem 3 (30 points) You are given an undirected graph G= (V;E). Suppose you are able to prove that, for any other problem Q in NP, there is an algorithm B that solves Q by making use of a polynomially bounded number of calls to A. The goal is to better understand the theory and to train to recognize to construct reductions. 1(b): If vertices v and w are Ch. undirected graph is in P. Note: usp[s] = True. The undirected edge-disjoint paths problem is NP-complete, even if G + H is Eulerian and IE(ii)( = 3. Run TSP algorithm. In this paper, we describe applications of the acyclic 2-coloring problem. Prove that MAX-CUT is NP-complete. algorithms graph-theory graph-algorithms. It is natural to wonder whether all problems can be solved in polynomial time. NOTE: CLIQUE is NP-complete (one of Karp's original 21 problems). Prove that in this case P=NP. NP-complete problems AmirHossein Ghamarian December 2, 2008 1. Successfully studied and implemented a few solutions to various NP-Complete Problems. (Less formally, think of NP-complete problems as the NP problems that are as 'hard' as possible. NP-, PSPACE-, and NEXP-complete variants, an NL-complete one. 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. Since the reducibility relation is transitive, to prove that a problem Ain NP is NP-complete, it su ces to prove that some other NP-complete problem Bis polynomially reducible to it. of computer and communication networks. 1 - Prove Lemma 10. 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. The computational problem is called the minimum spanning tree (MST) problem. 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?. The input to Clique is an undirected graph G and a non. Likewise, prove in the reverse direction, i. Prove that Hamiltonian cycle problem in NP-C using Traveling Salesman problem. Let I be an instance of NAE-3SAT such that all literals are positive and every variable x has dx 3. Prove the following problem is NP-Complete using a reduction from the subset-sum problem. You can use the fact that the Hamiltonian path problem is NP-complete. False Any NP-hard problem can be solved in polynomial time if there is an algorithm that can solve the satisfiability problem in polynomial time. 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. Polynomial-time reduction: convert an instance of A to an instance of another decision problem B in polynomial-time so that answer to A is “yes” if and only if the answer to B is “yes” If you can do this for all instances of A, then it proves that B is HARDER than A w. For each of the problems below, prove that it is NP-complete by showing that it is a generalization of an NP-complete problem. (Feedback set) Given an undirected graph G = (V;E), a feedback set is a set X V with the property that G X has no cycles. If we want to check a tour for credibility, we check that the tour contains each vertex once. As it should not be too hard to show that the latter reduces to the former, and the former reduces to the latter. Show the following problem is NP-complete: Degree12Tree: Given an undirected graph G= (V,E), does there exist a subgraph T = (V,F) of G(i. Given a positive weight function defined on , the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut such that is maximum. Be careful in cases where you may need to prove both. Therefore, NP-Complete set is also a subset of NP-Hard set. This is given by the following theorem. To prove that this problem is NP-hard, we will show that it is equivalent to nding an undirected Hamiltonian cycle in a graph. The size of the cut is the number of edges with one end in S and the other end in S. ! 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. Math Basics. 2 CIRCUIT-SAT: The First NP-Complete Problem CIRCUIT-SAT is a decision problem that asks the following: Given a Boolean circuit with n. 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. The Fireﬁghter problem is NP-. An undirected graph G and a positive integer K. Polynomial-time reduction: convert an instance of A to an instance of another decision problem B in polynomial-time so that answer to A is “yes” if and only if the answer to B is “yes” If you can do this for all instances of A, then it proves that B is HARDER than A w. 3 In this problem, all graphs are undirected. Prove that LastToken is NP-complete. A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices. Various polynomial time reductions are also been studied between these problems and and methods have been worked on. Any answer found by the TSP solution must also therefore be a valid Hamilton Circuit. [10 pts] Suppose someone gives you a black-box B that takes in any undirected graph G = (V;E). Traveling Salesman is NP-complete Thm. 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. – Paul Apr 9 '11 at 20:51. , UHAMPATH= is an undirected graph with a Hamiltonian path from to ). Show that NP is closed under union and concatenation. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. To show a gap-problem NP-hard via gap-producing reduction, we need to reduce from a NP-hard problem Lto the gap-problem in polynomial time. 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!. 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. We also show that it is unlikely. that Vertex Cover is NP- Complete by a reduction from Maximum Independent Set. SOME NP-COMPLETE PROBLEMS An undirected graph G is connected if for every pair (u,v) ∈ V × V,thereisapathfromu to v. Which of the following are true? [Check all that apply. We show that HamiltonianCycle / HamiltonianPath. 4 Reducing MISP to the maximum independent set Now that we know that MISP is NP-complete, we can use one of the vast number of algorithms already developed for solving various NP-hard problems, once we reduce MISP to that problem. 5-2, page 1101. 7 Reduction to 3-Coloring Given a graph G = (V;E), a valid 3-coloring assigns each vertex in the graph a color from. A triangle in an undirected graph is a 3-clique. It is a relatively new problem whose application is related to the timetabling problem [15]. A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. 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. S is an independent set of G if and only if S is a clique in G. The problem of determining whether there exists a cycle in an undirected graph is in P. , that the subgraph be a clique (fully-connected). Graph theory has abundant examples of NP-complete problems. An example of such a graph is depicted in Figure 2d, and the formal deﬁnition is the following. To prove that this problem is NP-hard, we will show that it is equivalent to nding an undirected Hamiltonian cycle in a graph. 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. In most cases, the problem is NP-complete. We are again given a graph, a simple, unweighted, undirected graph, together with the budget b. 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?. More specifically, assume that there is a polynomial algorithm A that solves a problem P in NP. Hence, to prove the NP-completeness of a problem, all we have to do is to 1) show that it is in NP and 2) show that an NP-complete problem can be reduced to it. neighborhood dihypergraph N(D) of D. Posted by Fredrik 2019-03-19 2019-03-19 Leave a comment on Proving the set-covering problem is NP-complete (using reduction from the vertex-cover problem) In this post, we will prove that the decision version of the set-covering problem is NP-complete, using a reduction from the vertex covering problem (which is NP-complete). The goal is to decide if H is a subgraph of G or not. True In an undirected graph with unit edge costs, a shortest-path tree found by breadth-first search is also a minimum spanning tree. 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-. (In other words, a clique is a complete subgraph. 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’. If Q just satisﬁes (2) then it's called NP-hard. However, despite the similarities between the Graph Isomorphism Problem and many 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. It's NP-complete (even for 0-1 weights) by a reduction from feedback arc set in directed graphs. polynomial-time reduction. Given a graph G with vertices V, a cut is a subset S ⊂ V. • Prove that the problem of ﬁnding an interior lattice point in an integer polytope is NP complete. independent set Is NP-Complete † This problem is in NP: Guess a set of nodes and verify that it is independent and meets the count. Solve problem 35. (1) The problem of determining whether there exists a cycle in an undirected graph is in P. 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. 1 illustrates three edge-disjoint paths P 1, P 2 and P 3 in a series-parallel graph. Now, let us consider an approximation algorithm for NP-Hard problem, Vertex Cover. , a weighted directed graph. This should be feasible for bipartite graphs with up to, say, 30+30 vertices. Then does not belong to any minimum spanning tree. NP-complete: Both NP-hard (bad news) and in NP (good news) How to prove that a problem is NP-complete? Example: MaxIndependentSet Input: A graph G, an integer k Question: Does G admit an independent set of size k? 1. Making statements based on opinion; back them up with references or personal experience. 1 Proving NP-completeness In general, proving NP-completeness of a language L by reduction consists of the following steps. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific. 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. NP Certification algorithm intuition. It is natural to wonder whether all problems can be solved in polynomial time. No graph with an articulation point can have a Hamiltonian cycle. I have developed the following: Given an undirected graph, G = (V,E), we can construct a directed graph, D =(V, E'). This new graph trivially has a clique of size k now. Unknown; this is generally believed to be true, but it might not be. a: Create a mapping that runs in polynomial time 2. Figure 1: An instance of Vertex Cover problem. A Sample Proof of NP-Completeness The following is the proof that the problem VERTEX COVER is NP-complete. 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. (A) 1,2 and 3 (B) 1 and 2 only (C) 2 and 3 only (D) 1. 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. The complete graph with n vertices is denoted by K n. You may assume that G is connected. Y X Class unknown Reduction Known NP-C. NP complete problems •problem A is NP-complete if-A is in NP (poly-time to verify proposed solution) -any problem in NP reduces to A •second condition says: if one solves pb A, it solves via polynomial reductions all other problems in NP •CIRCUIT SAT is NP-complete (see book)-and so the other problems discussed here, because they reduce to it. A linear time algorithm, MEDIAN PLACEMENT, due to [2] provides. 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. Recall that a clique in a graph is a subset S of the vertex set such that the members of S are pairwise adjacent. Problem Given a graph G = (V,E)and an undirected path, does it have a Hamilton path, a path visiting each node exactly once? Theorem HAMILTON PATH is NP-complete. Explain how the following can be ascertained by the representation. INSTANCE : Given a graph G and an integer k. Assuming HAMPATH is NP-complete, prove that HAMCYCLE is NP-complete. problem asks whether all sites in the heap can be properly assigned types from the given type set. To prove that this problem is NP-hard, we will show that it is equivalent to nding an undirected Hamiltonian cycle in a graph. 1 The traveling salesperson problem The traveling salesperson problem is a famous example of an NPC prob-lem. Complexity: It is NP-complete, it contains as a special case (D 1 =(V,t 1 s 1), D 2 =(V,t 2 s 2) and D 3 =D) the following directed 2-commodity integral flow problem that is NP-complete : Given a directed graph D and two pairs of vertices, s 1,t 1 and s 2,t 2, decide whether there exist a path from s 1 to t 1 and a path from s 2 to t 2 that are. =NP, then no NP-complete problem can b e solv ed in p olynomial time. The goal is to decide if there is a clique on kvertices, that is, a complete subgraph on kvertices. Tractability Difference between tractability and intractability can be slight is an undirected graph, u,v. 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. Argue that if an instance x was in B, then f(x) ∈ A. neighborhood dihypergraph N(D) of D. The DOMINATING-SET problem is as follows: given a graph Gand a number k, determine if Gcontains a dominating set of size kor less. HINT: rst gure out the following reduction involving a single edge: for two Boolean. , that the subgraph be a clique (fully-connected). NP-complete, so is IS. A TS-tour is a circuit that visits each vertex exactly once. We say that a directed edge points from the first vertex in the pair and points to the second vertex in the pair. (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,. However, in this problem we work with an array of arbitrary numbers (in particular, unsorted values). ・Problem X is a set of strings. This shows that this problem is NP-complete. algorithms graph-theory graph-algorithms. b) Infer the social relationships between. Some Easy Reductions: Next, let us consider some closely related NP-complete problems: Clique (CLIQUE): The clique problem is: given an undirected graph G = (V;E) and an integer k, does G have a subset V0 of k vertices such that for each distinct u;v 2V0, fu;vg2E. 1(c): If a Ch. The problem of determining whether there exists a cycle in an undirected graph is in P. Each vertex v2V has a label l v 2f 1;0;1g. Corollary 1 : The VERTEX COVER problem is NP-complete. The maximum independent set problem is NP-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(·),. 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. Prove that the clique problem is not NP-complete for such graphs. 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. Hartnell [20], and there has been much work on this problem; see, e. A dominating set is minimal if S cannot be contracted further; that is, there exists no vertex w 2S such that S f wgis also a dominating set in G. I welcome you to read and look through some of the proofs for yourself! Je Linderoth IE418 Integer Programming. NP-hardness. feedback vertex set in an undirected graph is a subset of vertices whose deletion results in an acyclic graph. i) The graph is completed ii) The graph has a loop iii) The graph has an isolated vertex Answer for each of the representation separately. Following is a simple. Show that the language A is in NP 2. Output: does G have a Hamiltonian circuit? This is a famous NP-complete problem. •Given a graph with nodes representing people, and there is an edge between A in B if A and B are enemies, find the largest set of people such that no two are enemies. THE DENSE K SUBGRAPH PROBLEM Abstract. Recently, the Surjective ( ;H)-Homomorphism problem has been shown to be NP-complete when H is a 4-vertex cycle with a self-loop in every vertex [20]. Let the Minimum Dominating Set problem be the task of determining whether there is a dominating set of size k. 2 DO: (a) Prove the correctness of Borůvka's algorithm. Equivalently, a graph. 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. 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. Problem: In the CLIQUE problem, we are given an undirected graph G and an integer K and have to decide whether there is a subset S of at least K vertices such that every two distinct vertices u,v ∈ S have an edge between them (such a subset is called a clique of G). Given an undirected graph, the vertex cover problem is to find minimum size vertex cover. David Johnson also runs a column in the journal Journal of Algorithms (in the HCL; there is an on-line bibliography of all issues). (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?. Hartnell [20], and there has been much work on this problem; see, e. BFS essentially finds the shortest path between a vertex and all other vertices in a graph and therefore doesn't work for the longest path problem. If G has an independent set of size k, then the corresponding vertices in D are an acyclic subgraph. , every edge ∈E is incident to at least one vertex in C. Which is not NPC. Q ∈NP, and 2. (10 Points) Show that the following problem is NP-complete: • Input: An undirected graph G and an edge e. This problem has been intensively studied, in part because of its many applications, and in part because it is one of the few problems in NP that. 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. TSP seems a lot like Hamiltonian Cycle. 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. Each vertex v2V has a label l v 2f 1;0;1g. (a)In the Sub-graph problem, we are given two graphs G and H. 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. [12] A graph is outerplanar if and only if the dimension of the graph is at most three. Algorithm Note, Oct 14 Wenjian Yang. Decision problem: I Input: Graph G = (V;E) and integer k I Question: Does G have a clique of size k? Optimization problem: Find the size of the largest clique in the input graph G. INSTANCE : Given a graph G and an integer k. 5 The subset-sum problem We next consider an arithmetic NP-complete problem. Definition of NP-Complete • For some NP-complete problems, it is possible to develop algorithms that have average-case. 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. Problem: CLIQUE Instance: An undirected graph G and integer k. NP-, PSPACE-, and NEXP-complete variants, an NL-complete one. (c) T rue or false: 3-CNF-SA T p 2-CNF-SA T (assume that P 6 = NP). This new graph trivially has a clique of size k now. Show that this problem is NP-complete. Problem 3 (30 points) You are given an undirected graph G= (V;E). You must give the time complexity of each algorithm, assuming $ n $ vertices and $ m $ edges. Some Easy Reductions: Next, let us consider some closely related NP-complete problems: Clique (CLIQUE): The clique problem is: given an undirected graph G = (V;E) and an integer k, does G have a subset V0 of k vertices such that for each distinct u;v 2V0, fu;vg2E. And our goal is to find at most b vertices that cover all edges of our graph. In particular,. 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). Show that the language A is in NP 2. , to make a directed graph strongly connected or to make an undirected graph bridge-connected or. Choose 4 out of the 5 problems, and for each one, prove that it is NP-complete, or prove that it is in P. 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. 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. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. Theorem 4 (Vygen [9]). A linear time algorithm, MEDIAN PLACEMENT, due to [2] provides. In other words, either the node is matched (appears in an edge e. prove that this problem is NP-complete by reducing the 3SAT Given an undirected network G (N, L), where N is the An auxiliary graph for an instance. An Annotated List of Selected NP-complete Problems. 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. Assume problem P is NP Complete. Speci cally, choose some known NP-complete problem B, and reduce Bto A(note: not the other way around!), i. 638 CHAPTER 10. ) Take any NP-complete problem Y (pick wisely!) 2. Suppose we have a black box which takes as input a pair (G,k) where G is a graph and k is a positive. ; Suppose the edge is the most expensive edge contained in the cycle. In particular,. NP-complete ,and remains NP-complete for bipartite undirected graphs with maximum degree six. Then we sum the total cost of the edges and finally. 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. Assuming HAMPATH is NP-complete, prove that HAMCYCLE is NP-complete. A closed path, or cycle,isapathfromsomenodeu to itself. One example is the following problem: Hamiltonian Circuit Problem: Input: an undirected graph G. The k-Connectedsubgraph Problem Zeev Nutov The Open University of Israel E-mail: [email protected] It takes time proportional to V + E in the worst case. First, note that this problem is in NP, because if there is a valid arrangement we can easily guess and check it in polynomial time. Full credit will be given for correct answers. 1996 n) using polynomial space. Construct a Peterson graph of 10 vertices. Thus by the examples above, CSP(K3)b is polynomial time solvable for any b ≤ 3 and NP-complete for any b ≥ 4. 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. We give a polynomial reduction from 3SAT. Given an undirected graph G, does G have a spanning tree in which every node has degree at most 374?. Prove that the following problem, the Non-Bored Jogger Problem (NBJ), is NP-complete. Prove that Vertex. 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?. • Question: Does G have a Hamiltonian cycle that passes. 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!. The undirected feedback set problem asks: given G and k, does G contain a feedback set of size at most k? Prove that the undirected feedback set problem is NP-complete. Make graph complete by adding edges with weight 1. We prove it is NP-complete. The problem of determining whether there exists a cycle in an undirected graph is in NP. Limaye, Mahajan and Nimbhorkar [19] prove that longest paths in planar DAGs can be computed in UL\coUL. Give veriﬁcation algorithms to show that the following problems are in NP. Show that determining whether a graph has a tonian cycle is NP-complete. In other words, either the node is matched (appears in an edge e. that are NP-complete will lead toa further understanding of the charac-terization of NP-complete problems. 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. NP complete problems •problem A is NP-complete if-A is in NP (poly-time to verify proposed solution) -any problem in NP reduces to A •second condition says: if one solves pb A, it solves via polynomial reductions all other problems in NP •CIRCUIT SAT is NP-complete (see book)-and so the other problems discussed here, because they reduce to it. 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. Traveling-Salesman Problem (TS): Start a complete undirected graph G=(V,E), and for each pair of vertices i and j a cost function c(i,j)>=0. , a graph G and a bound b), we produce a list of b zeros and n-b ones where n is the number of vertices in the graph. In particular,. For all s 1, s-Club is NP-hard to approximate within we prove that on graphs of diameter at most three, 2-Club. Given an undirected graph G, does G contain a simple path that visits all but 374 vertices? 2. Definition: Independent Set: For an undirected graph , is an. 1: The traveling salesman problem is NP-complete. Given an undirected graph G wi 1) Consider the clique problem: given a graph G (V, E) and a positive integer k, determine whethe Let G -(V, E) be a graph. • This was the first problem proved to be NP-complete. † If a graph contains a triangle, any independent set can contain at most one node of the triangle. Reducing TSP to graph problem InstanceA complete weighted undirected graph G = (V,E) with non-negative edge weights. Complete Graphs. The NP-hard 2-Club problem is, given an undirected graph G= (V;E) and a positive integer ', to decide whether there is a vertex 2-Club is NP-complete even on split graphs and, thus, also on chordal graphs [3]. Solution: False. , 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. Also design a decrease-by-one algorithm for finding all the prime factors of a given number n. The computational problem is called the minimum spanning tree (MST) problem. • TSP NP • Then, we will reduce the undirected Hamiltonian cycle, which is a known NP-complete problem, to TSP: • HAM-CYCLE ≤ 𝑝 TSP 5. 3 Decision problems Decision problem. of computer and communication networks. Steiner Problem in Petersen Graph is NP-Complete. Our result implies NP=P. 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. HAMILTONIAN CYCLE Input: Undirected graph G = (V;E). 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. (ii) Very likely, the answer is NO. The associated yes/no question is: Does the given graph have a clique of the given size? To prove NP-completeness: If A P B and Ais NP-complete. (b)Is this problem in EXP? 2. Assume problem P is NP Complete. 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. 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. QUESTION: Does there exist a zero cost K-way partition ofG by deleting k nodes or less? Theorem 1. • Prove that the traveling salesman problem is NP complete. In other words, for any yes instance of X, there exists a certiﬁcate that. The following are the examples of complete graphs. 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. Various NP-complete graph problems. [Hint: Use part (a). Which of the following are true? [Check all that apply. It follows that the Steiner k-cut problem is NP-complete and APX-hard for all k 3. Call this problem HAMILTONIAN CYCLE-2. To the best of our knowledge, the capacitated p-hub center allocation problem (as well. 1 Introduction Two paths in a graph are internally disjoint if no internal node of one of the paths belongs to the other. (2) The problem of determining whether ther e exists a cy cle in an undir ected graph is in NP. To prove NP-hardness, you may reduce from any problem that has been shown, in class or in CLRS, to be NP-complete. In weighted complete graphs with non-negative edge weights, the weighted longest path problem is the same as the Travelling salesman path problem, because the longest path always includes all. 1(a): If G is a connected graph, Ch. Really that only shows the problem is NP-hard but this problem is obviously in NP so if it's NP-hard it's NP-complete $\endgroup$ - DRF Apr 1 '15 at 12:39. Hamiltonian Path is a path in a directed or undirected graph that visits each vertex exactly once. , 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. We settle this question by proving that the problem, similarly to the undirected version, is indeed NP-complete. • Prove that ﬁnding a directed or undirected Hamiltonian path or cycle in a directed or undirected graph is NP complete. A TS-tour is a circuit that visits each vertex exactly once. Given an undirected graph G, does G have a spanning tree in which every node has degree at most 374?. Give the language that belongs to this problem and show that it is NP-complete. Show that the language A is in NP 2. [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. 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. Prove that this problem is NP-complete. The DOMINATING-SET problem is as follows: given a graph Gand a number k, determine if Gcontains a dominating set of size kor less. The K-CRITICAL NODE PROBLEM is NP. (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. A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices. It is easy to see that ANOTHER HAMILTON CIRCUIT is FNP-complete. Making statements based on opinion; back them up with references or personal experience. Decision problem: I Input: Graph G = (V;E) and integer k I Question: Does G have a clique of size k? Optimization problem: Find the size of the largest clique in the input graph G. 7 (a) When k = 2, this problem becomes exactly the (s,t. The goal is to find a cut of maximum weight. =NP, then no NP-complete problem can b e solv ed in p olynomial time. We define a cut (S, V − S) of G as a partition of V, and the weight of a cut as the number of edges crossing the cut. We ask whether there exists a subset S0 S whose elements sum to t. Show that if every component of a graph is bipartite, then the graph is bipartite. Next we need to show that CLIQUE is NP-hard; that is we need to show that CLIQUE is at least as hard any other problem in NP. 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. Prove Set-Cover, Hitting-Set and Dominating-Set are polynomial-time reducible to each other. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. Problem HC is known to be NP-complete. Reduction Basics Assume A reduces to B in polynomial time. In other words, given an undirected graph, find the largest set of vertices such that no two are connected with an edge. Therefore, the longest path problem is NP-hard. Prove that the following problems are NP-complete. Some of these problems are traveling salesperson, optimal graph coloring, the knapsack problem, Hamiltonian cycles, integer programming, finding the longest simple path in a graph, and satisfying a Boolean formula. 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. The Hamiltonian cycle (HC) problem has many applications such as time scheduling, the choice of travel routes and network topology (Bollobas et al. Show that, if P=NP, then every language A ЄP, except A=ø and A=∑*,is NP-complete. BFS essentially finds the shortest path between a vertex and all other vertices in a graph and therefore doesn't work for the longest path problem. 2 CIRCUIT-SAT: The First NP-Complete Problem CIRCUIT-SAT is a decision problem that asks the following: Given a Boolean circuit with n. NP-complete, so is IS. (c) Determine the chromatic number of the following graph, and prove your answer is indeed optimal. Except for some problems. Each of the following languages is either in P, or it is NP-complete. 10% Bonus Marks { The subset sum problem is known to be NP-complete. prove that the Max-Cut problem is NP-Complete. ・Problem X is a set of strings. Each edge e2Ehas a weight w e >0. In the subset-sum problem, we are given a ﬁnite set S of positive integers and an integer target t>0. Recap: Problem: To prove clique is a NP-complete problem Solution: We show the following: Proof that clique is in NP. In the MAX CUT problem, we are given an undirected graph G and an integer K and have to decide whether there is a subset of vertices S such that there are at least K edges that have one endpoint in S and one endpoint in S. A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices. 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. Question: Is ˝(G) k? Does it follow that this problem NP-complete? Why or why not? 3. There is a specific node, v and a positive integer distance l. Argue that if an instance x was in B, then f(x) ∈ A. • Prove that the traveling salesman problem is NP complete. Some of these problems are traveling salesperson, optimal graph coloring, the knapsack problem, Hamiltonian cycles, integer programming, finding the longest simple path in a graph, and satisfying a Boolean formula. 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. Each edge e2Ehas a weight w e >0. ・Certifier doesn't determine whether s ∈ X on its own; rather, it checks a proposed proof t that s ∈ X. A Sample Proof of NP-Completeness The following is the proof that the problem VERTEX COVER is NP-complete. – Paul Apr 9 '11 at 20:51. The Hamiltonian Cycle Problem is NP-Complete Karthik Gopalan CMSC 452 November 25, 2014 Karthik Gopalan (2014) The Hamiltonian Cycle Problem is NP-Complete November 25, 2014 1 / 31. Denition 4 (Complete-Communication topological graph). Denote K(G). 1 DO: Given an undirected graph, count its connected components in linear time. ) • Next show that a known NP-Complete language B can be reduced to C in polynomial. Prove that Hamiltonian cycle problem in NP-C using Traveling Salesman problem. Prove that in this case P=NP. Assumes that way given an undirected graph and it contains a Eulerian cycle if and only if, it is connected and the degrees of all its vertices is even. In general, you wouldn't expect there to be a "natural" direct reduction between two arbitrary different NP-complete problems (i. Many problems are hard to solve, but they have the property that it easy to authenticate the solution if one is provided. for the clique problem which is well-known to be NP-complete, unlike the graph isomorphism one. An undirected graph G and an integer K Question: Is there a vertex cover of size K. 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. 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. 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. 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. That is, a vertex cover of Gis a subset. Consider the following algorithm to check connectivity of a graph defined by its adjacency matrix. What it means for a problem to be in P, NP, or NP Complete; You will definitely be given a problem where you prove, using reduction from a known NP-Complete problem, that some other problem is NP-Complete. Select problem Y that is know to be in NP-Complete. 1: The traveling salesman problem is NP-complete. Prove that the following problems are NP-complete. In most cases, the problem is NP-complete. We need the following lemma. In the MAX-CUT problem, we are given an unweighted undirected graph G = (V, E). COMPUTER SCIENCE TRIPOS Part IB { 2014 { Paper 6 Complexity Theory (AD) (a)State precisely what it means for a language (i) to be co-NP-complete, (ii) to be in NL and (iii) to be in PSPACE. Spanning trees with restricted degrees Show that the following problem is NP-complete: Given an undirected graph G = (V;E) and an integer k, determine if G contains a spanning tree T such that each vertex of the tree has maximum degree k. Given a positive weight function defined on , the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut such that is maximum. • Prove that the problem of ﬁnding an interior lattice point in an integer polytope is NP complete. Some Easy Reductions: Next, let us consider some closely related NP-complete problems: Clique (CLIQUE): The clique problem is: given an undirected graph G = (V;E) and an integer k, does G have a subset V0 of k vertices such that for each distinct u;v 2V0, fu;vg2E. In the maximum clique problem, we are given a graph G= (V;E) and an integer k. In the Maximum Clique Problem, given an undirected graph. The goal is to decide if H is a subgraph of G or not. The Undirected Feedback Set Problem asks: Given G and k, does G contain a feedback set of size at most k? Prove that Undirected Feedback Set is NP-complete. In other words, does G have a k vertex subset whose induced subgraph is 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. Present correct and efficient algorithms to convert an undirected graph $ G $ between the following graph data structures. A linear time algorithm, MEDIAN PLACEMENT, due to [2] provides. SOME NP-COMPLETE PROBLEMS An undirected graph G is connected if for every pair (u,v) ∈ V × V,thereisapathfromu to v. 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. (b) T F [4 points] Let G = (V,E) be a ﬂow network, i. The problem of determining whether there exists a cycle in an undirected graph is in NP. Another related problem is to compute longest paths. Let the Steiner problem in graph is NP, itsufficient is to show that R- Restricted Steiner problem in Petersen graph is infact NP-complete. The Hamiltonian cycle problem is a special. The Fireﬁghter problem is NP-. We want to argue that there is a polytime veriﬁer for X. All NP problems are reducible to this problem. a given weighted directed graph into a discrete circle which minimizes the total weighted arc length.
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