In other words, if we could e ciently solve any NP-hard problem, we could e ciently solve any problem in NP. 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. A closed path, or cycle,isapathfromsomenodeu to itself. Assume problem P is NP Complete. This handout reviews the key steps in constructing a proof of NP-completeness for a problem. Garey , Johnson, and Stoclcmeyer [2] have proved that the problem remains NP-complete also in the case of undirected graphs with degrees at most 3. If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve A. (a) A double-Eulerian circuit in an undirected graph G is a closed walk that traverses every edge in G exactly twice. • Prove that the traveling salesman problem is NP complete. The K-CRITICAL NODE PROBLEM is NP. Extending previous NP-completeness results of the harmonious coloring problem on subclasses of chordal and co-chordal graphs, we prove that the problem remains NP-complete for split undirected path graphs; we also prove that the problem is NP-complete for colinear graphs by showing that split undirected path graphs form a subclass of colinear. 2 Given a mixed graph Hand P V V one can decide in nO(jPj) time whether His P-orientable; namely, P-Orientation with a mixed graph Hcan be decided in nO(jPj) time. 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. Prove these problems are NP-Complete: (a) SET-COVER: Given a ﬁnite set , a collection of subsets of, and an integer , determine whether there is a sub-collection of with cardinality that covers. The 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. the ﬁrst n Fibonacci numbers? Prove your answer. 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. If both are satisfied then it is an NP complete problem. The complete graph with n vertices is denoted by K n. 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. Hamiltonian cycle problem:-Consider the Hamiltonian cycle problem. We prove the following: • Let L = {3,4,5,} \ L be the set of forbidden cycle lengths. Prove that if uis a vertex of odd degree in a graph, then there exists a path from uto another. Easy to check that a given set is independent and of size k (thus the problem is in NP X) 2. Problem HC is known to be NP-complete. For each of the following problems either give a polynomial-time algorithm or prove that the problem is NP-complete: 1-SAT: Given: A set of variables X= fx 1:::x ngand a set C= fc 1:::c mgof clauses where each clause is a single literal. INSTANCE : Given a graph G and an integer k. I have developed the following: Given an undirected graph, G = (V,E), we can construct a directed graph, D =(V, E'). Prove that the following problem is NP-complete. Reducibility is a transitive relation. The Fireﬁghter problem is NP-. In Exercises 1–3 find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. ] Suppose the edge is not the cheapest edge that crosses the cut. is NP-complete) is De nition 0. The decision question is: Given an integer k, is there a TS-tour with total cost less than k?. We prove it is NP-complete. Complete Graphs. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. Unknown; this is generally believed to be true, but it might not be. A Sample Proof of NP-Completeness The following is the proof that the problem VERTEX COVER is NP-complete. 1 - Prove Lemma 10. And our goal is to find at most b vertices that cover all edges of our graph. Prove that the Sub-graph problem is NP-complete by a reduction from Clique. , 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. We consider here the family of Toeplitz graphs. 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. 346 19 Complexity Theory and NP-Completeness 19. We now prove that the recognition version of the CNP is NP-complete. 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. Problem 1: Given an undirected graph G = (V;E) with jVjeven,. Prove Hamiltonian cycle is NP-C from Hamiltonian Path (look it up). (b) T F [4 points] Let G = (V,E) be a ﬂow network, i. Problem: CLIQUE Instance: An undirected graph G and integer k. 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. This problem is NP-hard, as shown by René Peeters in 2003. 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. † We consider graphs whose nodes can be partitioned into m disjoint triangles. 5 The subset-sum problem We next consider an arithmetic NP-complete problem. 22 Let HALF-CLIQUE G is an undirected graph having a complete sub- graph with at least m/ '2 nodes, where m is the number of nodes in G}. Thanks for contributing an answer to Theoretical Computer Science Stack Exchange! Please be sure to answer the question. Draw a Venn diagram to depict the commonly believed relationship between P, NP, NP-complete and NP-hard problems. b) Infer the social relationships between. Introduction and preliminaries. A problem Z is NP-complete if Z is in NP and if every NP-problem reduces to Z. Recap: Problem: To prove clique is a NP-complete problem Solution: We show the following: Proof that clique is in NP. As it should not be too hard to show that the latter reduces to the former, and the former reduces to the latter. 1(a): If G is a connected graph, Ch. Given an undirected graph, the vertex cover problem is to find minimum size vertex cover. The INDSET problem is NP-complete. Reducing TSP to graph problem InstanceA complete weighted undirected graph G = (V,E) with non-negative edge weights. If Q just satisﬁes (2) then it's called NP-hard. n-1] of an undirected graph G. 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. Be careful about the degree of detail required for the timing analysis. 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. (a) Prove that if Lis regular, then L y is regular. The baseball card collector problem is as follows: Given packets P1, P2,. ALGORİTHM Connected (A[0. The goal is to better understand the theory and to train to recognize to construct reductions. Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). 3 Decision problems Decision problem. Notes: For this class is is generally not necessary to validate input in your pseudocode: if a problem says the input will be in a certain form then write your pseudocode under the assumption that the input is in that form. 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. 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. A spanning path in a graph G is called a Hamiltonian path. If we want to check a tour for credibility, we check that the tour contains each vertex once. 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. Problem HC is known to be NP-complete. Denition 4 (Complete-Communication topological graph). Max-Cut is the following problem: Given an undirected graph G = (V, E) with nonnegative edge capacities wu,v for (u, v) ∈ E and a number c, decide if there exists a cut in G with capacity at least c. Vertex sets and are usually called the parts of the graph. In Section 2 we will begin with the de nition of the basic form of the problem and prove some basic results. Given an undirected graph G with n vertices. If G has an independent set of size k, then the corresponding vertices in D are an acyclic subgraph. More formally, Z is NP-complete if Z ∈ NP and if for every X ∈ NP, X≤ P Z. Formally, we are given a connected, undirected graph G = (V, E ) Each edge (u, v) has numeric weight of cost. We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. goes via SAT or some other unrelated NP-complete problem). Select problem Y that is know to be in NP-Complete. Name: ID: 3. Draw a Venn diagram to depict the commonly believed relationship between P, NP, NP-complete and NP-hard problems. • This was the first problem proved to be NP-complete. komargodski,moni. Call this problem HAMILTONIAN CYCLE-2. If G has an independent set of size k, then the corresponding vertices in D are an acyclic subgraph. In Section 3 we. (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. java uses depth-first search to determine whether a graph has a cycle, and if so return one. We will show that the Clique problem is NP-complete. I have developed the following: Given an undirected graph, G = (V,E), we can construct a directed graph, D =(V, E'). If not, give a counterexample. 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. 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. Prove that Maximum Independent Set problem is NP-Complete by reducing it to Max-Clique 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. The first one was Almost-SAT problem. Max-Cut is the following problem: Given an undirected graph G = (V, E) with nonnegative edge capacities wu,v for (u, v) ∈ E and a number c, decide if there exists a cut in G with capacity at least c. Explain how the following can be ascertained by the representation. 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. So the coresponding language. Min Cut vs. Our result implies NP=P. Deﬁnition 10. A graph is k-connected if it contains k pairwise internally disjoint paths from every node to any other node. 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. Given k, the problem is to find whether there is a feedback set of size at most k. Secret-Sharing for NP Ilan Komargodski?, Moni Naor , and Eylon Yogev Weizmann Institute of Science filan. The vertex-disjoint paths problem is similarly defined. A graph is k-connected if it contains k pairwise internally disjoint paths from every node to any other node. A Calculating Method About How May `comparable' Pairs of Elements Exist in a Certain Set. , a weighted directed graph. For each of the problems below, prove that it is NP-complete by showing that it is a generalization of an NP-complete problem. The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. Deﬁnition 10. Show that if every component of a graph is bipartite, then the graph is bipartite. This famous problem can be described as follows: Given an undirected graph G=(V, E), does G have a Hamilton Circuit, i. 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. This shows that this problem is NP-complete. 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. 3 In this problem, all graphs are undirected. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. Min Cut vs. [10 pts] Suppose someone gives you a black-box B that takes in any undirected graph G = (V;E). , 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. Prove that this problem is NP-complete. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. However, in this problem we work with an array of arbitrary numbers (in particular, unsorted values). The INDSET problem is NP-complete. Final Exam Consider a connected undirected graph with distinct edge costs. 1 illustrates three edge-disjoint paths P 1, P 2 and P 3 in a series-parallel graph. If both are satisfied then it is an NP complete problem. 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. Show that the Minimum Dominating Set problem is NP-Complete. We now prove that the recognition version of the CNP is NP-complete. Theorem 10. [35] The YourFace social networking site has a feature by which two users can \friend" each. NP-Completeness And Reduction. How could I prove the following problem is NP-Complete by reducuing from subset-sum problem? Instance: An undirected graph, G = (V, E), and an integer k. Testing whether two given graphs are identical. • Question: Does G have a Hamiltonian cycle that passes. That is, a vertex cover of Gis a subset. In other words, does G have a k vertex subset whose induced subgraph is complete. We prove the following: • Let L = {3,4,5,} \ L be the set of forbidden cycle lengths. Which of the following are true? [Check all that apply. Give a reduction from 4-COLORABILITY to 7-COLORABILITY. The classic minimum dominating set problem is a special case with k = 1. NP AND COMPUTATIONAL INTRACTABILITY 3. Show that HALF-CLIQUE is NP-complete. If P NP, then UHAMPATH is not in P. Construct a graph G as follows. The clique problem is as follows. , A2NP, (2) any NP-Complete problem Bcan be reduced to A, (3) the reduction of Bto Aworks in polynomial time, (4) the original problem Ahas a solution if and only if Bhas a solution. (a) For the following directed, edge-weighted graph and given source vertex s, write down. Reading: 26. Show that the following problem is NP-complete: Problem: Clique, no-clique Input: An undirected graph $ G=(V,E) $ and. The DOMINATING SET (DS) decision problem is the following. show that TRIANGLE Є powered TRIANGLE={IG contains a triangle}. Here are some facts: NP consists of thousands of useful problems that need to be solved every day. If it answers NO, then there cannot be an independent set of size k. The Clique problem is NP-Complete The algorithm above does not work. Solution: Easy to check that the problem is in NP: a. Prove that in this case P=NP. undirected graph is in P. 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. (10 Points) Prove that the following problem belongs to PSPACE: Given a graph G and an integer k, we want to know whether the number of independent sets in G is equal to k. 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. I welcome you to read and look through some of the proofs for yourself! Je Linderoth IE418 Integer Programming. Thanks for contributing an answer to Theoretical Computer Science Stack Exchange! Please be sure to answer the question. Assuming HAMPATH is NP-complete, prove that HAMCYCLE is NP-complete. n-1] of an undirected graph G. Deciding if a given planar graph is 3-colorable is NP-complete. • Prove that the traveling salesman problem is NP complete. Bounded Degree Spanning Tree Instance : An undirected graph G = (V, E) and a positive integer k ≤≤≤. Except for some problems. Notes: For this class is is generally not necessary to validate input in your pseudocode: if a problem says the input will be in a certain form then write your pseudocode under the assumption that the input is in that form. 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?. If G has an independent set of size k, then the corresponding vertices in D are an acyclic subgraph. 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. NP complete. We widely generalize the result of [1] as follows. A computer graph is a graph in which every two distinct vertices are joined by exactly one edge. 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. Reducibility is a transitive relation. – Paul Apr 9 '11 at 20:51. Problems CS 154 There are thousands of NP-complete problems Your favorite topic certainly has an NP-complete problem in it Even the other sciences are not safe: biology, chemistry, physics have NP-complete problems too! Theorem (Cook-Levin): G is an undirected graph with a k-clique }. 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. for the clique problem which is well-known to be NP-complete, unlike the graph isomorphism one. But assuming P 6= NP, then a problem Ain P (and therefore also in NP) cannot be NP complete. ・Algorithm A solves problem X: A(s) = yes iff s ∈ X. Prove that the bin packing problem is NP-hard. Output: Does $ G $ contain a subgraph with exactly $ k $ vertices and at least $ y $ edges? 9-11. On the Web the following sites may be of. Solution to Spanning trees with restricted degrees. If it answers NO, then there cannot be an independent set of size k. (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,. We settle this question by proving that the problem, similarly to the undirected version, is indeed NP-complete. Algorithms, Deluxe Edition, Fourth Edition These Algorithms Video Lectures cover the essential information that every serious programmer needs to know about algorithms and data structures, with emphasis on applications … - Selection from Algorithms: 24-part Lecture Series [Video]. Now run A on this augmented graph. , 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. 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. We rst show that the so-called Hamiltonian Path problem is NP-complete: Given an undirected graph G = (V;E), determine whether G contains a Hamiltonian path, i. There are literally thousands ∗[email protected] The ( k,n )-CLIQUE problem is NP-complete. Final Exam Consider a connected undirected graph with distinct edge costs. The problem of determining whether there exists a cycle in an undirected graph is in NP. Construct a Peterson graph of 10 vertices. NP Certification algorithm intuition. 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. I have developed the following: Given an undirected graph, G = (V,E), we can construct a directed graph, D =(V, E'). Max-Cut is the following problem: Given an undirected graph G = (V, E) with nonnegative edge capacities wu,v for (u, v) ∈ E and a number c, decide if there exists a cut in G with capacity at least c. 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. The function f : X -> Y must run in polynomial time. $\endgroup$ - Lasse Rempe-Gillen Sep 27 '15 at 11:21. • Prove that the problem of ﬁnding an interior lattice point in an integer polytope is NP complete. Keywords: NP-complete problem, Traveling Salesman Problem, Mutation operator. The first part of an NP-completeness proof is showing the problem is in NP. a graph (directed or undirected). Suppose that. The proof for the directed case is similar. The goal is to decide if there is a clique on kvertices, that is, a complete subgraph on kvertices. Given a problem X, prove it is in NP-Complete. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in. NP-complete problem. 2 Problem Set 8 Problem 8-2. Given an undirected graph G, does G contain a simple path that visits all but 374 vertices? 2. Given an undirected graph, we first apply the given algorithm and determine how many colors are used in its coloring. undirected graph is in P. That is, a vertex cover of Gis a subset. CHAPTER 36: NP-COMPLETENESS. Let G = (V,E)be an undirected graph with nvertices and k a pos-itive integer, as before. Given an undirected graph, the vertex cover problem is to find minimum size vertex cover. The following problem is considered: Given an undirected, connected graph G, find a spanning tree in G such that the sum of the lengths of the fundamental cycles (with respect to this tree) is minimum. Reducing TSP to graph problem InstanceA complete weighted undirected graph G = (V,E) with non-negative edge weights. , UHAMPATH= is an undirected graph with a Hamiltonian path from to ). If we know a single problem in NP-Complete that helps when we are asked to prove some other problem is NP-Complete. adjacency lists are used for representing the graph. Prove that a complete graph with nvertices contains n(n 1)=2 edges. In particular,. Various polynomial time reductions are also been studied between these problems and and methods have been worked on. 1(a): If G is a connected graph, Ch. Introduction to Complexity Theory: CLIQUE is NP-complete In this lecture, we prove that the CLIQUE problem is NP-complete. Prove these problems are NP-Complete: (a) SET-COVER: Given a ﬁnite set , a collection of subsets of, and an integer , determine whether there is a sub-collection of with cardinality that covers. A closed path, or cycle,isapathfromsomenodeu to itself. It takes time proportional to V + E in the worst case. 1987; Akhmedov and Winter 2014). As a counter-example imagine a graph that consists of one cycle including all the ver. The size of the cut is the number of edges with one end in S and the other end in S. Construction problem: Find the largest clique in the input graph G. ; For the rest, the fastest known algorithms run in exponential time. Make graph complete by adding edges with weight 1. There are approximate polynomial time algorithms to solve the problem though. Algorithm Note, Oct 14 Wenjian Yang. This is a second practice problem on NP-Completeness proofs. NP complete. a: Create a mapping that runs in polynomial time 2. You may assume that G is connected. Sam Buss TFNP. [S] , the problem can be found as NP-complete, which means that no polynomial time (in the size of input) algorithm is known and probably none can exist. Any directed graph with in-/out-degrees ≤ 1 which has a ver-tex of total degree 1 has another vertex of total degree 1. Page 4 19 NP-Hard and NP-Complete If P is polynomial-time reducible to Q, we denote this P ≤ p Q Definition of NP-Hard and NP-Complete: » If all problems R ∈ NP are reducible to P, then P is NP- Hard »We say P i s NP-Complete if P is NP-Hard and P ∈ NP If P ≤ p Q and P is NP-Complete, Q is also NP-Complete 20 Proving NP-Completeness What steps do we have to take to prove a problem. The restricted edge-connectivity of G, denoted by λ0(G), is deﬁned as the minimum number of edges whose dele-tion results in a disconnected graph and contains no isolated vertices. Graph Isomorphism Problem is NP-complete, it would have huge implication on complexity theory. , that the subgraph be a clique (fully-connected). However, a following greedy algorithm is known for finding the chromatic number of any given graph. 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. 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. Given an undirected graph G with n vertices. goes via SAT or some other unrelated NP-complete problem). 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. How toprovea problem is NPC I The reducibility relation \ T" is transitive, i. 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. Prove that the set packing problem is 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. The size of the cut is the number of edges with one end in S and the other end in S. This problem can be reformulated as a bipartite graph problem. ) The size of this clique is |S|. Show that, if P=NP, then every language A ЄP, except A=ø and A=∑*,is NP-complete. feedback vertex set in an undirected graph is a subset of vertices whose deletion results in an acyclic graph. In the Clique problem, you are given an undirected graph. 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). This problem, besides being interesting in its own right, is useful in a variety of situations It is shown. 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. • Prove that the traveling salesman problem is NP complete. Find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. So, it deals with undirected graph. Recall that a clique in a graph is a subset S of the vertex set such that the members of S are pairwise adjacent. If you include justiﬁcation for your answers, you may obtain partial credit for incorrect answers. For example, the NP-completeness proof due to Papadimitrou [21] for the problem of ﬁnding. Prove that the following problem, the Non-Bored Jogger Problem (NBJ), is NP-complete. We now give two toy examples that in particular will shut some light on how to prove the just mentioned serum. is a hamiltonian cycle in graph G. The Fireﬁghter problem is NP-. The Digraph Realization Problem asks whether a given dihypergraph H coincides with N(D) for some digraph D. (a)In the Sub-graph problem, we are given two graphs G and H. 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. To prove NP-hardness, you may reduce from any problem that has been shown, in class or in CLRS, to be NP-complete. Now given a different problem P' If we show P. •Instance: an undirected graph. Prove that Dense Subgraph is NP-complete. Except for some problems. 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. There are many problems for which no polynomial-time algorithms ins known. 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. Intuitively, a problem isin P1 if thereisan efﬁcient (practical) algorithm toﬁnd a solutiontoit. Corollary 1 : The VERTEX COVER problem is NP-complete. 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. 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. 1 Problem 8-1. 3-colorability is a known NP-complete. 7 (a) When k = 2, this problem becomes exactly the (s,t. NP-Hard Given that we have at least one NP-Complete problem, showing others are NP- hard becomes easier: For a problem under consideration X 1. [email protected] Deﬁne a polynomial time reduction from Y to X. In all cases, the graph is given implicitly by a Turing machine which computes the neighbors of a given vertex. Theorem 4 (Vygen [9]). We do this by addd the edges (u,v) and (v,u) for every edge in E. Corollary 1 : The VERTEX COVER problem is NP-complete. Problem 1: Given an undirected graph G = (V;E) with jVjeven,. In the following questions, you are asked to reduce various problems to other problems. minimal vertex covering, in a graph is NP complete. Give a good algorithm to assign spies to countries. Except for some problems Input: An (undirected) graph G= (V;E), and vertices s;t2V. ) • Next show that a known NP-Complete language B can be reduced to C in polynomial. Steps 2 through 5 seek to accomplish the latter. Polynomial Time Verification. Finding a cycle that goes through each vertex exactly once. • Question: Does G have a Hamiltonian cycle that passes. We ask whether there exists a subset S0 S whose elements sum to t. 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-. By definition, GV is in NP. This problem is NP-hard, as shown by René Peeters in 2003. (b) Show that the travelling salesman problem (TSP) defined as TSP { (G, w, k) : undirected graph G (V, E) with weight w : E —+ N contains a cycle visiting every vertex exactly once with total weight < k}, is NP-complete. Prove by reduction that TSP is NP-complete, assuming that HAMCIRCUIT is NP-complete. Prove that Dense Subgraph is NP-complete. A problem Q is NP-complete if: 1. minimal vertex covering, in a graph is NP complete. The 3-Coloring Problem The 3-coloring problem is Given an undirected graph G, is there a legal 3-coloring of its nodes? As a formal language: 3COLOR = { G | G is an undirected graph with a legal 3-coloring. 10% Bonus Marks { The subset sum problem is known to be NP-complete. Show that NP is closed under union and concatenation. Recall that a bipartite graph is one which has two sets of vertices, and edges only exist between vertices in each set. NP-complete ,and remains NP-complete for bipartite undirected graphs with maximum degree six. THE DENSE K SUBGRAPH PROBLEM Abstract. Math Basics. The Fireﬁghter problem was ﬁrst introduced by B. A computer graph is a graph in which every two distinct vertices are joined by exactly one edge. 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. 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. The first problem we'll consider is the independent set problem. This problem was introduced by Aigner and Triesch [2] as a natural generalization of the Open Neighborhood Realization Problem for undirected graphs, which is known to be NP-complete. edu Problems solvable in p-time are considered tractable NP-complete problems have no known p-time solution, considered intractable. We ask whether there exists a subset S0 S whose elements sum to t. Our result implies NP=P. It will be shown that there is a polynomial time algorithm to solve the problem, and that it is NP-complete to nd a kind of minimum solution. 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?. Show that the following problem is NP-complete: Problem: Clique, no-clique Input: An undirected graph $ G=(V,E) $ and. All of the algorithms we have studied thus far have been polynomial-time algorithms: on inputs of size n, their worst-case running time is O(n k) for some constant k. , 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. , [Balister. Greedy Algorithm- Step-01: Color first vertex with the first color. doesn't speak the language. Show that Graph-Value is NP-complete by showing that Vertex-Cover reduces to Graph-Value. But it is NL-complete for directed acyclic graphs (DAG). Show that determining whether a graph has a tonian cycle is NP-complete. However, in this problem we work with an array of arbitrary numbers (in particular, unsorted values). (Less formally, think of NP-complete problems as the NP problems that are as 'hard' as possible. 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’. 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. It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where ). 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. Algorithm Note, Oct 14 Wenjian Yang. [Hint: reduce from the Independent Set problem] 2. 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. The problem of determining whether there exists a cycle in an undirected graph is in NP. ) Take any NP-complete problem Y (pick wisely!) 2. Show that the following problems are NP-complete April 7, 2018 Below is a list of 30 exercises in which you are asked to prove that some problem is NP-complete. The k-Connectedsubgraph Problem Zeev Nutov The Open University of Israel E-mail: [email protected] ValueThe value of the solution is the sum of the weights associated with edges in the cycle, and the goal is to ﬁnd a cycle whose total weight is minimum. Show that, if P=NP, then every language A ЄP, except A=ø and A=∑*,is NP-complete. S is an independent set of G if and only if S is a clique in G. We rst show that the so-called Hamiltonian Path problem is NP-complete: Given an undirected graph G = (V;E), determine whether G contains a Hamiltonian path, i. For each problem below, either describe a polynomial-time algorithm or prove that the prob-lem is NP-complete. (2) The problem of determining whether there exists a cycle in an undirected graph is in NP. Proving a Problem is NP-Complete To prove a problem X is NP-complete, you need to show that it is both in NP and that it is NP-Hard. • Prove that ﬁnding a directed or undirected Hamiltonian path or cycle in a directed or undirected graph is NP complete. It is easy to see that ANOTHER HAMILTON CIRCUIT is FNP-complete. For example, the NP-completeness proof due to Papadimitrou [21] for the problem of ﬁnding. Prove that a connected undirected graph with n vertices and n-1 edges cannot have any cycles. If P=NP, then UHAMPATH 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. Discussion: This problem is also open. If P NP, then UHAMPATH is not in P. to s=cis NP-hard for any. 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. Prove that MAX-CUT is NP-complete. Finding a subset of vertices that has the minimum conductance in a given graph has been often stated to be an NP-complete problem in the literature [2, 3, 6, 8, 14, 16, 17]. We show that HamiltonianCycle / HamiltonianPath. ? Instance: An undirected graph, G = (V, E), and an integer k. On the other hand, a problem is in NP 2, if it is ﬁrst efﬁcient to guess a solution and then efﬁcient to check that this solution is correct. The complete graph with n vertices is denoted by K n. 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. Consider the following decision and optimization versions of the longest path problem: • Longest-Path: Given an undirected G with integer edge weights w(e) 1 and an integer L, does there exist a simple path (no repeated nodes) whose length is L? • Find-Longest-Path: Given an undirected graph G with integer edge weights w(e) 1,. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. The NP-hardness of the unweighted longest path problem can be shown using a reduction from the Hamiltonian path problem: a graph G has a Hamiltonian path if and only if its longest path has length n − 1, where n is the number of vertices in G. The ( k,n )-CLIQUE problem is NP-complete. [12] A graph is outerplanar if and only if the dimension of the graph is at most three. 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. Figure 1: An instance of Vertex Cover problem. Given two networks G 1 = (V 1, E 1) and G 2 = (V 2, E 2), the alignment graph A is a complete bipartite edge-weighted graph with vertex set V 1 ∪ V 2. [6 marks] (b)Consider the following two decision problems. 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. Output: does G have a Hamiltonian circuit? This is a famous NP-complete problem. Shortest path query is an important problem over graphs and has been well studied. 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. (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,. 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. Prove that this problem is NP-complete. The following two corollaries are immediate from the above theroem. Traveling Salesman is NP-complete. , 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. Exact algorithms. Let's first recall the definition of an independent set. In the Clique problem, you are given an undirected graph. In particular, Biedl et al. 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. Give a good algorithm to assign spies to countries. 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. NP-complete problem. The DOMINATING-SET problem is as follows: given a graph Gand a number k, determine if Gcontains a dominating set of size kor less. [10 pts] Suppose someone gives you a black-box B that takes in any undirected graph G = (V;E). In Exercises 1–3 find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. 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). Various polynomial time reductions are also been studied between these problems and and methods have been worked on. In other words, does G have a k vertex subset whose induced subgraph is complete. We now convert I to an instance of the balanced vertex-ordering problem. Show that HALF-CLIQUE is NP-complete. 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 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. Let G = (V,E) by an undirected graph and suppose S ⊆ V. The following two corollaries are immediate from the above theroem. Definition: Independent Set: For an undirected graph , is an. c jEj, we have for our algorithm: E[number of satis ed edges] = 2 3 jEj 2 3 c. Prove that Dense Subgraph is NP-complete. (green text added 7 August 2003) This algorithm can be used to decide the NP-complete language 3-COLOR. But assuming P 6= NP, then a problem Ain P (and therefore also in NP) cannot be NP complete. Show that NP is closed under union and concatenation. An algorithm is given for solving the Hamiltonian cycle problem for any undirected graph G = (V; E) with a worst-case running time bounded by O(jV j 4); provided jEj V 2: This development of a polynomial time algorithm that decides the Hamiltonian cycle problem, which is known to belong to the NP complete class of languages, will prove that P = NP. 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 problem of determining whether there exists a cycle in an undirected graph is in P. There is a specific node, v and a positive integer distance l. The following should be obvious. We want to argue that there is a polytime veriﬁer for X. (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. Show that this problem is NP-complete. The function f : X -> Y must run in polynomial time. For series-parallel graphs it is complete for L [16]. Exercise 34. It takes time proportional to V + E in the worst case. Corollary 1 : The VERTEX COVER problem is NP-complete. 1: The traveling salesman problem is NP-complete. Show that for any problem in NP, there is an algorithm which solves in time O(2p(n) ), where n is the. How to prove that the problem is NP. In this paper, we describe applications of the acyclic 2-coloring problem. Give a reduction from 4-COLORABILITY to 7-COLORABILITY. , to make a directed graph strongly connected or to make an undirected graph bridge-connected or. R-Restricted Steiner problem in Petersen graph is NP-complete. Denote K(G). The question "does there exist a simple path in a given graph with at least k edges" is NP-complete. 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. For example, the NP-completeness proof due to Papadimitrou [21] for the problem of ﬁnding. The size of the cut is the number of edges with one end in S and the other end in S. Prove that if uis a vertex of odd degree in a graph, then there exists a path from uto another. 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 other words, either the node is matched (appears in an edge e. In the Maximum Clique Problem, given an undirected graph. 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). 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. It is natural to wonder whether all problems can be solved in polynomial time. The dominating set problem that is NP-Complete is minimum-size-dominating-set, not just if a graph has a dominating set or not. From the PCP theorem, we know that for any NP-complete L, there exists an (O(logn);O(1))-restricted veri er V. 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?. Another related problem is to compute longest paths. 1 Proving NP-completeness In general, proving NP-completeness of a language L by reduction consists of the following steps. The graph isomorphism problem (GI) consists in determining whether two given graphs are isomorphic—in other words, whether there is a bijec-tion between the nodes of the graphs preserving the edges. 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. Both problems are NP-complete. For each of the problems below, prove that it is NP-complete by showing that it is a generalization of an NP-complete problem. Output: does G have a Hamiltonian circuit? This is a famous NP-complete problem. 3 Decision problems Decision problem. Specifically, NP Complete problems can only possibly. Note: usp[s] = True. Given a graph with vertices of degree at most 3, and a parameter , the reduction leaves the graph and the parameter unchanged: clearly the output of the reduction is a possible input for the clique problem. a: Create a mapping that runs in polynomial time 2. , there is no polynomial time solution for this unless P = NP. In Exercises 1–3 find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. 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. Recap: Problem: To prove clique is a NP-complete problem Solution: We show the following: Proof that clique is in NP. 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. Max-Cut is the following problem: Given an undirected graph G = (V, E) with nonnegative edge capacities wu,v for (u, v) ∈ E and a number c, decide if there exists a cut in G with capacity at least c. Limaye, Mahajan and Nimbhorkar [19] prove that longest paths in planar DAGs can be computed in UL\coUL. NP-complete problems AmirHossein Ghamarian December 2, 2008 1. (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?. Identify all isolated and pendant vertices. [35] The YourFace social networking site has a feature by which two users can \friend" each. Authors: Yasushi Ieno Comments: 15 Pages. a directed graph which has exactly one edge between each pair of vertices. Be careful in cases where you may need to prove both. 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. 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. Our result implies NP=P. Both problems are NP-complete. • Sometimes, we can only show a problem NP-hard = "if the problem is in P, then P = NP," but the problem may not be in NP. Give a good algorithm to assign spies to countries. In his seminal paper, Grover points out the prospect of faster solutions for an NP-complete problem like SAT. For each of the following problems either give a polynomial-time algorithm or prove that the problem is NP-complete: 1-SAT: Given: A set of variables X= fx 1:::x ngand a set C= fc 1:::c mgof clauses where each clause is a single literal. CS502 – Final term (Fall 2012) Q:A sequence of a value in a column of the dynamic programming table for an instance of the. Give the language that belongs to this problem and show that it is NP-complete. 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. (In other words, a clique is a complete subgraph. [S] , the problem can be found as NP-complete, which means that no polynomial time (in the size of input) algorithm is known and probably none can exist. Choose an NP-complete B language from which the reduction will go, that is, B ≤ p A. P, NP, and NP-Completeness Siddhartha Sen Questions: [email protected] 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. NP-hard: a problem to which all other problems in NP reduce. The dominating set problem that is NP-Complete is minimum-size-dominating-set, not just if a graph has a dominating set or not. 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. komargodski,moni. The Miserly Satis ability problem asks the following: given a collection of monotone clauses and a number k, is it possible to satisfy all of the clauses without setting more than k variables to 1? Prove that Miserly Satis ability is NP-complete. Show that Graph-Value is NP-complete by showing that Vertex-Cover reduces to Graph-Value. Prove that this problem is NP-complete. The following problems deﬁned for G and k are NP-complete. For a given graph G, take a di erent beer bottle b i for each. a graph (directed or undirected). Topic 24 C: NP Complete Problems We illustrate the range of NP Complete problems and how they are shown to be NPC by sketching proofs for several problems in logic, graph theory, and arithmetic. } This problem is known to be NP-complete by a reduction from 3SAT. In other words, does G have a k vertex subset whose induced subgraph is complete. In particular,. , that the subgraph be a clique (fully-connected). ALGORİTHM Connected (A[0. Now given a different problem P' If we show P. Prove that this problem is NP-complete. Any answer found by the TSP solution must also therefore be a valid Hamilton Circuit. The graph K n is regular of degree n-1, and therefore has 1/2n(n-1) edges, by consequence 3 of the handshaking lemma. (2) The problem of determining whether there exists a cycle in an undirected graph is in NP. Find a vertex-cover of maximum size in a given undirected graph. However, a following greedy algorithm is known for finding the chromatic number of any given graph. NP complete. ) Give your solution in simple and elegant pseudocode. Independent set in NP, 2. 1 (Decision Version of Max-Cut). This is given by the following theorem. If both are satisfied then it is an NP complete problem. NP-hard: a problem to which all other problems in NP reduce. In Section 3 we. Reduction Basics Assume A reduces to B in polynomial time. 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. Reducing TSP to graph problem InstanceA complete weighted undirected graph G = (V,E) with non-negative edge weights. In other words, either the node is matched (appears in an edge e. 1(c): If a Ch. Williamson NP-Completeness Proofs. Key words: Algorithm, MSP problem, HC problem, NP complete problem 1. , show that B T A The logic is as follows:. , that the subgraph be a clique (fully-connected). NP-Completeness : Proofs Show the following two problems NP- complete by restriction to Hamiltonian Path and Partition, respectively. To prove that this problem is NP-hard, we will show that it is equivalent to nding an undirected Hamiltonian cycle in a graph. I welcome you to read and look through some of the proofs for yourself! Je Linderoth IE418 Integer Programming. In other words, for any yes instance of X, there exists a certiﬁcate that. Given two graphs G and H, the question is to determine whether there. (a)Explain why this problem is in NP. Assuming that the HC problem is NP-Complete, prove that the Traveling Salesman problem is NP-Complete. Usually easy to convert to decision problem! If we know how to solve the decision problem, then we can usually solve the original problem. If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve A. For-mulate each problem as a decision problem ﬁrst, if necessary. Use V as the sub-routing to construct a gap-producing reduction. The NP-hardness of the unweighted longest path problem can be shown using a reduction from the Hamiltonian path problem: a graph G has a Hamiltonian path if and only if its longest path has length n − 1, where n is the number of vertices in G. 1 NP-Completeness (16 points) As we saw in class, the following problem is NP-complete. Traveling Salesman is NP-complete. Question: Does G have a cycle that visits each vertex exactly once? Consider HAMILTONIAN CYCLE restricted to graphs in which every vertex has degree at most 2. 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. If Q just satisﬁes (2) then it's called NP-hard. [Hint: reduce from the Independent Set problem] 2. c jEj, we have for our algorithm: E[number of satis ed edges] = 2 3 jEj 2 3 c. It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where ). For example,. Describe the reduction function f 4. We have displayed several Hamiltonian paths. Each vertex v2V has a label l v 2f 1;0;1g. • Prove that ﬁnding a directed or undirected Hamiltonian path or cycle in a directed or undirected graph is NP complete. Prove X is in NP. How to prove that the problem is NP. 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. minimal vertex covering, in a graph is NP complete. Suppose that someone attempted to solve the problem e ciently if the input values are in order. respective deadlines. Observe that complete-communication graphs are reex-ive, undirected, connected graphs with= V V. Graph Coloring Algorithm- There exists no efficient algorithm for coloring a graph with minimum number of colors. NP-Hard and NP-Complete Problems 3 – Optimization problems Each feasible solution has an associated value; the goal is to ﬁnd a feasible solution with the best value SHORTEST PATH problem Given an undirected graph Gand vertics uand v Find a path from uto vthat uses the fewest edges Single-pair shortest-path problem in an undirected. Finding a subset of vertices that has the minimum conductance in a given graph has been often stated to be an NP-complete problem in the literature [2, 3, 6, 8, 14, 16, 17]. 1 illustrates three edge-disjoint paths P 1, P 2 and P 3 in a series-parallel graph. 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 problem of nding a minimal dominating set of minimum cardinality is a hard problem. Prove that if a walk in a graph contains a Ch. prove that the Max-Cut problem is NP-Complete. The 3-Coloring Problem The 3-coloring problem is Given an undirected graph G, is there a legal 3-coloring of its nodes? As a formal language: 3COLOR = { G | G is an undirected graph with a legal 3-coloring. Limaye, Mahajan and Nimbhorkar [19] prove that longest paths in planar DAGs can be computed in UL\coUL. We prove that the problem is NP-complete, even for oriented graphs. Bounded Degree Spanning Tree Instance : An undirected graph G = (V, E) and a positive integer k ≤≤≤. Thanks for contributing an answer to Theoretical Computer Science Stack Exchange! Please be sure to answer the question. Also design a decrease-by-one algorithm for finding all the prime factors of a given number n. NP-Hard Given that we have at least one NP-Complete problem, showing others are NP- hard becomes easier: For a problem under consideration X 1.

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