Consider the following problem: Long Enough Cycle Input: undirected simple graph G = (V, E) with |V| = n, positive integer k Question: Does G contain a simple cycle containing at least k vertices? a) Describe the language Llong for Long Enough Cycle. b) Show that Long Enough Cycle is in NP. Hint for b): Give a polynomial verifier Viong for Ltong Explain why Viong satisfies the definition of verifier and why it is is indeed a polynomial time verifier. Don't forget to describe what a certificate looks like.

Consider the following problem: Long Enough Cycle Input: undirected simple graph G = (V, E) with |V| = n, positive integer k Question: Does G contain a simple cycle containing at least k vertices? a) Describe the language Llong for Long Enough Cycle. b) Show that Long Enough Cycle is in NP. Hint for b): Give a polynomial verifier Viong for Ltong Explain why Viong satisfies the definition of verifier and why it is is indeed a polynomial time verifier. Don't forget to describe what a certificate looks like.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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True or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n)
Part 2: Random GraphsA tournament T is a complete graph whose edges are all oriented. Given a completegraph on n vertices Kn, we can generate a random tournament by orienting each edgewith probability 12 in each direction.Recall that a Hamiltonian path is a path that visits every vertex exactly once. AHamiltonian path in a directed graph is a path that follows the orientations of thedirected edges (arcs) and visits every vertex exactly once. Some directed graphs havemany Hamiltonian paths.In this part, we give a probabilistic proof of the following theorem:Theorem 1. There is a tournament on n vertices with at least n!2n−1 Hamiltonian paths.For the set up, we will consider a complete graph Kn on n vertices and randomlyorient the edges as described above. A permutation i1i2 ...in of 1,2,...,n representsthe path i1 −i2 −···−in in Kn. We can make the path oriented by flipping a coin andorienting each edge left or right: i1 ←i2 →i3 ←···→in.(a) How many permutations of the vertices…
True or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n) <= |h(n)-h(m)|. (b) If an A* heuristics is admissible then it satisfies the monotone restriction. (c) Best-first search guarantees optimality in its returned solution. (d) Least-cost-first search guarantees optimality in its returned solution. (e) If all edges are with unit cost, then Breadth-first search guarantees optimality in its returned solution.
3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)
The graph-coloring problem is usually stated as the vertex-coloring problem: assign the smallest number of colors to vertices of a given graph so that no two adjacent vertices are the same color. Consider the edge-coloring problem: assign the smallest number of colors possible to edges of a given graph so that no two edges with the same end point are the same color. Explain how the edge-coloring problem can be polynomial reduced to a vertex-coloring problem. Give an example.
5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
Q6. Given graph G = (V, E), the graph G'= (V', E') is defined such that (u,v) is in E' if and only if G contains a path with at most two edges between u and v. Describe an efficient algorithm for computing G' from G. Once develop the algorithm assuming G has an adjacency-matrix representation and once develop the algorithm assuming adjacency-list representation for the graph. Analyze the running times of your algorithms.
3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1
36. Let G be a simple graph on n vertices and has k components. Then the number m of edges of G satisfies n-k ≤m if G is a null graph. This statement is A. sometimes true B. always true C. never true D. Neither true nor false
A randomly matchable graph: if every matching of the graph can be extended to a 1-factor.Give examples of the following: a) an infinite class of bipartite graphs that is randomly matchable; b) an infinite class of non-bipartite graphs that is randomly matchable; c) an example of a graph that has a 1-factor, but is not randomly matchabl
5. Consider a directed graph G with n nodes. Write a function findUnreachableNode that takes a node and prints all the nodes that are unreachable from the given node. You can use either adjacency list or adjacency matrix to solve this problem. Function Signature: findUnreachableNode (int node) For example: In the following graph, findUnreachableNode (0) will return 4, 6, 7 as they are unreachable from node 0. 1 7 4 Good luck!!!
3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k. Show that the Strongly Independent Set Problem is NP-Complete.