2. Show that the following languages are in NP, by providing a non-deterministic algorithm. You are not allowed prove the membership in NP by providing certificate and verifier. • A = {(G, k) | G is k-colorable} • B = {G | G has a dominating set of size < n/2}, where n is the number of vertices of G. For an undirected graph G = (V, E), a set S C V is a dominating set if for every v ¢ S, there is a vertex u E D such that (v, u) e E. %3D

2. Show that the following languages are in NP, by providing a non-deterministic algorithm. You are not allowed prove the membership in NP by providing certificate and verifier. • A = {(G, k) | G is k-colorable} • B = {G | G has a dominating set of size < n/2}, where n is the number of vertices of G. For an undirected graph G = (V, E), a set S C V is a dominating set if for every v ¢ S, there is a vertex u E D such that (v, u) e E. %3D

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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Prove that 3-COL poly Independent set, demonstrate that the independent set is NP-competent. (Hint: Consider the combination u, c, where u is a G3-COL node and c is a colour, as a 3-coloring for the graph G3-COL. A portion of the vertices in the graph GInd are chosen by an independent set. Therefore, a node would be needed for each pair of u and c in order to build the graph Gind in the instance GInd, NInd = InstanceMap(G3-COL). A valid 3-coloring of the original graph must match to each valid independent set of size n in the constructed graph, so take care when defining the edges for the graph GInd = InstanceMap(G3-COL). If a surprising number of independent sets appear in the created graph, you may
Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true? Select one: a. If X is isomorph to F, then X is a complement of F. b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F. c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F. d. If X is NOT isomorph to F, then X is a complement of F. Question 2: Which statement is NOT true about Merge Sort Algorithm: Select one: a. Merge Sort time complexity for worst case scenarios is: O(n log n) b. Merge Sort is a quadratic sorting algorithm c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort. d. Merge Sort adopts recursive approach
Call graphs G and H isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = {< G, H > |G and H are isomorphic graphs}. Show that ISO ∈ NP. don't copy from other sources/websites .solve only if u knows or else I will downvote for sure
The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)First, give an example yes-input and an example no-input, where in each case, F has at leastfive nodes and G has at least three nodes.Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy.
(1) Like divide and conquer, dynamic programming (DP) solves problems by combiningsolutions to subproblems: true or false?_______; unlike divide and conquer, DPsubproblems are independent: true or false?_______. (2) For a graph G=<V, E>, the total running time of BFS is ?(____); the total running timeof DFS is ?(____). If G is an undirected graph, ∑? ∈ ???????(?) = ; if G is adirected graph, ∑? ∈ ???????(?) = __________. (3) Heap sort algorithm is stable or not? _____ , in-place or not? ______. Its worst case timecomplexity is ?(____). The height of a heap A[1..n] is ____. (4) For a sorting algorithm for an array of n elements, in the worst case the number ofinversions is _______ , the average number of inversions is _______.Shell sort algorithm is stable or not? ______ , in-place or not? _______ . (5) The worst case time complexity of BuildMaxHeap is ?(_______). The worst case timecomplexity of MaxHeapify is ?(______). The worst case time complexity of Heap-Extract-Max…
Answer True or False to the following claims: a. If G is graph on at most 5 vertices and every vertex has degree 2, then G is a cycle. b. Let G be a forest. If we add an edge to G, then G is no longer a forest. c. Let G be a graph, and let u, v, and w be vertices in G. Suppose that a shortest path from u to v in G is of length 3, andsuppose that a shortest path from v to w in G is of length 4. Then a shortest path from u to w in G is of length 7.
The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using qualifiers. You need to provide a clear expression using the qualifiers
The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using qualifiers. YOU ALSO NEED TO PROVIDE A CLEAR EXPRESSION USING THE QUALIFIERS.PLEASE GIVE CLEAR EXPLAINATION.
The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using quantifiers
The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy.
Prove that independent set is NP-compete by proving that 3-COL ≤ poly Independent set. (Hint: A 3-coloring for the graph G3-COL can be thought of as a subset of the pairs u, c where u is a node of G3-COL and c is a color. An independent set of the graph GInd selects a subset of its nodes. Hence, a way to construct the graph Gind in the instance GInd, NInd = InstanceMap(G3-COL) would be to have a node for each pair u, c. Be careful when defining the edges for the graph GInd = InstanceMap(G3-COL) so that each valid independent set of size n in the constructed graph corresponds to a valid 3-coloring of the original graph. If the constructed graph has unexpected independent sets, you may need to add more edges to it.
Let G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups Xand Y , and each edge has one end point in X and one end point in Y .A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions:1. Every vertex in X belongs to at most two edges in S.2. Every vertex in Y belongs to at most one edge in S.Give an algorithm to find the size (number of edges) of maximum 2-1 generalized matching