Prove that for every even n ≥ 2, there exists an instance of the stable matching problem with n jobs and n candidates such that the instance has at least 2n/2 distinct stable matchings.
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- make Membership Testing AlgorithmInput: a group G operating on f = 1,2,.....n; a permutation g of f = 1,2,.....n; a base and strong enerating set for G; Schreier vectors v I 1 I k, for the stabiliser chainOutput: a boolean result showing if g G is in the group; function is in group(p: permutation; I 1..k+l): boolean; (* return true if the permutation p is in the group G I *)Write The Rank-Maximal Matching Algorithm. Let G = (AUP, EU... UE). Initialize G₁ = G₁, and M to any maximum matching in G. For i = 1 tor-1 solve \ the steps, and output.Analysis of Linear search (Worst, Average and Best Cases)#include <stdio.h>// Linearly search x in arr[]. If x is present then return the index,// otherwise return -1int search(int arr[], int n, int x){ int i; for (i=0; i<n; i++) { if (arr[i] == x) return i; } return -1;}
- In a binary classification problem, assume we have a classifier R that guesses randomly (50/50 positive/negative). For data that has 70% positive and 30% negative instances, show that R gets 50% error.Let M be a matching in a given instance of POM. Then M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free.Given an instance 1 of POM, the relation forms a strict partial order over the set of matchings in I.Algorithm for Testing MembershipInput : a group G acting on f~ = { 1,2 ..... n };a permutation g of f~ = { 1,2 ..... n };a base and strong ~enerating set for G;Schreier vectors v (i) , 1 < i < k, for the stabiliser chain;Output: a boolean value answer, indicating whether g ~ G;function is_in_group(p : permutation; i : 1..k+l ) : boolean;(* return true if the permutation p is in the group G (i) *)
- Backtrack Search Algorithm Using Known Subgroupa group G with the base B=[51, 132,..... k] and a strong generating set S as the input; a subgroup L of G whose members meet the decidable condition P;a powerful generating set with respect to B for the subgroup K = H (P);Consider the following generalization of the maximum matching problem, which we callStrict-Matching. Recall that a matching in an undirected graph G = (V, E) is a setof edges such that no distinct pair of edges {a, b} and {c, d} have endpoints that areequal: {a, b} ∩ {c, d} = ∅. Say that a strict matching is matching with the propertythat no pair of distinct edges have endpoints that are connected by an edge: {a, c} ̸∈ E,{a, d} ̸∈ E, {b, c} ̸∈ E, and {b, d} ̸∈ E. (Since a strict matching is also a matching, wealso require {a, b} ∩ {c, d} = ∅.) The problem Strict-Matching is then given a graphG and an integer k, does G contain a strict matching with at least k edges.Prove that Strict-Matching is NP-complete.Find maximum Matching in the graph given below, where Dotted lines are given initial matchings
- Please don't use handwritting for this question How would you modify the dynamic programming algorithm for the coin collecting problem if some cells on the board are inaccessible for the robot? Apply your algorithm to the board below, where the inaccessible cells are shown by X’s. How many optimal paths are there for this board? You need to provide 1) a modified recurrence relation, 2) a pseudo code description of the algorithm, and 3) a table that stores solutions to the subproblems.Exercise c. Strict matchingConsider the following generalization of the maximum matching problem, which we callStrict-Matching. Recall that a matching in an undirected graph G = (V, E) is a setof edges such that no distinct pair of edges {a, b} and {c, d} have endpoints that areequal: {a, b} ∩ {c, d} = ∅. Say that a strict matching is matching with the propertythat no pair of distinct edges have endpoints that are connected by an edge: {a, c} ̸∈ E,{a, d} ̸∈ E, {b, c} ̸∈ E, and {b, d} ̸∈ E. (Since a strict matching is also a matching, wealso require {a, b} ∩ {c, d} = ∅.) The problem Strict-Matching is then given a graphG and an integer k, does G contain a strict matching with at least k edges.Prove that Strict-Matching is NP-complete.Question 3 Is it possible for all matching to be stable for the StableMarrige problem with N=2 (two men and two women)? What if N=3? I would say yes, as long as there are no unstable pairs that would cause an instability. But I am not confident lol Full explain this question and text typing work only thanks