We are given a set V of n variables {x1, x2, ... , Xn} and a set C of m weak and strict inequalities between the variables, i. e., inequalities of the form Xi < x; or Xi < xj. The set C of inequalities is called consistent over the positive integers Z+ {1, 2, 3, ...} iff there is an assignment of positive integer values to the variables that satisfies all the inequalities. For example, the set {x1 < X3, X2 < x1} is consistent, whereas {x1 < x3, X2 < X1, X3 < x2} is not consistent. Required:
a. Give an efficient algorithm to determine whether the set C of inequalities is consistent over the positive integers. State precisely the asymptotic running time of your algorithm in terms of n and m.
b. If the set of inequalities has a solution, then it has a unique minimum solution, i. e., a solution in which every variable has the minimum value among all possible solutions. Give an efficient algorithm to compute the minimum solution.

1. the program must provide following functions to extract some statistics. note that the data_list parameter specified in these functions may be the same for all functions or different for different functions—that is your choice. a skeleton file is provided on mirmir. a) open_file()prompts the user to enter a year number for the data file. the program will check whether the year is between 1990 and 2015 (both inclusive). if year number is valid, the program will try to open data file with file name ‘year.txt’, where is the year. appropriate error message should be shown if the data file cannot be opened or if the year number is invalid. this function will loop until it receives proper input and successfully opens the file. it returns a file pointer and year. i. hint: use string concatenation to construct the file name b) read_file(fp)has one parameter, a file pointer read. this function returns a list of your choosing containing data you need for other parts of this project. c) find_average(data_list) takes a list of data (of some organization of your choosing) and returns the average salary. the function does not print anything. hints: i. this is not the average of the last column of data. it is not mathematically valid to find an average by finding the average of averages—for example, in this case there are many more in the lowest category than in the highest category. ii. how many wage earners are considered in finding the average (denominator)

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