2. (20 points) Stable Matchings: K&T Ch 1, Ex 5. Consider a version of the stable matching problem where there are n students and n colleges as before. Assume each student ranks the colleges and vice versa), but now we allow ties in the ranking. In other words, we could have a school that is indifferent two students s1 and s2, but prefers either of them over some other student s3 (and vice versa). We say a student s prefers college c1 to c2 if c is ranked higher on the s's preference list and c1 and c2 are not tied. (a) Strong Instability. A strong instability in a matching is a student-college pair, each of which prefer each other to their current pairing. In other words, neither is indifferent about the switch Does there always exist a matching with no strong instability? Either give an example instance for which all matchings have a strong instability (and prove it), or give and analyze an algorithm that is guaranteed to find a matching with no strong instabilities (b) Weak Instability. A weak instability in a matching is a student-college pair where one party prefers the other, and the other may be indifferent. Formally, a student s and a college c with pairs c' and s' form a weak instability if either s prefers c to c' and c either prefers s to s' or is indifferent between s and s'. c prefers s to s, and s either prefers c to c, or is indifferent between c and d. Does there always exist a perfect matching with no weak instability? Either give an instance with a weak instability or an algorithm that is guaranteed to find one. 2. (20 points) Stable Matchings: K&T Ch 1, Ex 5. Consider a version of the stable matching problem where there are n students and n colleges as before. Assume each student ranks the colleges and vice versa), but now we allow ties in the ranking. In other words, we could have a school that is indifferent two students s1 and s2, but prefers either of them over some other student s3 (and vice versa). We say a student s prefers college c1 to c2 if c is ranked higher on the s's preference list and c1 and c2 are not tied. (a) Strong Instability. A strong instability in a matching is a student-college pair, each of which prefer each other to their current pairing. In other words, neither is indifferent about the switch Does there always exist a matching with no strong instability? Either give an example instance for which all matchings have a strong instability (and prove it), or give and analyze an algorithm that is guaranteed to find a matching with no strong instabilities (b) Weak Instability. A weak instability in a matching is a student-college pair where one party prefers the other, and the other may be indifferent. Formally, a student s and a college c with pairs c' and s' form a weak instability if either s prefers c to c' and c either prefers s to s' or is indifferent between s and s'. c prefers s to s, and s either prefers c to c, or is indifferent between c and d. Does there always exist a perfect matching with no weak instability? Either give an instance with a weak instability or an algorithm that is guaranteed to find one