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 c1 is ranked higher on the ss preference list andc1 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 of a set of colleges and students with preference lists for which every perfect matchings has a strong instability; or give an algorithm that is guaranteed to find a matching with no strong instability and prove that it is correct.

(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 c.

In other words, the pairing between c and s is either preferred by both, or preferred by one while the other is indifferent. Does there always exist a perfect matching with no weak instability? Either give an example of a set of colleges and students with preference lists for which every perfect matching has a weak instability; or give an algorithm that is guaranteed to find a perfect matching with no weak instability.