Consider a version of the stable matching problem where there are n students and n colleges...

 

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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 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 students 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'. 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. 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 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 students 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'. 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.

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