A necessary and sufficient condition for the existence of a complete stable matching

TL;DR: A new structure called a “stable partition” is defined, which generalizes the notion of a complete stable matching, and it is proved that every instance of the stable roommates problem has at least one such structure.

About: This article is published in Journal of Algorithms.The article was published on 1991-01-02 and is currently open access. It has received 198 citations till now. The article focuses on the topics: Stable roommates problem & Stable marriage problem.

Summary

Person

• Preference list 1 234 2 314 3 124 4 Arbitrary Knuth [ll] demonstrated that multiple solutions could exist and asked for an efficient algorithm to generate a solution, if one exists.
• The authors give a necessary and sufficient condition for the existence of a complete stable matching; namely, the non-existence of any odd party, which will be defined subsequently.
• For a given preference relation, it is not clear whether any stable partition exists.
• The authors will prove that any two stable partitions have exactly the same odd parties (not only having the same persons involved in a corresponding odd party, but also with the same party permutation).
• To complete this example, the authors just have to fill in all the other entries and follow the rule that whenever (a lb) is a superior entry, then @Ia) is inferior.

Proof

• Suppose that II is a stable partition without any odd party.
• The authors will prove that the inverse of Theorem 3.3 is also true.
• All the subscripts considered in the following are modulo r.
• Before the inferior entry (aim1 I bi) in the current list of a,-i, there must be a party entry in II.
• In stable partition II, a, (bj, respectively) is matched with the first person b, (the last person ai, respectively) on his list in table T. While in TI', ai is matched with the second person bi+I, and bj is matched with ajel whom he prefers to aj.

4. IRVING'S ALGORITHM

• The algorithm successively deletes entries from preference lists until either each person has only one entry on his list, or until someone has no entries.
• In the first case, the entries specify a complete stable matching, and in the second case, there are no solutions.

DEFINITION.

• Irving's algorithm is divided into two phases: Phase 1.
• This phase of the algorithm will terminate either (i) with every person holding a proposal, or (ii) with one person rejected by everyone.
• (They form an odd party.) (ii) Otherwise, eliminate this rotation.
• Remark 2. By Theorem 6.5, every stable partition contains the same odd parties as II does, where II is the stable partition contained in the final table.
• And the following theorem follows from the algorithm, Proposition 3.2, and Theorem 3.3.

Pro05

• In table T,, if every person has zero or one entry on his list, then the lists specify a stable partition.
• Thus, by Proposition 5.6, TI is also a stable partition for the original instance TO.
• So, checking the Phase 1 table at the end of this phase, (i) if every person has zero or one entry on his list, then the lists specify a stable partition, (ii) if someone has more than one entry on his list, this brings us to the second phase of the algorithm.

Proof:

• So, after step (21, each bi still has aiel on his list, and thus no list is caused to be empty.
• Therefore, it must be some person ai whose list becomes empty, and this person ai has only one entry left on his list after step (2).
• Then the lists of every person involved in this rotation become empty after eliminating R. From the above observation, if the elimination of rotation R makes some list empty, it indicates an odd party.
• For convenience, such a rotation is called an odd rotation (for persons in A).

7. CONCLUSIONS

• The authors notice that their results, Lemma 6.1 and Theorem 6.2, are already in [7] , though ours extend the idea to the stable partition.
• In that report, Irving also had the following result: an instance of the stable roommates problem admits a stable matching if and only if the shortlists are non-empty and there is no improper rotation.
• Translate this result into their terms: an instance of the stable roommates problem admits a complete stable matching if and only if there does not exist any single-person odd party and there is no odd rotation.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "A necessary and sufficient condition for the existence of a complete stable matching" ?

In this paper, the authors give a necessary and sufficient condition for the existence of a complete stable matching ; namely, the non-existence of any odd party, which will be defined subsequently. The authors define a new structure called a “ stable partition, ” which generalizes the notion of a complete stable matching, and prove that every instance of the stable roommates problem has at least one such structure. The authors also show that a stable partition contains all the odd parties, if there are any.