Hi Gabriel. Yes, the YRMH-IGYT algorithm arrives at the same outcome as the TTC algorithm.
Here is the theorem which proves this:
Theorem: For a given ordering f, the YRMH_IGYT algorithm yields the same outcome as the top trading cycles algorithm.
For any set J of agents and set G of houses remaining in the algorithm, YRMH_IGYT algorithm assigns the next series of houses in one of two possible ways.
Case 1: There is a sequence of agents i1 , i2 , …, ik (which may consist of a single agent) where agent i1 has the highest priority in J and demands house of i2 , agent i2 demands house of i3 , …, agent ik&1 demands house of ik, and ik demands an available house h. At this point agent ik is assigned house h, the next agent ik&1 is assigned house hik (which just became available),…, and finally agent i1 is assigned house hi2. Note that the ordered list (h, i1 , hi2 , i2 , …, hik , ik) is a (top trading) cycle for the pair (J, G).
Case 2: There is a loop (i1 , i2 , …, ik) of agents. When that happens agent i1 is assigned the house of i2 , agent i2 is assigned house of i3 , …, agent ik is assigned house of i1 . In this case (hi1 , i1 , hi2 , i2 , …, hik , ik) is a (top trading) cycle for the pair (J, G).
Hence the YRMH_IGYT algorithm locates a cycle and implements the associated trades for any sets of remaining agents and houses. This observation together with Remark 1 [There is at least one cycle in each step of the TTC algorithm] implies the desired result.
This theorem was consulted from the Article “House Allocation with Existing Tenants” by Abdulkadiroglu and Sonmez from the Journal of Economic Theory (1999).