@techReport{Abraham:2008:7815,
	      Author = {Abraham,D.J.;Manlove,D.F.;},
	       Institution = {University of Glasgow, Department of Computing Science},
                          Title = {Pareto optimality in the Roommates problem}, 
                          ISBN =
	      {TR-2004-182},
	      abstract = {We consider Pareto optimal matchings as a means of coping with instances of
the Stable Roommates problem (SR) that do not admit a stable matching.
Given an instance I of SR, we show that the problem of finding a maximum
Pareto optimal matching is solvable in O(\sqrt{n\alpha(m,n)}m\log^{3/2} n)
time, where n is the number of agents and m is the total length of the
preference lists in I.  By contrast we prove that the problem of finding a
minimum Pareto optimal matching is NP-hard, though approximable within 2.
We also show that the problem of finding a Pareto optimal matching with the
fewest number of blocking pairs is NP-hard.  However, for a fixed integer 
K, we give a polynomial-time algorithm that constructs a Pareto 
optimal matching with at most K blocking pairs, or reports that no such 
matching exists.  
		    
		  },Publisher =
	      {N/A},
	      Year = {2004}}