McDermid, Eric J. (2011) A structural approach to matching problems with preferences. PhD thesis, University of Glasgow.
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Abstract
This thesis is a study of a number of matching problems that seek to match together pairs or groups of agents subject to the preferences of some or all of the agents. We present a number of new algorithmic results for five specific problem domains. Each of these results is derived with the aid of some structural properties implicitly embedded in the problem.
We begin by describing an approximation algorithm for the problem of finding a maximum stable matching for an instance of the stable marriage problem with ties and incomplete lists (MAXSMTI). Our polynomial time approximation algorithm provides a performance guarantee of 3/2 for the general version of MAXSMTI, improving
upon the previous best approximation algorithm, which gave a performance guarantee of 5/3.
Next, we study the sexequal stable marriage problem (SESM). We show that SESM is W[1]hard, even if the men's and women's preference lists are both of length at most three. This improves upon the previously known hardness results. We contrast this with an exact, loworder exponential time algorithm. This is the first nontrivial exponential time algorithm known for this problem, or indeed for any hard stable matching problem.
Turning our attention to the hospitals / residents problem with couples (HRC), we show that
HRC is NPcomplete, even if very severe restrictions are placed on the input. By contrast, we give a lineartime algorithm to find a stable matching with couples (or report that none exists) when stability is defined in terms of the classical GaleShapley concept. This result represents the most general polynomial time solvable restriction of HRC that we are aware of.
We then explore the three dimensional stable matching problem (3DSM), in which we seek to find stable matchings
across three sets of agents, rather than two (as in the classical case). We show that under two natural definitions of stability, finding a stable matching
for a 3DSM instance is NPcomplete. These hardness results resolve some open questions in the literature.
Finally, we study the popular matching problem (POPM) in the context of matching a set of applicants to a set of posts. We provide a characterization of the set of popular matchings for an arbitrary POPM instance in terms of a new structure called the switching graph exploited to yield efficient algorithms for a range of associated problems, extending and improving upon the previously bestknown results for this problem.
Item Type:  Thesis (PhD) 

Qualification Level:  Doctoral 
Keywords:  stable matching, hospitals/residents, approximation algorithm, graph, lattice, partially ordered set, preference, fairness, onesided matching, GallaiEdmonds decomposition theorem, indifference, ties, bipartite graph, exponentialtime algorithm, couples 
Subjects:  Q Science > QA Mathematics > QA75 Electronic computers. Computer science 
Colleges/Schools:  College of Science and Engineering > School of Computing Science 
Supervisor's Name:  Irving, Dr. Robert W. 
Date of Award:  2011 
Depositing User:  Eric J McDermid 
Unique ID:  glathesis:20112371 
Copyright:  Copyright of this thesis is held by the author. 
Date Deposited:  23 Mar 2011 
Last Modified:  10 Dec 2012 13:54 
URI:  http://theses.gla.ac.uk/id/eprint/2371 
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