An empirical study of the stable marriage problem

1
An empirical study of the stable marriage
problem with ties and incomplete lists (SMTI)
Ian Gent Patrick Prosser
2
What is smti?

• SM with ties is in P
• SM with incomplete lists is in P
• SM with ties and incomplete lists is in NP
• so says David Rob in 1999

But how does it behave? Just because it is NPC,
does that mean it is hard? If so, where are the
hard problems
3
A problem generator
ltn,p1,p2gt

• n is the number of men and women
• p1 is the probability of incomplete lists
• p1 0 lists are complete
• p1 1 lists are empty
• p2 is the probability of ties
• p2 0 there are no ties
• p2 1 all ties, we have a maximum matching
• ltn,0,0gt is an SM!

This is new!
4
Complete algorithms for the decision and
optimisation problems coded up as constraint
programs
5
A complete algorithm

• For the decision problem Is there a stable
  matching of size n
• a simple extension of the earlier O(n4) encoding
• a constraint between every man and woman
• disallowing unstable pairs
• disallowing anything but a bijection
• domain is preference list
• variable ordering
• choosy person
• value ordering
• most preferred partner

This is new!
6
A complete algorithm

• For the optimisation problem
• find the largest stable matching
• find the smallest stable matching
• a simple extension the the earlier O(n4)
  encoding
• a constraint between every man and woman
• disallowing unstable pairs
• disallowing anything but a bijection
• domain is preference list
• a person that anyone can marry
• a set of 0/1 variables Z
• Zi 0 if Mi is unmarried
• maximise or minimise the sum of Z
• variable ordering
• choosy person
• ignore Z variables!

This is new!
7
We can generate instance We can answer the
decision problem We can find the optimum
matching, smallest or largest

• How do problem features influence the above with
  respect to
• The cost of answering the decision problem
• and if the answer is yes or no
• The cost of the optimisations
• and the size?

8
First question

• Given an SMTI lt10,p1,p2gt
• As I vary p2 (indifference) what is the effort
  to
• determine if there is a stable marriage of size
  n?
• (This is the decision problem)
• Will problems become harder or easier as p2
  increases?
• Will problems become harder or easier as p1
  increases?
• (p1 is incompleteness)
• Question is this independent of the algorithm?

9
(No Transcript)
10
Average search effort increases as p2
increases Why? Search effort falls as p1
(incompleteness increases) Why? Search effort is
erratic? Why?
Would allDiff help? What does that mean?
11
(No Transcript)
12
Average search effort was reduced, less
noisy allDiff helps!
How about solubility? Is it more likely that an
SMTI has a stable matching of size n as we
increase p1? What about p2?
13
(No Transcript)
14
What a mess! Yes, it is more likely that we can
find a stable matching as p2 increases. But
really! Call that a graph?
Rather than p2 on the x axis, how about kappa?
15
What is kappa?
ltSolgt is expected number of solutions N is log of
the size of the state space
K 0 if all states are solutions, therefore easy
soluble k infinity if no states are
solutions, therefore easy insoluble k 1 there
is one solution on average, on the knife edge
hard
16
Kappa for smti?
Treat it as a constraint satisfaction
problem measure tightness of constraints and
domain size estimate kappa on a problem by
problem basis
17
(No Transcript)
18
Kappa looks good for the decision problem! We
might be able to put it to use
What about the optimisation problem? What about
the size of the smallest and largest stable
matchings?
19
(No Transcript)
20
What is harder (a) finding the largest stable
matching? (b) finding the smallest stable
matching?
Betting ends
Place your bets!
21
(No Transcript)
22
(No Transcript)
23
Conclusion and future work

• You have just had a first taste of the smti
• what it feels like
• how it is influenced by p1 and p2
• I spared you n
  ?
• The experiments took in excess of 2 months cpu
• at least 766MHz
• Lots more to be done
• incorporate allDiff in the optimisation problem
• look at constant sized preference lists as we
  vary n
• higher dimensionality
• more than 2 sexes
• kappa for smti
• theory based heuristics
• real problems

24
Questions?