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I made a couple of minor optimizations to my solution.  It still will not handle large inputs, but it is a little better.  Thanks to Eric Ivancich, I removed some of the duplicate checks of potential solutions.  I hope I got them all.  Also, I added some memoization for values of solution sets, which has the downside of requiring more memory but seems to improve overall performance.

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require 'yaml'

#execute program by providing a file name with the data in YAML format
#Example YAML file contents:
#David: Helen Vicki Joseph
#Helen: David Vicki Joseph
#Joseph: Vicki Helen David
#Vicki: David Helen Joseph

class PairCalculator
  def initialize(filename)
    #read data into hash from YAML file
    @pairs_hash = YAML::load_file(filename)
    @num_people = @pairs_hash.length
    @num_choices = @num_people - 1
    @pair_set_value_hash = Hash.new
    @pair_value_hash = Hash.new
    
    #build array of all possible pairs
    @all_pairs = get_all_pairs
  end
 
  #finds the nth choice of a person for some value n, 0 would be
  #the first choice, 1 second, etc.
  def get_nth_choice(person, n)
    @pairs_hash[person].scan(/\w+/)[n]
  end
 
  #this method returns 0 if the pair is only one name
  #otherwise it looks up the pair value in a hash
  #preference indexes, 0=best, 1=next best, etc.
  def get_pair_value(pair)
    pair_array = pair.split(" ")
    return 0 if(pair_array.length==1)
    
    #get the value from the pair value hash
    return @pair_value_hash[pair] + @pair_value_hash[pair_array[1] + " " + pair_array[0]]
  end

  #this method calculates the value of a possible solution of pairs
  def get_pair_set_val(pair_set)
    #check the hash first
    return @pair_set_value_hash[pair_set] if(@pair_set_value_hash.include?(pair_set))
    
    #not in the hash calculate it
    val = 0
    pair_set.each {|next_pair| val += get_pair_value(next_pair)}
    @pair_set_value_hash[pair_set] = val
    return val
  end
 
  #this method calculates a list of pairs from the pair array minus the
  #head_pair and any similar pairs (see similar? method for more details).
  def get_tail(pair_array, head_pair)
    tail_array = []
    pair_array.each do |next_pair|
      #remove all pairs with common partners
      tail_array += [next_pair] unless(similar?(head_pair, next_pair))
    end
    return tail_array
  end
 
  #pairs are considered similar if they share a common name or they
  #are both a single name (this case makes sure that only one single
  #name can be selected in any given solution
  def similar?(pair1, pair2)
    #returns true if they are both length of 1
    return true if(pair1.split(" ").length==1 and
                  pair2.split(" ").length==1)

    #returns true if the pairs have common partners
    pair1.split(" ").each do |next_name|
      if(pair2.include?(next_name))
        return true
      end
    end
    return false
  end

  #this method generates a list of all possible pairs
  def get_all_pairs
    all_pairs = []
    @pairs_hash.each_key do |p1|
      1.upto(@num_choices) do |cnt|
        #select the next person in the list
        p2 = get_nth_choice(p1, cnt-1)
        
        #add to all pairs list
        all_pairs += [p1 + " " + p2] unless(all_pairs.include?(p2 + " " +1))
        
        #add to pair_value_hash for later lookup
        @pair_value_hash[p1 + " " + p2] = (cnt-1)**2
      end
      #add individual names if there are an odd number of people
      all_pairs += [p1] if (@num_people & 1 == 1)
    end
    return all_pairs
  end
 
  #this method executes an exhaustive search of all pair sets for
  #each set it calculates the value with the best found up to that
  #point. if the value is lower then it saves the new solution.
  def get_preferred_pairs(pair_array=@all_pairs)
    solution = []
    solution_value = 1.0/0.0
    #base case: pair_array is empty
    if(pair_array.length==0)
      return solution
    end
   
    #iterate over all pairs in the array use each one as the head
    #in a potential solution
    pair_array.each do |pair|
      #get the next potential solution
      pair_set = [pair] +
        get_preferred_pairs(get_tail(pair_array, pair)) #recursive call
      #calculate the value of the next potential solution
      pair_set_val = get_pair_set_val(pair_set)
      if(solution_value>pair_set_val)
        #better solution has been found... save it!
        solution = pair_set
        solution_value = pair_set_val
      end
    end
    return solution
  end
end

if(ARGV.length == 0)
  puts "Usage: provide YAML file name as the argument."
  exit 0
end

t1 = Time.now
pc = PairCalculator.new("list.yaml")
pair_set = pc.get_preferred_pairs
pair_set.each {|pair| puts pair}
t2 = Time.now
puts "Pair set value: #{pc.get_pair_set_val(pair_set)} \nTime: #{t2 - t1}"


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