Chapter 1

• I am going to go through chapter 1 in order.
  • Well, you know, as much as I can.
  • But you should read it anyway.
• They introduce a number of problems which will be used for illustration in the later chapters.
  • They show how these problems are related.
  • They show how subtle changes in the problems lead to radically more difficult problems.
• A quick side note: The knapsack problem.
  • A thief breaks into a museum with a fixed size knapsack. They want to steal the maximum amount of treasure that will fit into this knapsack.
  • Items can be measured in a single dimension (weight for example)
  • This problem is probably $O(n!)$ when each item has a distinct size and value.
    • This problem dates from 1897.
    • The 0-1 knapsack problem
    • Consider a knapsack of capacity 5.
    • There are three items,

      Item	Capacity	Value
      A	6	100
      B	2	50
      C	2	51

  • This problem not bad when there an unlimited number of each item type. all items are the same size.
    • And all types are the same size.
    • This is the unbounded fractional knapsack problem.
    • Think piles of coins in a dragon horde , where there is no chance to exhaust the supply.
      • There are n different types of items.
      • Each item i has a weight, w_i
      • Same size.
      • Each item i has a price, p_i

    • for each item type  i
         compute the value, vi = pi/wi
      sort items by value
      for each item type ordered by the value (large to small) until the knapksack is full
         take as many of that item as will fit into the knapsack 

    • Do the big three
      • How does this work?
      • Is it correct?
      • What is the performance?
  • Notice how a slight change in the problem changes the performance?
• The stable matching problem.
  • This is a problem "discovered" in the 60s by economists.
  • Can they design a process where college admissions, job recruiting or even dating can be self enforcing?
• For this problem we consider
  • a set of n men $M = \{m_1, m_2, ... m_n\}$
  • a set of n women $W = \{w_1, w_2, ... w_n\}$.
  For each man and woman
  • there is an ordered list of there preferences from the other set.
    • $m_1: w_1, w_2, w_3$
    • $m_2: ...$
    • $w_1: m_2, m_1, m_3$
  • they wish to be paired with exactly one member of the other set.
  • they wish to be pared with the highest choice from their ordered list possible.
• A matching S is a set of ordered pairs of the form (m_i, w_j)
• A match is perfect for each element M each element from W appear exactly once in a pair in the match.
• An instability with respect to S occurs when (m,w) and (m',w') are matches within S, but m prefers w' to w and w' prefers m to m'
  • Or A pair (m,w') is an instability with respect to S: (m,w') is not an element of S, but but both m and w' prefer each other to their partner is in S.
• Finally a match is stable if
  • It is perfect (or 1 to 1)
  • There is no instability
• Does an algorithm exist to find a stable match for a given set of men and women?