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. (The 0-1 knapsack problem)
  • This problem not bad when there an unlimited number of each item type.
    • This is the unbounded 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
      • Each item i has a price, p_i

    • for each item i
         compute the value, vi = pi/wi
      sort items by value
      for each item type ordered by the value 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
• 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?