Exhaustive Search

• Exhaustive search means trying every possible combination for an optimal solution.
• IT tends to be very expensive.
• Three examples:
  • Traveling Salesperson Problem
    • Assume a connected weighted graph.
    • What is the lowest cost Hamiltonian circuit of the graph.
      • A Hamiltonian circuit visits every vertex, except the first, exactly once.
    • Generate all permutations of the verticies.
    • Check to see if each is optimal
    • This is O(n!)
  • Knapsack Problem
    • Assume n items
    • Each has value v_i, 0 < i ≤ n
    • Each has weight w_i
    • And a Knapsack which can hold a total weigh of W.
    • Find a subset S of the n items, such that the total weight of the items in S is less than W, with the maximum value.
    • Essentially, set up a bit vector, either an item is in (1) or not (0) .
    • Evaluate if this vector is
      • Under the given weight
      • And has maximum value.
    • This is O(2ⁿ)