Definitions

• Warning, this is the 10,000 foot view of this topic.
• An algorithm A is polynomial if the timing function T_A(n) ∈ O(n^k), k constant.
• We define the set of problems P to be the problems for which a polynomial time algorithm exists to solve the problem for all input.
• Problem Z ∈ P if there exists an algorithm A_z such that A_Z ∈ O(n^k) and for all input i, s= A(i), is a solution to Z for i.
• 
• An algorithm V(i,c) verifies a solution c (or certificate) for an input i and a problem Z
  • a = V(i,c)
  • If a = true, then c is a solution to Z for input i.
  • If a = false, then c is not a solution to Z for the input i.
  • For Sorting, i = (3, 6, 2, 1) , c= (1, 2, 3, 6)

  •  
    VERIFY_SORT(I, C)
    
     S ← SORT(I)
     if S = C then
       return true
     else
       return false
     
    
        

• We define the set of problems NP to be the problems for which a polynomial time algorithms exists which can verify any solution to the problem.
• Problem Z ∈ NP if there exists V_Z(i,c) such that V_Z ∈ O(n^k) for constant k and V_z verifies Z.
• CONVEX_HULL ∈ NP

  VERIFY_CONVEX_HULL(S,C)
  // remember, S is the set of points, and C is points supposedly on the hull.
  
   C' = GRAHAM_SCAN(S)
   C' = SORT(C')
   C = SORT(C)
   return C = C'
  
  

• CLAIM: P∈ NP
• A verification algorithm is an example of a decision algorithm.
• A decision problem is a problem which takes an input and returns true if the input satisfies a criteria and false if it does not.
• Given a circuit C, with n inputs, is there a set of inputs such that the circuit will produce a true. (CIRCUIT_SAT), review this problem.
• We deal with many optimization problems, or problems which ask us for the best possible answer.
  • Given a complete weighted graph G = (V,E), and a starting vertex v_s∈ V find the lowest cost path from v_s visiting all other v∈V and only traversing an edge once that returns to v_s (TSP)
• In general, we can create a decision problem that is no harder, and possibly easier than a given optimization problem.
  • SHORTEST_PATH: given a graph G=(V,E), and a∈V, b∈V, find the shortest path from a to b.
  • PATH: Given a graph G, a and b as above, and a cost c, is the cost from a to b equal to c?
• If we can show that the decision problem is hard, then the optimization problem is at least as hard.
• A reduction
  • If A and B are decision problems.
  • α and β are inputs to these problems.
  • If the result r_A is the result for A on α
  • If the result r_B is the result for B on β
  • Algorithm R(α) → β is a reduction from A to B if r_A = r_B for all α
  • written A ≤ B
  • If R ∈ P then R is a polynomial time reduction.
  • written A ≤_P B
  • This is a way to say that we can solve A by transforming the input to an input for B and solving B.
• We saw a polynomial time reduction from sorting to convex hull.
• A problem A ∈ NPC (the set NP-Complete) if
  • A ∈ NP
  • for all problems S ∈ NP , S ≤_PA
• So NPC ∈ NP
• But an open question: Is NPC ∈ P? (or does P = NP?)
• Showing a problem is in NPC
  • Show it is in NP
    • IE Show that there exists an algorithm in P which will verify a solution to the problem.
  • Reduce all problems in NP to it in polynomial time.
• The second step would be easier if we had a problem in NPC to start with,
  • Then we could reduce that problem to our problem.
  • And by the transitive property, all other problems in NP would be reduced to our problem.