NPC part 2

Objectives

Notes

• Input/Solutions and representation
  • In the end everything we are interested in can be represented in binary.
• So is there anything in the set NPC?
  • Remember, everything in NP must map the problem in polynomial time.
  • This seems an impossible task since we don't even know if there is one problem in the set!
• Circuit Satifiability (CIRCUIT-SAT)
  • Given a circuit, can you find a set of inputs that makes the circuit true?
  • There will be a set of n inputs
  • And one way to solve this is try all 2ⁿ combinations.
    • We don't know any really good way to do this otherwise.
  • But if we have a circuit and a set of input, we can fairly easily see that evaluating that circuit for a given set of input can be done in polynomial time.
  • Therefore CIRCUIT-SAT ∈ NP
• Can we show it is in NPC?
  • Remember, this is an overview.
  • But for every problem in NPC, there is a algorithm A that can check the certificate C for correctness in Polynomial time.
  • A computation consists of
    • A set of hardware, that executes in polynomial time
    • A state or set of memory.
      • This includes input
      • variables
  • To perform a computation, we need to execute machine
    • on state 0 to produce state 1
    • and on state 1 to produce state 2
    • and so on.
  • In particular we want to execute A with C as the input.
  • This will have a polynomial number of states (Since the problem we are testing in in NP)
    • And in the end, there will be an answer in the final state if the solution is correct or not.
  • This can all be done in polynomial time.
  • We have constructed a (big) circuit in polynomial time that
    • Given an algorithm we know must exist
    • Will answer yes or know if a solution is correct
  • So if we can find an input (in polynomial time) that produces the answer "yes" in this circuit, we have an answer (in polynomial time) to any problem in the set NPC!
  • Thus every problem in NPC has been reduced in polynomial time to CIRCUIT-SAT
  • Thus CIRCUIT-SAT ∈ NPC
• After this, a large number of problems have been shown to be NP-Complete.
  • This is usually done by a reduction proof.
  • For example, SAT
    • Given a boolean expression consisting of and, or and not, can you find a input that results in a true?
    • Showing this is in NP is easy, again just checking the value of an equation.
    • You can show that CIRCUIT-SAT ≤_PSAT
      • Given an circuit, construct the equation from that circuit
      • If the equation is satifiable, the circuit is then satisifable from the same input.
      • And Finding the input for the equation provides the answer tot he input for the circuit
      • Some care needs to be used to make sure this is a polynomial construction
      • But it is doable.
    • This SAT ∈ NPC
• There is a fairly good modern overview of this problem here.
  • While he doesn't dismiss this problem.
  • He points out how we are working around the problem.
  • In some way, P=NP is perhaps a temporary detour.
  • This is worth a read/watch.
• Why is this true?
  • Heuristics, restrictions, ...
  • The traveling salesperson problem (TSP)
    • Given a complete weighted graph
    • What is the least cost path to visit each city exactly once and return to the starting city?
  • TSP ∈ NPC
  • But
    • If we can restrict it so that the triangle inequality holds
      • Given u,v,w ∈ G.V, w(u,v) ≤ w(u,v) + w(v,w)
      • This is true for distances (that don't involve mountains)
      • But not necessarily true for air fair.
    • If this is true, we can solve the TSP problem by creating a MST
    • Thus TSP become O(V+E)
  • By just selecting the lowest cost unvisited vertex, an algorithm can be derived that produces a solution that is no more than 25% more expensive than the optimal for most cases.
    • However some cases exist where it will produce the worst path.
    • Think Quicksort
  • Christofides and Serdyukov have produced an algorithm that will guarantee 1.5 times optimal.
  • And the list of improvements grows.