COMP3010: Algorithm Theory and Design

Daniel Sutantyo, Department of Computing, Macquarie University

11.0 - P vs NP

The Plan

11.0 - P vs NP

  • Introduction to P vs NP
  • NP-Completeness and Reduction
  • Approximation Algorithms

11.0 - P vs NP

Tractable vs Intractable Problems

  • In week 3, we spent a bit of time discussing brute-force methods
    • the focus was to find overlapping subproblems to then construct a dynamic programming algorithm
    • in the examples, we often went from \(O(2^n)\) to \(O(n^2)\)
    • does this mean we can always use dynamic programming to find a polynomial-time solution to a problem?

11.0 - P vs NP

Tractable vs Intractable Problems

https://xkcd.com/399/

11.0 - P vs NP

Tractable vs Intractable Problems

  • Tractable problems are problems that we can solve using a polynomial-time algorithm, i.e. \(O(n^k)\)
  • Intractable problems are problems for which there are no known polynomial-time algorithm to solve it
    • not polynomial doesn't always mean exponential, it can be factorial, i.e. \(O(n!)\), or quasi-polynomial, i.e. \(O(n^{\log n})\)
    • you may have heard of P vs NP but there is also something called QP!

11.0 - P vs NP

Tractable vs Intractable Problems

  • In fact there is a lot of complexity classes:

11.0 - P vs NP

Tractable vs Intractable Problems

  • Although you already know the more important ones:

11.0 - P vs NP

Tractable vs Intractable Problems

  • The main complexity classes you should know:
    • logarithmic - \(O(\log n)\)
    • linear - \(O(n)\)
    • polynomial - \(O(n^k)\)
    • quasi-polynomial - \(O(n^{\log n})\)
    • subexponential - \(O(2^{n^\varepsilon})\) for a small \(\varepsilon >  0\), e.g. \(O(2^{\sqrt{n}})\)
    • exponential - \(O(2^n)\)
    • factorial - \(O(n!)\)
    • double exponential - \(O(2^{2^n})\)

11.0 - P vs NP

Tractable vs Intractable Problems

  • The main complexity classes you should know:
    • logarithmic - \(O(\log n)\)
    • linear - \(O(n)\)
    • polynomial - \(O(n^k)\)
    • quasi-polynomial - \(O(n^{\log n})\)
    • subexponential - \(O(2^{n^\varepsilon})\) for a small \(\varepsilon >  0\), e.g. \(O(2^{\sqrt{n}})\)
    • exponential - \(O(2^n)\)
    • factorial - \(O(n!)\)
    • double exponential - \(O(2^{2^n})\)

superpolynomial

11.0 - P vs NP

Tractable vs Intractable Problems

  • polynomial - \(O(n^k)\)
  • exponential - \(O(2^n)\)

11.0 - P vs NP

Tractable vs Intractable Problems

  • subexponential - \(O(2^{n^\varepsilon})\)
  • polynomial - \(O(n^k)\)
  • exponential - \(O(2^n)\)

11.0 - P vs NP

  • polynomial - \(O(n^k)\)
  • exponential - \(O(2^n)\)
  • subexponential - \(O(2^{n^\varepsilon})\)
  • can you prove that some problems can only be done in polynomial time?
    • yes
  • can you prove that some problems can only be done in subexponential time?
    • yes
  • can you prove that some problems can only be done in exponential time?
    • yes
  • i.e. we can prove that this is the best that we can do

Where We Are

11.0 - P vs NP

  • polynomial - \(O(n^k)\)
  • exponential - \(O(2^n)\)
  • subexponential - \(O(2^{n^\varepsilon})\)
  • can you prove that some problems can ONLY be done in exponential time?
  • can you prove that some problems can be done in exponential time?
  • can you prove that some problems CANNOT be done in polynomial time?
  • can you prove that some problems CANNOT be done in subexponential time?

Where We Are

11.0 - P vs NP

Where We Are

  • Let's say that the only algorithm I know that can solve a problem \(Q\) is exponential in complexity
    • is there absolutely NO WAY to do it in subexponential time, or am I just too stupid to work it out?
    • is there absolutely NO WAY to do it in polynomial time, or am I just too stupid to work it out?
  • This is the problem that scientists have:
    • they cannot show that \(Q\) can be done in polynomial time
    • but they also cannot show that \(Q\) cannot be done in polynomial time

11.0 - P vs NP

Where We Are

  • To repeat, there are problems that are definitely exponential in complexity, but, there are also a bunch of problems where we just don't know if we can do them in polynomial time or not or subexponential complexity) 
  • Let \(\text{EXP}\) be the class of problems that can be solved in exponential time
  • Let \(\text{P}\) be the class of problems that can be solved in polynomial time
    • i.e. if a problem \(Q\) can be done in polynomial time, then usually we just say that \(Q \in P\)
    • do you remember that we also defined complexity classes as sets?
    • is \(Q\) also in \(\text{EXP}\)?
  • Is \(\text{P} \subseteq \text{EXP}\) or \(\text{P} \subsetneq \text{EXP}\)?

11.0 - P vs NP

Where We Are

  • Let's say that \(Q \in \text{P}\)
    • is \(Q \in \text{EXP}\)? 
      • yes
    • is \(Q \in \text{SUBEXP}\)? 
      • \(\text{SUBEXP}\) the class of problems solvable in subexponential time
      • yes
  • Let's say that \(Q \in \text{EXP}\)
    • is \(Q \in \text{P}\)?
      • well, can you prove it?

11.0 - P vs NP

The Class NP

  • So far we mentioned the classes \(\text{P}\), \(\text{EXP}\), and \(\text{SUBEXP}\), and we know that
    • \(\text{P} \subsetneq \text{SUBEXP} \subsetneq \text{EXP}\)
  • We are going to study another class of problems called \(\text{NP}\)
    • \(\text{NP}\) does NOT stand for non-polynomial
    • this is a special class of problems for which there are no known polynomial time algorithm to solve them, but if you do have a solution, then it can be verified in polynomial time
    • in other words, I can work out if the answer is correct or not in polynomial time, but I cannot produce the answer in polynomial time

11.0 - P vs NP

The Class NP

  • Why are they so special?
    • if I can verify a solution in polynomial time, does it mean that I can produce a solution in polynomial time as well?
    • in practice, these problems are very similar to the problems in \(\text{P}\)
      • examples:
        • Shortest Path vs Longest Path
        • Euler cycle vs Hamiltonian cycle

11.0 - P vs NP

Shortest Path vs Longest Path

11.0 - P vs NP

Shortest Path vs Longest Path

11.0 - P vs NP

Shortest Path vs Longest Path

11.0 - P vs NP

Shortest Path vs Longest Path

11.0 - P vs NP

Shortest Path vs Longest Path

  • If I tell you a possible solution (and assume I know what the longest path is), is it easy to check if it's correct?

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once
  • Hamiltonian Cycle: start and end at same vertex, visit every vertex once

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once
  • Hamiltonian Cycle: start and end at same vertex, visit every vertex once

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once
  • Hamiltonian Cycle: start and end at same vertex, visit every vertex once

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once
  • Hamiltonian Cycle: start and end at same vertex, visit every vertex once

11.0 - P vs NP

Euler Cycle vs Hamiltonian Cycle

  • Euler Cycle: start and end at the same vertex, use every edge once
  • Hamiltonian Cycle: start and end at same vertex, visit every vertex once
    • if I tell you a solution, is it easy to check if it's correct?

11.0 - P vs NP

The Class NP

  • One more time, we don't know there if is a polynomial-time algorithm to solve the problems in \(\text{NP}\), but we know that we can verify a solution in \(\text{P}\)
  • Let's make sure the setting is clear to you:
    • if \(Q \in \text{NP}\), is \(Q \in \text{EXP}\)?
      • yes
    • if \(Q \in \text{P}\), is \(Q \in \text{NP}\)?
      • yes
    • if \(Q \in \text{NP}\), is \(Q \in \text{P}\)?
      • well, this is what we are wondering, are these \(\text{NP}\) problems just problems in \(\text{P}\), but we don't know how to solve them yet?

11.0 - P vs NP

The Class NP

\(\text{P}\)

\(\text{EXP}\)

11.0 - P vs NP

The Class NP

\(\text{P}\)

\(\text{NP}\)

\(\text{EXP}\)

11.0 - P vs NP

The Class NP

\(\text{P}\)

\(\text{NP} =\)

\(\text{EXP}\)

11.0 - P vs NP

The Class NP

  • The main question in the P vs NP problem is whether or not P and NP are just the same set
    • that is, is every single problem in NP just a problem in P?
    • for now, make sure that you are clear about the differences between P, EXP, and NP (and maybe SUBEXP)
    • one question you can ask yourself:
      • what do we have to do to prove that P = NP?

11.0 - P vs NP

Why Should You Care?

  • Do I think that P = NP?
    • No
  • Are there people smarter than me who thinks that P = NP?
    • Yes
  • Why should you care?
    • you should know that there are no efficient algorithm to solve NP problems, so don't try finding one
    • you should also know that two problems can be very similar, but completely different in terms of complexity

11.0 - P vs NP

Why Should You Care?

Illustration by Stefan Szeider, https://www.ac.tuwien.ac.at/people/szeider/cartoon/

11.0 - P vs NP

Why Should You Care?

Illustration by Stefan Szeider, https://www.ac.tuwien.ac.at/people/szeider/cartoon/

11.0 - P vs NP

Why Should You Care?

Illustration by Stefan Szeider, https://www.ac.tuwien.ac.at/people/szeider/cartoon/