CS 411 Fall 2025  >  Outline for November 21, 2025

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CS 411 Fall 2025
Outline
for November 21, 2025

Outline

P, NP, NP-Complete [L 11.3]

• Tractability
  • A problem specifies, for each of a number of possible legal inputs, one or more correct outputs. Each legal input an an instance of the problem.
  • An algorithm solves a problem if, for each instance, it returns a correct output.
  • When discussing this topic, we explicitly consider an integer to be a collection of digits. Its contribution to the size of the input is the number of digits. And simple integer operations like addition and comparing equality are no longer considered constant-time, but rather linear in the number of digits.
  • An algorithm is polynomial-time if its time complexity is \(O(p(n))\), for \(p\) a polynomial.
    • Equivalently, an algorithm is polynomial-time if its time complexity is \(O(n^k)\), for some constant \(k\).
  • A problem is tractable if it is solved by a polynomial-time algorithm. Other problems are intractable.
  • The theory of computational complexity concerns, among other things, which problems are tractable.
• \(\mathrm{P}\)
  • A decision problem is a problem whose answers are all “yes” or ”no”.
    • Instance that give the answers “yes” and “no” are yes-instances and no-instances, respectively.
  • \(\mathrm{P}\) is the class of all decision problems that can be solved by a polynomial-time algorithm, i.e., the class of all tractable decision problems.
• \(\mathrm{NP}\)
  • The notion of algorithm we have discussed is, more precisely, a deterministic algorithm.
  • A nondeterministic algorithm has an intial guessing stage, in which an arbitrary guess is generated. This can then be used in later processing.
    • A nondeterministic algorithm solves a decision problem if,
      • For each yes-instance, there is some guess that will result in the nondeterministic algorithm answering “yes”.
      • The nondeterministic algorithm never gives an incorrect “yes” answer.
    • Note that we do not require a nondeterministic algorithm to halt eventually. We also say it solves a decision problem even if it sometimes incorrectly answers “no”.
    • The time complexity of a nondeterministic algorithm gives its greatest best-guess running time over all yes-instances of a certain size.
  • Example: nondeterministic algorithm to solve TSP.
    • Reformulate TSP as a decision problem. Given a graph with weights on the edges, along with a budget, is there a cycle that hits every vertex whose total weight is at most the budget?
    • Nondeterministic algorithm: check the guess to see whether it is the cycle we want. If so, answer “yes”.
    • It is not known whether TSP is tractable. However, the time complexity of this nondeterministic algorithm is linear!
  • \(\mathrm{NP}\) is the class of all decision problems that can be solved by a polynomial-time nondeterministic algorithm.
  • It is easy to turn a polynomial-time deterministic algorithm into a polynomial-time nondeterministic algorithm; simply ignore the guess. So every problem in \(\mathrm{P}\) is also in \(\mathrm{NP}\).
  • Does every problem in \(\mathrm{NP}\) also lie in \(\mathrm{P}\)? No one knows. TSP is in \(\mathrm{NP}\). It is not known whether TSP lies in \(\mathrm{P}\).

\(\mathrm{P}\), \(\mathrm{NP}\), \(\mathrm{NP}\)-Complete will be continued next time.