Lecture 34: P and NP

$ \def\verify{ {\mathrm{verify}} } $

COSC 311 Algorithms, Fall 2022

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Last Time

Two Problems:

  • Minimum Vertex Cover (MVC)
  • Maximum Indpendent Set (MaxIS)

Polynomial-time reductions between them:

  • MVC $\leq_P$ MaxIS
  • MaxIS $\leq_P$ MVC

Consequence

  • MVC can be solved efficiently $\iff$ MaxIS can be solved efficiently

Today

  1. Decision Problems
  2. The Classes P and NP

A Technicality

Objective. Understand relationships between computational problems.

Technical issue. Desired outputs for different problems can be vastly different:

  • matching
  • independent set
  • spanning tree
  • …

Convenience. Focus on decision problems:

  • output is “yes”/”no”

MVC vs VC

Minimum Vertex Cover (MVC)

  • Input: Graph $G$

  • Output: A vertex cover $C$ of smallest possible size

Vertex Cover (VC)

Input: Graph $G$, number $k$

Output:

  • “yes” if $G$ has a vertex cover of size $k$
  • “no” otherwise

MaxIS vs IS

Maximum Independent Set (MaxIS)

Input: Graph $G$

Output: an indpendent set of the largest possible size

Independent Set (IS)

Input: Graph $G$, number $k$

Output:

  • “yes” if $G$ has an indpendent set of size $k$
  • “no” otherwise

Complexity of Decision Problems

Goal. Classify (decision) problems according to their relative complexities:

  • which problems can be solved efficiently?
  • which problems cannot be solved efficiently?
  • which problems can be reduced to other problems?

The Class P

Definition. The class P consists of all decision problems that can be solved in polynomial time.

  • P = “polynomial time”

  • a problem $A$ is in P if there is an algorithm that given any instance $X$ of $A$

    • the algorithm correctly outputs “yes”/”no”
    • the running time is $O(N^c)$ for some constant $c$, $N$ = size of input

Which Problems Are In P?

The Class NP

Verifying Output

Consider IS$(G, k)$:

  • “yes” if $G$ has an indpendent set of size $k$
  • “no” if $G$ does not have an indpendent set of size $k$

Question. How could you be convinced that IS$(G, k)$ = “yes?”

NP, Informally

NP = “nondeterministic polynomial time”

Informal Definition. The class NP consists of decision problems whose solution can be verified in polynomial time.

Setup

  • $A$ is a decision problem, $X$ an instance (input) of $A$
  • If $X$ is a “yes” instance, there should be some way to convince me this is the case
  • If $X$ is a “no” instance, there should be no way to convince me $X$ is a “yes” instance

Verifier

Definition. Given a decision problem A, a verifier for A is a polynomial time algorithm $\verify(X, C)$ that takes as input

  • an instance $X$ of A, and
  • a certificate $C$ (size polynomial in size of $X$)

and returns a value “accept” or “reject,” subject to two conditions:

  1. completeness if $X$ is a “yes” instance, then there exists a certificate $C$ such that $\verify(X, C)$ returns “accept”

  2. soundness if $X$ is a “no” instance, then for every certificate $C$, $\verify(X, C)$ returns “reject”

A Verifier for IS

Consider IS$(G, k)$:

  • “yes” if $G$ has an indpendent set of size $k$
  • “no” if $G$ does not have an indpendent set of size $k$

What is a verifier for IS?

  • what should be the certificate?
  • how do we verify a certificate?

NP, Formally

Definition. The class NP consists of all decision problems that admit a polynomial time verifier.

  • By previous example, IS is in NP

Conceptually NP can be thought of the class of puzzles

  • a puzzle may be hard to solve
  • you can easily verify if you (or someone else) solved the puzzle

P vs NP

Open Question. Is there any problem in NP that is not in P?

Informal statement. Are there problems that are hard to solve, but whose solutions are easy to verify?

  • one of deepest mathematical challenges of our time

Activity

Which of the following problems are in NP:

  1. BipartiteMatching$(G, k)$
  2. NoFlow$(G, k)$
  3. GeneralizedChess$(n, C)$
  4. BooleanSatisfiability$(\varphi(x_1, x_2, \ldots,x_n))$

Bipartite Matching

Question. Is BipartiteMatching in NP?

More Generally

If a problem $A$ is in P, then $A$ is in NP:

  • P $\subseteq$ NP

Why?

NoFlow

Question. Is NoFlow in NP?

Observation

Even if did not know about the Ford-Fulkerson MaxFlow algorithm, we could still identify NoFlow is in NP.

How?

GeneralizedChess

Question. Is GeneralizedChess in NP?

Chess Remarks

A feature of chess games:

  • a game may last exponentially many rounds in the size of the board
  • therefore: winning strategy might require exponential time to describe/verify

Fact. GeneralizedChess requires exponential time to solve (in $n$).

Consequence. Showing GeneralizedChess is in NP would imply that P $\neq$ NP.

Boolean Satisfiability

Question. Is BooleanSatisfiability in NP?

Next Time

  • NP Completeness: characterizing the “hardest” problems in NP