Lecture 35: NP Completeness

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

COSC 311 Algorithms, Fall 2022

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Lauren Biernacki, University of Michigan

• 4:00 in SCCE A131
• Refreshments at 3:30 in SCCE C209

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Two Classes of Problems:

P: decision problems solvable in polynomial time

NP: decision problems with a polynomial time verifier

• verifier takes as input
  1. instance $X$ of a problem
  2. a certificate $C$
• returns “accept”/”reject” subject to
  • completeness if $X$ is “yes” instance, then some certificate is accepted
  • soundness if $X$ is “no” instance, then no certificate is accepted

We Showed

1. P $\subseteq$ NP: every problem in P is in NP

2. IndpendentSet (IS) is in NP

   Input: Graph $G$, number $k$

   Output: “yes” $\iff G$ has indpendnet set of size $k$

   Certificate?:

   Verification?:

Examples

NoFlow

Input:

• directed graph $G = (V, E)$, source $s$, sink $t$, all edge capacities 1
• positive integer $k$

Output:

• “yes” if $G$ does not admit a flow of value at least $k$
• “no” if $G$ does admit a flow of value at least $k$

Question. Is NoFlow in NP?

NoFlow, Again?

What if we did not know that MaxFlow can be solved in polynomial time?

• How could we infer that NoFlow is in NP?

GeneralizedChess

Input: $n \times n$ chessboard, configuartion

Output: “yes” $\iff$ player 1 can force a win

Question. Is GeneralizedChess in NP?

Boolean Formulae

• variables are Boolean variables, $x, y, z, \ldots$
• logical connectives
  • $\wedge = $ “and”
  • $\vee = $ “or”
  • $\neg = $ “not”
    • also $\bar x \equiv \neg x$

Example. $\varphi(x, y, z) = (x \wedge y) \vee (\bar y \wedge z)$.

• $\varphi(F, F, T) = $

• $\varphi(F, T, F) = $

BooleanSatisfiability

Input: a Boolean formula $\varphi(x_1, x_2, \ldots, x_n)$

Output: “yes” $\iff \varphi$ has a satisfying assignment

Question. Is BooleanSatisfiability in NP?

Reducibility in NP

Main Question. What are the hardest problems in NP?

The Hardest Problems in NP

Definition. We say that a decision problem $A$ is NP complete if for every problem $B \in $ NP, we have $B \leq_P A$

• $A$ is NP complete if every instance of every problem in NP can be reduced to solving an instance of $A$

NP and Verification

Observation. Every problem in NP has a polynomial time verifier

• suppose $A$ a problem in NP
• $\verify$ is a verifier for $A$:
  • $\verify(X, C) \mapsto $ “accept”/”reject”
• $X$ is “yes” instance $\iff$ there exists a certificate $C$ such that $\verify(X, C) = $ “accept”

• solving $A$ can be reduced to answering:

  • “Is there a certificate $C$ that is accepted by $\verify(X, C)$?”

Idea of Cook-Levin Proof

Suppose $A \in$ NP

• Given (1) verifier $\verify$ for $A$, (2) instance $X$ of $A$

• Construct: a Boolean formula $\varphi(x_1, \ldots, x_n)$ such that $\varphi$ is satisfiable $\iff$ there is a certificate $C$ accepted by $\verify(X, C)$

• certificates for $\verify(X, \cdot)$ correspond to variable assignments for $\varphi(\cdot)$

• determining if there is a certificate $C$ accepted by $\verify(X, C)$ is equivalent to determining if some assignment $x_1, \ldots, x_n$ satisfies $\varphi(x_1,\ldots,x_n)$.

Formal proof requires formal definition of algorithm (e.g., Turing machines)

Conclusion?

BooleanSatisfiability (SAT) is NP complete!

• every problem $A$ in NP satisfies $A \leq_P $ SAT
• an efficient algorithm for SAT would imply P $=$ NP

Simpler Boolean Formulae

Terminology:

• a literal is a variable or its negation: $x, \bar{x}$
• a clause is an expression of the from
  1. $(z_1 \wedge z_2 \wedge \cdots \wedge z_k)$ (conjuctive clause) where each $z_i$ is a literal, or
  2. $(z_1 \vee z_2 \vee \cdots \vee z_k)$ (disjunctive clause) where each $z_i$ is a literal
• a conjunctive normal form (CNF) expression is an expression of the form $C_1 \wedge C_2 \wedge \cdots \wedge C_\ell$ where each $C_i$ is a disjunctive clause

Observation: a CNF formula evaluates to true $\iff$ all clauses evaluate to true

3-SAT

Definition. A 3-CNF formula is a Boolean formula in conjunctive normal form such that every clause contains 3 literals.

Example.

$\varphi(w, x, y, z) = (x \vee y \vee z) \wedge (y \vee \bar z \vee w) \wedge (\bar x \vee \bar y \vee \bar w)$

3-SAT:

• Input: a 3-CNF formula $\varphi$
• Output: “yes” $\iff \varphi$ is satisfiable

3-SAT is NP-Complete

Theorem (Tseytin 1970). Any Boolean formula $\varphi$ can be efficiently (in polynomial time) transformed into a 3-CNF formula $\psi$ such that:

1. if $\varphi$ is satisfiable, then so is $\psi$
2. if $\varphi$ is not satisfiable, then neither is $\psi$

Relationships

IS is NP Complete

Theorem. IS in NP Complete.

Question. What do we need to show?

Example

$\varphi(w, x, y, z) = (x \vee y \vee z) \wedge (y \vee \bar z \vee w) \wedge (\bar x \vee \bar y \vee \bar w)$

Next Time

1. IS Completed
2. Coping with NP Completeness