Lecture 20 - NP-complete problems and reductions I

Date Pre-lecture slides Post-lecture scribbles Lecture recording
November 07 2023 Lecture 20 - NP-complete problems and reductions I Lecture 20 - NP-complete problems and reductions I Lecture 20 - NP-complete problems and reductions I




For a set of boolean variables $x_1,x_2,…,x_n$

(i.e. $x_3 \lor \neg x_7 \lor x_8$)

(i.e. $(x_1 \lor x_4) \land (x_3 \lor \neg x_7 \lor x_8) \land x_5$)

Problem: SAT

For a given CNF (3CNF for 3SAT) formula $\phi$, is there a truth assignment of the variables such that $\phi$ evaluates to true?


Reducing SAT to 3SAT:

To reduce from an instance of SAT to an instance of 3SAT all clauses must be made to have exactly 3 literals. To do this dummy variables are introduced such that the original formula is satisfiable if and only if the formula with the dummy variables is satisfiable. Short clauses are padded with dummy variables to have 3 literals, long clauses are broken apart using dummy variables to get 3 literals. Proof in Prof. Har-Peled’s lectures

Complexity Classes


Problems with no known P solution:


An algorithm $C(\cdot,\cdot)$ is a certifier for a problem $X$ if

The string $s$ is the problem instance (i.e. a particular graph for vertex cover problem, a CNF formula for SAT problem). The string $t$ is called the certificate or proof for $s$.

Efficient Certifier:

A certifier $C$ is an efficient certifier for problem $X$ if there is a polynomial $p(\cdot)$ such that


Cook-Levin Theorem


A problem $X$ is NP-Hard if for any $Y \in$ NP, $Y \leq_P X$


A problem $X$ is NP-Complete if $X$ is both NP and NP-Hard


Suppose $X$ is NP-Complete. Then $X$ can be solved in polynomial time if and only if P=NP

Theorem (Cook-Levin):

SAT is NP-Complete

SAT $\leq_P X$ implies that every NP problem $Y\leq_P X$. $Y\leq_P$ SAT (Cook-Levin) and SAT $\leq_P X$ implies $Y\leq_P X$.

Example NP-Hard reductions in the lecture slides

Reductions example

Your very own, Mr. Kevin Lim, did an animation of a classic 374 reduction:

Additional Resources


Nicholas Bampton