More Polynomial Reduction

$All~{}NP\leq_p Circuit-SAT\leq_p 3SAT\leq IS\leq VC$ $IS\leq_p Clique$ $3SAT\leq_p Directed~{}HC\leq_p Hamiltonian~{}Cycle(HC)\leq_p TSP$ $HC\leq_p Hamiltonian~{}Path$

Circuit

Gates

• We can construct circuits from gates

e.g.

Theorm: A polynomial time alg. can be simlulated with a series of circuits $C_1, C_2, \cdots, C_n, \cdots,$, where $C_n$ correspondds to the alg. running on isntances of size $n$

Proof:

In total $P(n)$ bits get involved in the computation

• We can convert any algorithm to a circuit consisted of gates given a certain input bits

Circuit-SAT Example

• Is the circuit satisfiable by some truth assignments of the unkown inputs?

Consider:

Let $x$ be an instance of $X$. $|x|=n_i$. let the maximum size of the proof instances be $n_2$, $n_2$ is a polynomial of $n$. Thus, $V$ is simulated by a circuit.

Therefore we can say that

Theorem: Let $X\in NP$. Then $X\leq_p Circuit-SAT$ Corollary: Circuit-SAT is NPC

Theorem: Circuit-SAT $\leq_p$ 3SAT

Proof:

Cook-Levin Theorem: 3-SAT $\in$ NPC

Set-Cover Problem

$S=\{1, \cdots, n\}$ $S_i \subset S, i=1,\cdots m$ to find min numbe of subset to cover $S$ Can we find $k$ subsets to cover $S$?

Set-Packing

Find max number of subsets that do not overlap