Hamiltonian Path I

Hamiltonian Path I

For some problems it is difficult to find an algorithm in P

We first discuss an example of finding a Hamiltonian Path

Definition: for a given directed graph find a path going through all nodes once

Fig 7.17

Hamiltonian Path II

s t

HAMPATH = {hG , s, ti | G : directed, a Hamiltonian path from s to t}

A brute-force way: checking all possible paths But the number is exponential

Polynomial verification

Hamiltonian Path III

This is an example where verification is easier than determination

Compositeness I

We discuss another example where verification is easier than determination

An integer is composite if

x = pq, p > 1, q > 1 Given x , difficult to find p, q

Given x , p, q easily verify x = pq or not

Not polynomial verifiable I

Some problems are difficult so even a polynomial verifier cannot be easily obtained

HAMPATH: given hG , s, ti no Hamiltonian path from s to t

Verification may still be difficult

Given s and t it seems we still need to check all paths

Verifier I

Definition: an algorithm V is a verifier of a language A if

A = {w | V accepts hw , ci for some strings c}

Example: compositeness. V accepts hw , ci = hx, pi, where p is a divisor Example: Hamiltonian path. V accepts

hw , ci = hhG , s, ti, a path from s to ti

Verifier II

Definition: a polynomial verifier if it takes polynomial time of |w |

A: polynomially verifiable if ∃ a polynomial verifier Note that we measure time on |w | without

considering |c|

For a polynomial verifier, |c| should be in polynomial of |w |

Otherwise, reading |c| already non-polynomial

NP I

NP is a class of languages

Definition: a language ∈ NP if it has a polynomial verifier

We will prove that this definition is equivalent to that the language is decided by nondeterministic polynomial TM

This is where the name comes from Some use this as the definition

Note that for nondeterministic TM the running time

NP II

Definition:

NTIME(t(n))

={L | L decided by O(t(n)) nondeterministic TM}

NP = ∪_kNTIME(n^k)

NTM for HAMPATH I

A list p₁· · · p_m is nondeterministically chosen For each list:

1 Check repetitions

2 Check s = p₁; t = p_m

3 Check that for i = 1 . . . m − 1, (p_i, p_{i +1}) is an edge of G

Cost on each list is polynomial:

repetitions: O(m²) s = p , t = p : O(m)