NPC

Table of Contents

TIME(t(n)): the collection of languages that are decidable by an O(t(n)) time TM. ==> P

NTIME(t(n)): the collection of languages that are decidable by an O(t(n)) time NTM. ==> NP

Assume t(n)>n. Every t(n) time single tape NTM equals to a `\(2^{O(t(n))}\)` time single tape Deterministic TM

Assume t(n)>n. Every t(n) time multi-tape TM equals to a `\(O(t^2(n))\)` time single tape TM.

a language is in NP <=> decided by some polynomial time NTM

1 Prove clique is in NP

1.1 Proof 1:

``` V=on input <<G,k>,c>:

  1. Test whether c is a subgraph of k nodes in G
  2. Test whether G contains all edgesconnecting nodes in c
  3. if both pass accept, otherwise reject.

```

1.2 Proof 2:

``` N=on input <G,k>:

  1. non-deterministically select a subset c of k nodes in G
  2. Test whether c contains all edges connecting nodes in c
  3. if yes, accept. Otherwise reject.

```

language B is in NPC if:

  1. B is NP
  2. every A in NP is polynomial time reducible to B

Author: Hebi Li

Created: 2017-06-14 Wed 22:43

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