* Problems in P
* The class NP
* Introduction to NP-completeness

PROBLEMS IN P

The vast majority of the problems that you must have studied in an
advanced algorithms course are in P.  Many of those problems are
formulated as optimization problems.  There is a distinction, but you
can formulate decision versions of the problems and consider their
complexity following the models we have developed.

SHORTEST_PATH = { <G, s, t, k>: G is a directed graph with integer
weights on edges such that the shortest path from s to t is of length
<= k}

NETWORK_FLOW = { <G, s, t, k>: G is a directed graph with integer
capacities on edges such that the maximum flow from s to t is of value
>= k}

STRONGLY_CONNECTED = { <G>: G is a directed graph in which there is a
path between any pair of vertices}

THE CLASS NP

Two definitions: one a natural generalization of P, and the other in
terms of verifiers.

NTIME(t(n)) = {L : L is a language decided by an O(t(n)) time
nondeterministic Turing machine}

NP = U_k NTIME(n^k)

A verifier for a language L is an algorithm V, where 

L = {w: V accepts <w,c> for some string c}.

The verifier is a special TM whose time is expressed only in terms of
the length of w.

NP = Class of languages that have polynomial time verifiers.

Theorem: Both of the above definitions are equivalent.

Proof: Consider a language L.  Let V be a poly time verifier for L.
The NDTM for deciding L simply selects a candidate certificate c of
length n^k, and calls V on <w,c>.  Accept if V accepts, reject
otherwise.

Let L be decided by NDTM N.  Then for any w, there exists a branch of
length <= n^k that accepts w.  The verifier V simply interprets a
certificate as a branch of N and simulates it.  Accepts if N accepts
on that branch, rejects otherwise.

End Proof

Theorem: P is subset of NP

Trivial Proof.

What other problems are there?

A clique is a complete subgraph.

CLIQUE = {<G, k>: G is an undirected unweighted graph and has a clique
of size k}

A vertex cover is a set of vertices such that every edge is adjacent
to at least one vertex in the cover.

VERTEX_COVER = {<G, k>: G has a vertex cover of size k}

Given a bipartite graph, does there exist a 2-D matching?

3D_MATCHING

We have already shown that a t(n)-time NDTM can be simulated by a
2^{O(t(n))}-time DTM.  So we have:

NP is a subset of U_k TIME(2^{n^k}) = EXPTIME

NP-COMPLETENESS

A is poly-time many-one reducible to B if there exists a poly-time
computable function f such that

w is in A iff f(w) is in B.

Theorem: If A <=P B and B is in P, then A is in P.

A language B is NP-complete if it satisfies two conditions: