Transcript Approximation Algorithms Lecture for CS 302 What is a NP problem?  Given an instance of the problem, V, and a ‘certificate’, C, we.

Approximation Algorithms
Lecture for CS 302
What is a NP problem?

Given an instance of the problem, V, and a
‘certificate’, C, we can verify V is in the
language in polynomial time

All problems in P are NP problems
–
Why?
VERTEX-COVER

Given a graph, G, return the smallest set of
vertices such that all edges have an end
point in the set
HAMILTONIAN PATH


Given a graph, G, find a path that visits every
vertex exactly once
Alt version: Find the path with the minimum
weight
What is NP-Complete?

A problem is NP-Complete if:
–
–

It is in NP
Every other NP problem has a polynomial time
reduction to this problem
NP-Complete problems:
–
–
–
–
3-SAT
VERTEX-COVER
CLIQUE
HAMILTONIAN-PATH (HAMPATH)
Applications
Applications


SAT is used to verify circuit design
NEED MORE EXAMPLES. WANT
PICTURES
Dilemma



NP problems need solutions in real-life
We only know exponential algorithms
What do we do?
Accuracy



NP problems are often optimization problems
It’s hard to find the EXACT answer
Maybe we just want to know our answer is
close to the exact answer?
Approximation Algorithms




Can be created for optimization problems
The exact answer for an instance is OPT
The approximate answer will never be far
from OPT
We CANNOT approximate decision problems
k-approximation


S is the approx. answer, OPT is optimal
Maximization
–

k*OPT ≤ S ≤ OPT
Minimization
–
OPT ≤ S ≤ k*OPT
Approximate Vertex-Cover





Let S be the cover
Pick an edge (u,v) in the graph
Add it’s end-points u and v to S
Remove any edge that neighbors u or v
Repeat until there are no edges left
Approximate Vertex-Cover



OPT must cover every edge so either u or v
must be in the cover => OPT > ½|S|
=> 2*OPT ≥ |S|
We have a 2-approximation
Traveling Salesperson Problem

Given a graph, find a minimum weight
hamiltonian path

There is a 2-approximation based on
MINIMUM SPANNING TREES
Minimum Spanning Tree


Given a graph, G, a Spanning Tree of G is a
subgraph with no cycles that connects every
vertex together
A MST is a Spanning Tree with minimal
weight
Finding a MST



Finding a MST can be done in polynomial
time using PRIM’S ALGORITHM or
KRUSKAL’S ALGORITHM
Both are greedy algorithms
Details can be found on Wikipedia
MST vs HAMPATH


Any HAMPATH becomes a Spanning Tree by
removing an edge
cost(MST) ≤ cost(min-HAMPATH)
Approximate TSP

Given a complete graph G
Compute G’s MST, M
The tour is the pre-order traversal of M

This is a 2-approximation


Approximating 3-SAT

f is a Boolean formula in 3-CNF form if:
f  ( x1  x2  x3 )(x2  x4  x5 )...


What’s the optimization version of 3-SAT?
Satisfy as many clauses as you can
Approximating 3-SAT

Algorithm:
–
–
For each variable xi assign True with probability
½, False with probability ½
This satisfies 7/8ths of the clauses in expectation
Approximating 3-SAT
x1  x2  x3
x1 x2 x3
T
T
T
T
T
T
T
F
T
T
F
T
T
T
F
F
T
F
T
T
T
F
T
F
T
F
F
T
F
F
F
F
The only way we don’t
satisfy the clause is if we
select the last
assignment. This
happens only 1/8th of the
time.