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Approximation Algorithms

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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 ... – PowerPoint PPT presentation

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Title: Approximation Algorithms


1
Approximation Algorithms
  • Lecture for CS 302

2
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?

3
VERTEX-COVER
  • Given a graph, G, return the smallest set of
    vertices such that all edges have an end point in
    the set

4
HAMILTONIAN PATH
  • Given a graph, G, find a path that visits every
    vertex exactly once
  • Alt version Find the path with the minimum weight

5
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)

6
Applications
7
Applications
  • SAT is used to verify circuit design
  • NEED MORE EXAMPLES. WANT PICTURES

8
Dilemma
  • NP problems need solutions in real-life
  • We only know exponential algorithms
  • What do we do?

9
Accuracy
  • NP problems are often optimization problems
  • Its hard to find the EXACT answer
  • Maybe we just want to know our answer is close to
    the exact answer?

10
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

11
k-approximation
  • S is the approx. answer, OPT is optimal
  • Maximization
  • kOPT S OPT
  • Minimization
  • OPT S kOPT

12
Approximate Vertex-Cover
  • Let S be the cover
  • Pick an edge (u,v) in the graph
  • Add its end-points u and v to S
  • Remove any edge that neighbors u or v
  • Repeat until there are no edges left

13
Approximate Vertex-Cover
  • OPT must cover every edge so either u or v must
    be in the cover gt OPT gt ½S
  • gt 2OPT S
  • We have a 2-approximation

14
Traveling Salesperson Problem
  • Given a graph, find a minimum weight hamiltonian
    path
  • There is a 2-approximation based on MINIMUM
    SPANNING TREES

15
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

16
Finding a MST
  • Finding a MST can be done in polynomial time
    using PRIMS ALGORITHM or KRUSKALS ALGORITHM
  • Both are greedy algorithms
  • Details can be found on Wikipedia

17
MST vs HAMPATH
  • Any HAMPATH becomes a Spanning Tree by removing
    an edge
  • cost(MST) cost(min-HAMPATH)

18
Approximate TSP
  • Given a complete graph G
  • Compute Gs MST, M
  • The tour is the pre-order traversal of M
  • This is a 2-approximation

19
Approximating 3-SAT
  • f is a Boolean formula in 3-CNF form if
  • Whats the optimization version of 3-SAT?
  • Satisfy as many clauses as you can

20
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

21
Approximating 3-SAT
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 dont satisfy the clause is if we
select the last assignment. This happens only
1/8th of the time.
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