Download Algorithm Design & Analysis: NP-Complete Problems & Approximation Algorithms (CS 410/584) and more Study notes Computer Science in PDF only on Docsity!
2009 David Maier
Algorithm Design & Analysis
Exhibiting an NP-Complete Problem
Need to show one
problem Q NP-complete
via a direct prvia a direct proof. f
That is, show how to
reduce every other
problem in NP to it
(in )
Oth bl th
Q
Lecture Notes 10 David Maier (^1)
Other problems can then
be shown NP-hard by
reducing Q to them
Q1 Q
NP
Algorithm Design & Analysis
Initial Problem
There are different initial problems in
the literature.
They are all based on encoding the
computation of polynomial-time
algorithms.
Rely on the “computational history” of
a poly-time algorithm is polynomial in
th i f th i t
Lecture Notes 10 David Maier (^2)
the size of the input.
2009 David Maier
Algorithm Design & Analysis
Configurations of an Algorithm
Suppose A is a verification algorithm
with time complexity bounded by c⋅n k
- On each step of the underlying machineO h t f th d l i hi model, it can write at most d bits
- So A uses at most d⋅c⋅n k^ bits of memory in any computation with an input of length
- If there are e bits of other state ( i ) h
Lecture Notes 10 David Maier (^3)
(registers, program counter), then one configuration of the machine needs
Algorithm Design & Analysis
All Configurations
If we represent the configuration at
each step in one computation, need
at most c⋅n k(e + d( ⋅c⋅n k) =)
There is a bit more. Have to have
- input x, |x| =
- certificate y |y|
Lecture Notes 10 David Maier (^4)
certificate y, |y|
- instructions of A – size is
2009 David Maier
Algorithm Design & Analysis
Example
Is there a satisfying assignment for
this circuit?
B
A
Lecture Notes 10 David Maier (^7)
D
C
Algorithm Design & Analysis
Circuit Satisfiability is in NP
Assume an instance is represented as a
directed graph
- edges as wiresd i
- nodes are labeled with the kind of gate
- number of inputs can grow linearly with representation: O( ) to enumerate inputs
Problem is in NP
Certificate = assignment of 0 or 1 N d t h k
Lecture Notes 10 David Maier (^8)
Need to check:
- Each gate behaves appropriately given its input and output
- output of whole circuit is
2009 David Maier
Algorithm Design & Analysis
Circuit Sat is NP-Hard
Given any problem Q in NP
Need a polynomial-time function f that maps a string x to a circuit C such that x is in Qa string x to a circuit C, such that x is in Q if and only if
Must have a verification algorithm
A(x, y) for Q with running time T(n)
in
using certificates bounded by S(n) in
Lecture Notes 10 David Maier (^9)
Algorithm Design & Analysis
Configurations of Computation of A
A configuration is essentially wires
A PC Aux state x y Memory
Initial configuration
Lecture Notes 10 David Maier (^10)
A PC Aux state x y Memory
2009 David Maier
Algorithm Design & Analysis
Formula Satisfiability
Input: Boolean formula using:
PlPl us parens and Boolean variablesd B l i bl
Is there an assignment of true, false to
the variables that makes the formulas
true
((x ∧ ¬y) ∨ z) ↔ (w → x)
Lecture Notes 10 David Maier (^13)
Algorithm Design & Analysis Reducing Circuit Sat to Formula Sat Might seem like we can just work backwards from the output of the circuit, and write down formula
B
A
B
D
C
Lecture Notes 10 David Maier (^14)
2009 David Maier
Algorithm Design & Analysis Instead, Variable per Wire, Sub-formula per Gate Plus, output wire is true
B
A w
w
w w7 w
D
C
w
w w
(w1 ↔ (A ∧ B)) (w2 ↔ (C ∧ D)) (w3 ↔ ¬w1) (w4 ↔ ¬w2)
w
Lecture Notes 10 David Maier (^15)
(w5 ↔ (w1 ∨ w2)) (w6 ↔ (w3 ∨ w4)) (w7 ↔ (w5 ∧ w6)) (w8↔ ¬w7) w
Algorithm Design & Analysis
Properties of Resulting Formula
1. Linear in the size of the circuit,
hence
2. Can be constructed from the
circuit in polynomial time
3. Is satisfiable if and only if the
original
Polynomial-time reduction from circuit
Lecture Notes 10 David Maier (^16)
y
sat to formula sat
So formula sat is NP-Hard Is formula sat in NP?
2009 David Maier
Algorithm Design & Analysis
Each Component Formula Has 3 Literals
Can convert each to 3-CNF
How? List cases that are false
(b2 ↔ (x ∧ b1)) F T T (¬b2 ∧ x ∧ b1) ∨ T F T (b2 ∧ ¬x ∧ b1) ∨ T T F (b2 ∧ x ∧ ¬b1) ∨ T F F (b2 ∧ ¬x ∧ ¬b1)
Then use DeMorgan’s Laws
Lecture Notes 10 David Maier (^19)
Then use DeMorgan s Laws
¬(x ∧ y) = ¬x ∨ ¬y ¬(x ∨ y) = ¬x ∧ ¬y
Algorithm Design & Analysis
Result is in 3-CNF
(b2 ∨ ¬x ∨ ¬b1) ∧ (¬b2 ∨ x ∨ ¬b1) ∧ ((¬¬b2b2 ∨ ¬∨ ¬xx ∨∨ b1)b1) ∧∧ (¬b2 ∨ x ∨ b1)
What about 2-literal clauses?
(x ∨ ¬z) Add a new variable p (x ∨ ¬z ∨ p) ∧ (x ∨ ¬z ∨ ¬p)
Wh t b t 1 lit l l?
Lecture Notes 10 David Maier (^20)
What about 1-literal clauses?
2009 David Maier
Algorithm Design & Analysis
Things to Check
- Translation can be done in polynomial
time
The 3-SAT problem is NP-complete
Lecture Notes 10 David Maier (^21)
Algorithm Design & Analysis
Branching Out from Logic
Graph problems
Clique of an undirected graph G = (N, E)q g p
Set of nodes M such that (m1, m2) in E for every
Does this graph have a clique of 4
A B C
Lecture Notes 10 David Maier (^22)
nodes? Does it have a clique of 5 nodes?
D E F
2009 David Maier
Algorithm Design & Analysis Equivalence
Suppose formula is satisfiable
Must make at least one literal true in each clause These literals must be compatible, so there are
So there must be a
Suppose there is a clique of size k
Must have exactly one node corresponding to each clause (why?) T ll hi h lit l t k t h
Lecture Notes 10 David Maier (^25)
Tells us which literals to make true, hence which variables to make true or false
Must be compatible literals
Algorithm Design & Analysis
How Big is This Graph?
Number of nodes =
Number of edges
So we can reduce 3-SAT to CLIQUE in
polynomial time
Lecture Notes 10 David Maier (^26)
Is CLIQUE in NP?
2009 David Maier
Algorithm Design & Analysis
Vertex Cover
Given an undirected graph G = (V, E)
and integer j, is there a set of j
n des P such that f r each ed enodes P such that for each edge
{v, w} in E,
Does this graph have a vertex cover
A B C
Lecture Notes 10 David Maier (^27)
of 2 nodes? Does it have a vertex cover of 1 node?
D E F
Algorithm Design & Analysis
Complement Graph
This last graph is the complement of
the example for clique
HHas exactly the missing edges tl th i i d
A B C
Lecture Notes 10 David Maier (^28)
D E F
2009 David Maier
Algorithm Design & Analysis
Examples
S = {3104, 11015, 22010, 117, 8506,
target t = 22324
Second example: same S
target t = 2324
Lecture Notes 10 David Maier (^31)
Algorithm Design & Analysis
Reduction from 3-SAT
Use decimal numbers, divide into
positions
- Some for variables: Make sure
- Some for clauses: Make sure each
has
Lecture Notes 10 David Maier (^32)
2009 David Maier
Algorithm Design & Analysis
Approximation Algorithms for NP-Complete Problems
If Q is NP-complete, then what can we
do?
- Limit ourselves to small inputsLimit ourselves to small inputs
- Find special cases that can be
solved in polynomial time
- Find poly-time algorithms that give
good approximations
For example for 3 SAT is there a poly time
Lecture Notes 10 David Maier (^33)
For example, for 3-SAT, is there a poly-time algorithm that is guaranteed to find an assignment (if it exists) that
Algorithm Design & Analysis
Approximating Vertex Cover
If we could find the minimum size
vertex cover, M, for a graph G, we
c uld answer the questi n if G has acould answer the question if G has a
vertex cover of size k.
Will show a poly-time algorithm that
finds a cover C of G such that |C|
Lecture Notes 10 David Maier (^34)
2009 David Maier
Algorithm Design & Analysis
Approximation Bound
Why is C within a factor of two of
minimum M?
L t F bLet F be chosen edges. So C = h d S C
- No two edges in F have a node in common. Suppose it has {u, v} and {u, w}. Assume {u, v} chosen first. Then
- So, no cover can have a node x that covers
Lecture Notes 10 David Maier (^37)
- M must have at least one node for each edge in F, so |M| ≥ |F|, |C| =
