Approximation Algorithms

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Transcript Approximation Algorithms

What is a NP problem?

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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?
What is NP-Complete?
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A problem is NP-Complete if:
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It is in NP
Every other NP problem has a polynomial time
reduction to this problem
NP-Complete problems:
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3-SAT
VERTEX-COVER
CLIQUE
HAMILTONIAN-PATH (HAMPATH)
Dilemma
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NP problems need solutions in real-life
We only know exponential algorithms
What do we do?
A Solution
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There are many important NP-Complete
problems
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There is no fast solution !
But we want the answer …
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If the input is small use backtrack.
Isolate the problem into P-problems !
Find the Near-Optimal solution in
polynomial time.
5
Accuracy
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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
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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
Performance ratios
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We are going to find a Near-Optimal
solution for a given problem.
We assume two hypothesis :
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Each potential solution has a positive cost.
The problem may be either a maximization
or a minimization problem on the cost.
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Performance ratios …
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If for any input of size n, the cost C of
the solution produced by the algorithm
is within a factor of ρ(n) of the cost C*
of an optimal solution:
Max ( C/C* , C*/C ) ≤ ρ(n)
We call this algorithm as an ρ(n)approximation algorithm.
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Performance ratios …
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In Maximization problems:
C*/ρ(n) ≤ C ≤ C*
In Minimization Problems:
C* ≤ C ≤ ρ(n)C*
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ρ(n) is never less than 1.
A 1-approximation algorithm is the optimal
solution.
The goal is to find a polynomial-time
approximation algorithm with small constant
approximation ratios.
Approximation scheme
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Approximation scheme is an
approximation algorithm that takes Є>0 as an
input such that for any fixed Є>0 the scheme
is (1+Є)-approximation algorithm.
Polynomial-time approximation scheme
is such algorithm that runs in time polynomial
in the size of input.

As the Є decreases the running time of the
algorithm can increase rapidly:
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For example it might be O(n2/Є)
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Approximation scheme
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We have Fully Polynomial-time
approximation scheme when its
running time is polynomial not only in n
but also in 1/Є
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For example it could be O((1/Є)3n2)
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Some examples:
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Vertex cover problem.
Traveling salesman problem.
Set cover problem.
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VERTEX-COVER
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Given a graph, G, return the smallest set
of vertices such that all edges have an end
point in the set
The vertex-cover problem
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A vertex-cover of an undirected graph G
is a subset of its vertices such that it
includes at least one end of each edge.
The problem is to find minimum size of
vertex-cover of the given graph.
This problem is an NP-Complete
problem.
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The vertex-cover problem …
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Finding the optimal solution is hard (it’s
NP!) but finding a near-optimal solution
is easy.
There is an 2-approximation algorithm:
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It returns a vertex-cover not more than
twice of the size optimal solution.
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The vertex-cover problem …
APPROX-VERTEX-COVER(G)
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2
3
4
5
6
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C←Ø
E′ ← E[G]
while E′ ≠ Ø
do let (u, v) be an arbitrary edge of E′
C ← C U {u, v}
remove every edge in E′ incident on u or v
return C
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The vertex-cover problem …
Near Optimal
size=6
Optimal
Size=3
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The vertex-cover problem …
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This is a polynomial-time
2-aproximation algorithm. (Why?)
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Because:
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Optimal
APPROX-VERTEX-COVER is O(V+E)
|C*| ≥ |A|
Selected Edges
|C| = 2|A|
|C| ≤ 2|C*|
Selected Vertices
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Minimum Spanning Tree
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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
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Finding a MST can be done in polynomial
time using PRIM’S ALGORITHM or
KRUSKAL’S ALGORITHM
Both are greedy algorithms
HAMILTONIAN CYCLE
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Given a graph, G, find a cycle that visits
every vertex exactly once
TSP version: Find the path with the
minimum weight
MST vs HAM-CYCLE
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Any HAM-CYCLE becomes a Spanning
Tree by removing an edge
cost(MST) ≤ cost(min-HAM-CYCLE)
Traveling salesman problem
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Given an undirected complete weighted
graph G we are to find a minimum cost
Hamiltonian cycle.
Satisfying triangle inequality or not this
problem is NP-Complete.
The problem is called Euclidean TSP.
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Traveling salesman problem
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Near Optimal solution
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Faster
Easier to implement.
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Euclidian Traveling Salesman Problem
APPROX-TSP-TOUR(G, W)
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2
3
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select a vertex r Є V[G] to be root.
compute a MST for G from root r using Prim Alg.
L=list of vertices in preorder walk of that MST.
return the Hamiltonian cycle H in the order L.
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Euclidian Traveling Salesman Problem
MST
root
Pre-Order walk
Hamiltonian
Cycle
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Traveling salesman problem
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This is polynomial-time 2-approximation
algorithm. (Why?)
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Because:
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Pre-order
Solution
APPROX-TSP-TOUR is O(V2)
C(MST) ≤ C(H*)
C(W)=2C(MST)
C(W)≤2C(H*)
C(H)≤C(W)
C(H)≤2C(H*)
Optimal
H*: optimal soln
W: Preorder walk
H: approx soln &
triangle inequality
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EULER CYCLE
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1.
2.
Given a graph, G, find a cycle
that visits every edge exactly
once
Necessary & Sufficient
Conditions: G is connected and
every vertex is even degree.
Algorithm (O(n2)):
Repeatedly use DFS to find and
remove a cycle from G
Merge all the cycles into one
cycle.
Min-Weight Matching
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Given a complete weighted graph of even
nodes, G, find a perfect matching of
minimum total weight
Algorithm (O(n3)): Formulated as a linear
programming problem, but solved using
a special algorithm.
Euclidian Traveling Salesman Problem
APPROX-TSP-TOUR2(G, c)
1 Select a vertex r Є V[G] to be root.
2
Compute a MST T for G from root r using Prim Alg.
3
Find a minimal-weight matching M for vertices of
odd degree in T.
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Find an Euler cycle C in G’ = (V, T U M).
5 L=list of vertices in preorder walk of C.
6 return the Hamiltonian cycle H in the order L.
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Euclidian Traveling Salesman Problem
MST
Min Matching
Euler Cycle
HAM-Cycle
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Time Complexity
APPROX-TSP-TOUR2(G, c)
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Select a vertex r Є V[G] to be root.
Compute a MST T for G from root r using
Prim Alg.
Find a minimal-weight matching M for
vertices of odd degree in T.
Find an Euler cycle A in G’ = (V, T U M).
L=list of vertices in preorder walk of A.
return the Hamiltonian cycle H in order L.
O(1)
O(n lg n)
O(n3)
O(n2)
O(n)
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Traveling salesman problem
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This is polynomial-time 3/2-approximation
algorithm. (Why?)
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Because:
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Min match
Euler cycle
Solution
APPROX-TSP-TOUR2 is O(n3)
C(MST) ≤ C(H*) Optimal
C(M) ≤ 0.5C(H*)
C(A) = C(MST)+C(M)
C(H) ≤ C(A)
C(H) ≤ 1.5C(H*)
H*: optimal soln
M: min matching
A: Euler cycle
H: approx soln &
triangle inequality
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Proof of C(M)≤ 0.5C(H*)
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Let optimal tour be H*: j1…i1j2…i2j3…i2m
{i1,i2,…,i2m}: the set of odd degree vertices in T.
Define 2 matchings: M1={[i1,i2],[i3,i4],…,[i2m-1,i2m]}
M2={[i2,i3],[i4,i5],…,[i2m,i1]}
M is min matching: C(M)  C(M1) and
C(M)  C(M2)
By triangle inequality:
C(H*)  C(M1) + C(M2)
 2 C(M)
 C(M) 1/2 C(H*)
TSP In General
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Theorem: If P ≠ NP, then for any constant ρ>1, there
is no polynomial time ρ-approximation algorithm.
Proof: If we have a polynomial time ρ-approximation
algorithm for TSP, we can find a tour of cost ρH*.
 c(u,w) = if ((u,w) in E) then 1 else ρ|V|+1
 The optimal cost H* of a TSP tour is |V|.
 G has a Ham-cycle iff TSP has a tour of cost  ρ|V|.
 If a TSP tour has one edge of cost ρ|V|+1, then the
total cost is (ρ|V|+1)+|V|-1>ρ|V|
Selected edge not in E
Rest of edges
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The Set-Cover Problem
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Instance (X, F) :
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X : a finite set of elements.
F : family of subsets of X.
Solution C : subset of F that includes all
the members of X.
Set-Cover is in NP
Set-Cover is NP-hard, as it’s a
generalization of vertex-cover problem.
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An example: |X| = 12, |F| = 6
Minimal
Covering set
size=3
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A Greedy Algorithm
GREEDY-SET-COVER(X,F)
1M←X
2C←Ø
3 while M ≠ Ø do
4
select S Є F that maximizes |S ‫ ח‬M|
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M←M–S
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C ← C U {S}
7 return C
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Not optimal …
1st chose
3rd chose
2nd chose
Greedy
Covering set
size=4
4th chose
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Set-Cover …
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This greedy algorithm is polynomialtime ρ(n)-approximation algorithm
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ρ(n)=lg(n)
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The bin packing problem
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n items a1, a2, …, an, 0 ai  1, 1  i  n, to
determine the minimum number of bins of unit
capacity to accommodate all n items.
E.g. n = 5, {0.3, 0.5, 0.8, 0.2 0.4}
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The bin packing problem is NP-hard.
APPROXIMATE BIN PACKING
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Problem: fill in objects each of size<= 1, in
minimum number of bins (optimal) each of
size=1 (NP-complete).
Online problem: do not have access to the full set:
incremental;
Offline problem: can order the set before starting.
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Theorem: No online algorithm can do better than
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4/3 of the optimal #bins, for any given input set.
NEXT-FIT ONLINE BIN-PACKING
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If the current item fits in the current bin put it
there, otherwise move on to the next bin.
Linear time with respect to #items - O(n), for n
items.
Theorem: Suppose, M optimum number of bins
are needed for an input. Next-fit never needs
more than 2M bins.
Proof: Content(Bj) + Content(Bj+1) >1, So,
Wastage(Bj) + Wastage(Bj+1)<2-1, Average
wastage<0.5, less than half space is wasted,
so, should not need more than 2M bins.
FIRST-FIT ONLINE BIN-PACKING
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Scan the existing bins, starting from the first bin,
to find the place for the next item, if none exists
create a new bin. O(N2) naïve, O(N logN) possible,
for N items.
Obviously cannot need more than 2M bins!
Wastes less than Next-fit.
Theorem: Never needs more than Ceiling(1.7M).
Proof: too complicated.
For random (Gaussian) input sequence, it takes
2% more than optimal, observed empirically.
Great!
BEST-FIT ONLINE BIN-PACKING
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Scan to find the tightest spot for each item
(reduce wastage even further than the previous
algorithms), if none exists create a new bin.
Does not improve over First-Fit in worst case in
optimality, but does not take more worst-case
time either! Easy to code.
OFFLINE BIN-PACKING
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Create a non-increasing order (larger to smaller) of
items first and then apply some of the same algorithms
as before.
Theorem: If M is optimum #bins, then First-fit-offline
will not take more than M + (1/3)M #bins.
Polynomial-Time
Approximation Schemes
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A problem L has a fully polynomial-time approximation
scheme (FPTAS) if it has a polynomial-time (in n and 1/ε)
(1+ε)-approximation algorithm, for any fixed ε >0.
0/1 Knapsack has a FPTAS, with a running time that is
O(n3 / ε).
Knapsack Problem
Knapsack problem.
Given n objects and a "knapsack."
we'll assume wi  W
Item i has value vi > 0 and weighs wi > 0.
Knapsack can carry weight up to W.
Goal: fill knapsack so as to maximize total value.
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Ex: { 3, 4 } has value 40.
W = 11
Item
Value
Weight
1
1
1
2
6
2
3
18
5
4
22
6
5
28
7
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Knapsack is NP-Complete
KNAPSACK: Given a finite set X, nonnegative weights wi, nonnegative
values vi, a weight limit W, and a target value V, is there a subset S  X
such that:
 wi  W
iS
 vi
 V
iS
SUBSET-SUM: Given a finite set X, nonnegative values ui, and an integer
U, is there a subset S X whose elements sum to exactly U?
Claim. SUBSET-SUM  P KNAPSACK.
Pf. Given instance (u1, …, un, U) of SUBSET-SUM, create KNAPSACK
instance:
vi  wi  ui
V  W U
 ui  U
iS
 ui
 U
iS
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Knapsack Problem: Dynamic Programming 1
Def. OPT(i, w) = max value subset of items 1,..., i with weight limit w.
Case 1: OPT does not select item i.
– OPT selects best of 1, …, i–1 using up to weight limit w
Case 2: OPT selects item i.
– new weight limit = w – wi
– OPT selects best of 1, …, i–1 using up to weight limit w – wi


 0
if i  0

OPT(i, w)   OPT(i 1, w)
if w i  w
 max OPT(i 1, w), v  OPT(i 1, w  w ) otherwise


i
i 
Running time. O(n W).
W = weight limit.

Not polynomial in input size!
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
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Knapsack Problem: Dynamic Programming II
Def. OPT(i, v) = min weight subset of items 1, …, i that yields value
exactly v.
Case 1: OPT does not select item i.
– OPT selects best of 1, …, i-1 that achieves exactly value v
Case 2: OPT selects item i.
– consumes weight wi, new value needed = v – vi
– OPT selects best of 1, …, i-1 that achieves exactly value v


 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 
if v  0
if i  0, v > 0
if v i  v
otherwise

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Knapsack: FPTAS
 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 
if v  0
if i  0, v > 0
if v i  v
otherwise
Item
Value
Weight
1
1
1
0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1
0 1 2 3 4 5
2
1
2
3
3
5
0 0 x x x x x x x x x x x x x x x
4
4
6
1
5
6
7
 i = 0 or v = 0
0
2 0
3 0
W = 11
4 0
5 0
53
Knapsack: FPTAS
 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 
if v  0
if i  0, v > 0
if v i  v
otherwise
Item
Value
Weight
1
1
1
0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1
0 1 2 3 4 5
2
1
2
3
3
5
0 0 x x x x x x x x x x x x x x x
4
4
6
1
5
6
7
 i = 1 , v = …
0
1
x x x x x x x x x x x x x x
2 0
3 0
W = 11
4 0
5 0
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Knapsack: FPTAS
 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 
if v  0
if i  0, v > 0
if v i  v
otherwise
Item
Value
Weight
1
1
1
0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1
0 1 2 3 4 5
2
1
2
3
3
5
0 0 x x x x x x x x x x x x x x x
4
4
6
1
5
6
7
 i = 2 , v = …
0
1
x x x x x x x x x x x x x x
2 0
1
3 x x x x x x x x x x x x x
3 0
W = 11
4 0
5 0
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Knapsack: FPTAS
 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 

if v  0
if i  0, v > 0
if v i  v
otherwise
Item
Value
Weight
1
1
1
0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1
0 1 2 3 4 5
2
1
2
3
3
5
0 0 x x x x x x x x x x x x x x x
4
4
6
1
5
6
7
i=3,v=…
0
1
x x x x x x x x x x x x x x
2 0
1
3 x x x x x x x x x x x x x
3 0
1
3 5 6 8 x x x x x x x x x x
W = 11
4 0
5 0
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Knapsack: FPTAS
 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 

if v  0
if i  0, v > 0
if v i  v
otherwise
Item
Value
Weight
1
1
1
0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1
0 1 2 3 4 5
2
1
2
3
3
5
0 0 x x x x x x x x x x x x x x x
4
4
6
1
5
6
7
i = 4, v = …
0
1
x x x x x x x x x x x x x x
2 0
1
3 x x x x x x x x x x x x x
3 0
1
3 5 6 8 x x x x x x x x x x
4 0
1
3 5 6 7 9
11
12
14
W = 11
x x x x x x
5 0
57
Knapsack: FPTAS
 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 

if v  0
if i  0, v > 0
if v i  v
otherwise
Item
Value
Weight
1
1
1
0 1 2 3 4 5 6 7 8
8 9 1 1 1 1 1 1
0 1 2 3 4 5
2
1
2
3
3
5
0 0 x x x x x x x x x x x x x x x
4
4
6
1
5
6
7
i = 5, v = …
8
0
1
x x x x x x x x x x x x x x
2 0
1
3 x x x x x x x x x x x x x
3 0
1
3 5 6 8 x x x x x x x x x x
4 0
1
3 5 6 7 9
11
12
14
x x x x x x
5 0
1
3 5 6 7 7 8
10
12
13
14
16
18
19
21
W = 11
S={1, 2, 5}
58
Knapsack: FPTAS Tracing Solution
// first call: pick_item(n, v) where M[n,v] <= W and v is max
pick_item( i, v) {
if (v == 0) return;
if (M[i,v] == wi + M[i-1, v-vi]) { print i; pick_item(i-1, v-vi); print i; }
else pick_item(i-1, v);
}
Item
Value
Weight
1
1
1
0 1 2 3 4 5 6 7 8
8 9 1 1 1 1 1 1
0 1 2 3 4 5
2
1
2
3
3
5
0 0 x x x x x x x x x x x x x x x
4
4
6
1
5
6
7
i = 5, v = …
8
0
1
x x x x x x x x x x x x x x
2 0
1
3 x x x x x x x x x x x x x
3 0
1
3 5 6 8 x x x x x x x x x x
4 0
1
3 5 6 7 9
11
12
14
x x x x x x
5 0
1
3 5 6 7 7 8
10
12
13
14
16
18
19
21
W = 11
S={1, 2, 5}
59
Knapsack Problem: Dynamic Programming II
Def. OPT(i, v) = min weight subset of items 1, …, i that yields value
exactly v.
Case 1: OPT does not select item i.
– OPT selects best of 1, …, i-1 that achieves exactly value v
Case 2: OPT selects item i.
– consumes weight wi, new value needed = v – vi
– OPT selects best of 1, …, i-1 that achieves exactly value v


 0

 
OPT (i, v)  
 OPT (i 1, v)

 min  OPT (i 1, v), wi  OPT (i 1, v  vi ) 
if v  0
if i  0, v > 0
if v i  v
otherwise
V*  n vmax

Running time. O(n V*) = O(n2 vmax).
V* = optimal value = maximum v such that OPT(n, v)  W.
Not polynomial in input size!


60
Knapsack: FPTAS
Intuition for approximation algorithm.
Round all values up to lie in smaller range.
Run dynamic programming algorithm II on rounded instance.
Return optimal items in rounded instance.



Item
Value
Weight
Item
Value
Weight
1
934,221
1
1
1
1
2
5,956,342
2
2
1
2
3
17,810,013
5
3
3
5
4
21,217,800
6
4
4
6
5
27,343,199
7
5
6
7
W = 11
original instance
W = 11
rounded instance
S = { 1, 2, 5 }
61
Knapsack: FPTAS
Knapsack FPTAS. Round up all values:
v 
v 
vi   i   , vˆi   i 
 
 
vmax = largest value in original instance
– 
= precision parameter
– 
= scaling factor =  vmax / n
–
Observation. Optimal solution to problems with
v or vˆ are equivalent.
Intuition. v close to v so optimal solution using v is nearly optimal;
vˆ small and integral so dynamic programming algorithm is fast.
 
Running time. O(n3 / ).
 Dynamic program II running timeis O(n2 vˆmax) , where

 vmax   n 
vˆmax      
    
62
Knapsack: FPTAS
 vi 
vi    
 
Knapsack FPTAS. Round up all values:
Theorem. If S is solution found by our algorithm and S* is any other
feasible solution of the original problem, then (1  )  vi   vi
iS
i  S*
Pf. Let S* be any feasible solution satisfying weight constraint.
 vi

i  S*
 vi 
always round up
 vi
solve rounded instance optimally
i  S*

iS

 (vi  )
never round up by more than 
 vi  n
|S|  n
iS

i S
DP alg can take vmax
 (1 )  vi
i S
n  =  vmax, vmax  iS vi
63