Christofides Algorithm
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- Lester Reeves
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4 2. compute minimum perfect matching of odd nodes
5 2. compute minimum perfect matching of odd nodes
6 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order
7 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 1
8 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 1 2
9 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 1 2 3
10 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order
11 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order
12 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order
13 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order
14 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order
15 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
16 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
17 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
18 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
19 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
20 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
21 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
22 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
23 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order
24 2. compute minimum perfect matching of odd nodes 3. find Eulerian walk node order 4. construct TSP tour along node order Theorem Christofides algorithm is a 3 -approximation algorithm 2 for Metric TSP.
25 Proof Sketch
26 Proof Sketch
27 Proof Sketch
28 Proof Sketch
29 Proof Sketch
30 Proof Sketch
31 Approximation Schemes
32 Approximation Schemes Definition (PTAS, FPTAS) optimization problem nonnegative objective function f algorithm A :(I, Á) æ (1+Á)-approximate solution
33 Approximation Schemes Definition (PTAS, FPTAS) optimization problem nonnegative objective function f algorithm A :(I, Á) æ (1+Á)-approximate solution PTAS / polynomial time approximation scheme: running time Æ polynomial(size(i))
34 Approximation Schemes Definition (PTAS, FPTAS) optimization problem nonnegative objective function f algorithm A :(I, Á) æ (1+Á)-approximate solution PTAS / polynomial time approximation scheme: running time Æ polynomial(size(i)) FPTAS / fully polynomial time approximation scheme: running time Æ polynomial(size(i), 1 Á )
35 Recap: Dynamic Programming for Knapsack recursion values: C j ( ) :=minimum total weight of a feasible knapsack on [j] that has exactly value
36 Recap: Dynamic Programming for Knapsack recursion values: C j ( ) :=minimum total weight of a feasible knapsack on [j] that has exactly value recursion start: Y ] 0, for = 0 C 0 ( ) = [ Œ, otherwise
37 Recap: Dynamic Programming for Knapsack recursion values: C j ( ) :=minimum total weight of a feasible knapsack on [j] that has exactly value recursion start: Y ] 0, for = 0 C 0 ( ) = [ Œ, otherwise recursion formula: C j ( ) =min {C j 1 ( ), C j 1 ( c j )+a j } C j ( ) =Œ, if it exceeds
38 Example Knapsack Problem a = c = = 10
39 Example Modified Knapsack Problem rounding down: c Õ i := 7 8 ci t for i = 1,...,n
40 Example Modified Knapsack Problem rounding down: c Õ i := 7 8 ci t for i = 1,...,n a = c = = 10
41 FPTAS for Knapsack Input: Knapsack instance (a, c, ) on n items, error bound Á Output: total value of an (1 + Á)-approximate solution
42 FPTAS for Knapsack 1 Let c max Ω max {c i : i œ [n]} and set t = Á cmax n 2 Set ci Õ Ω Í Î c i t for i = 1,...,n 3 Set C 0 (0) Ω 0andC 0 ( ) ΩŒfor all œ {0, 1,...,c max } 4 for j = 1,...,n, œ {0, 1,...,c max } do 5 if c j < 0 then 6 Set C j ( ) Ω C j 1 ( ) 7 else 8 if C j 1 ( c j )+a j > then 9 Set C j ( ) Ω C j 1 ( ) 10 else 11 Set C j ( ) Ω min {C j 1 ( ), C j 1 ( c j )+a j } 12 end 13 end 14 end
43 FPTAS for Knapsack Theorem The algorithm is a fully polynomial-time approximation scheme (FPTAS) for Knapsack.
Theorem 2.9: nearest addition algorithm
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