Christofides Algorithm

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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.

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