1. Design Techniques for Approximation Algorithms and Approximation Classes

2. Summary -Sequential algorithms -Greedy technique -Local search technique -Linear programming based algorithms -Relaxation and rounding -Primal-dual -Dynamic programming -Approximation classes -The NPO world

3. Performance ratio -Given an optimization problem P, an instance x and a feasible solution y, the performance ratio of y with respect to x is R(x,y) = max(m(x,y)/m*(x), m*(x)/m(x,y)) -An algorithm is said to be an r-approximation algorithm if, for any instance x, returns a solution whose performance ratio is at most r

4. MINIMUM BIN PACKING -INSTANCE: Finite set I of rational numbers {a 1,…,a n } with a i  (0,1] -SOLUTION: Partition {B 1,…,B k } of I into k bins such that the sum of the numbers in each bin is at most 1 -MEASURE: Cardinality of the partition, i.e., k

5. Sequential algorithm -Polynomial-time 2-approximation algorithm for MINIMUM BIN PACKING -Next Fit algorithm begin for each number a if a fits into the last open bin then assign a to this bin else open new bin and assign a to this bin return f end.

6. Proof -Number of bins used by the algorithm is at most 2A, where A is the sum of all numbers -For each pair of consecutive bins, the sum of the number included in these two bins is greater than 1 -Each feasible solution uses at least A bins -Best case each bin is full (i.e., the sum of its numbers is 1) -Performance ratio is at most 2 -Theorem: First Fit Decreasing computes solution whose measure is at most 1.5m*(x)+1

7. Tightness -Let I={1/2,1/2n,1/2,1/2n,…,1/2,1/2n} contain 4n items

8. Gavril’s algorithm for vertex cover -Theorem: Gavril’s algorithm is a polynomial-time 2- approximation algorithm begin U=ø; for any edge (u,v) do if (u is not in U) and (v is not in U) then insert u and v in U; return U end.

9. MINIMUM GRAPH COLORING -INSTANCE: Graph G=(V,E) -SOLUTION: A coloring of V, that is, function f such that, for any edge (u,v), f(u)  f(v) -MEASURE: Number of colors, i.e., cardinality of the range of f

10. Sequential algorithm: bad -Sequential algorithm for MINIMUM GRAPH COLORING begin order V with respect to the degree; for each node v do if there exists color not used by neighbors of v then assign this color to v else create new color and assign it to v end.

11. Example The performance ratio is 2 Generalizing, the performance ratio is n/2, where n is the number of nodes

12. Greedy technique: good -Polynomial-time 2-approximation algorithm for MAXIMUM SAT begin for any variable v begin p := number of clauses, which contain v; n := number of clauses, which contain not v; if p  n then f(v) := TRUE else f(v) := FALSE; simplify the formula; end; return f end.

13. Proof -By induction on the number n of variables, we prove that the algorithm satisfies at least half of the m clauses -n=1: trivial -Inductive step. Let v be the first variable to which a value has been assigned. Assume p  n (that is, f(v)=TRUE). By induction hypothesis at least p+(m-p-n)/2  m/2 clauses are satisfied

14. MAXIMUM INDEPENDENT SET -INSTANCE: Graph G=(V,E) -SOLUTION: A subset V’ of V such that, for any edge (u,v), either u is not in V’ or v is not in V’ -MEASURE: Cardinality of V’

15. Greedy technique: bad -Greedy algorithm for MAXIMUM INDEPENDENT SET begin V’:=ø; U:=V; while U is not empty do begin x := vertex of minimum degree in graph induced by U; insert x in V’; eliminate x and all its neighbors from U end; return V’ end.

16. Example The performance ratio is 2 Generalizing, the performance ratio is n/4, where n is the number of nodes

17. MAXIMUM CUT -INSTANCE: Graph G=(V,E) -SOLUTION: Partition of V into disjoint sets V 1 and V 2 -MEASURE: Cardinality of the cut, i.e., the number of edges with one endpoint in V 1 and one endpoint in V 2

18. Local search technique -Polynomial-time 2-approximation algorithm for MAXIMUM CUT begin V 1 :=ø; repeat if exchanging one node between V 1 and V 2 =V- V 1 improves the cut then perform the exchange; until a local optimum is reached; return f end.

19. Proof -We prove that any local optimum contains at least half of the m edges -Notation: -c= # of edges of the cut -i= # of edges inside V 1 -o= # of edges outside V 1 -m=c+i+o, that is, i+o=m-c -For any node v, -i(v)= # of edges between v and a node in V 1 -o(v)= # of edges between v and a node not in V 1

20. Proof (continued) -V 1 is a local optimum: for any v  V 1, i(v)-o(v)  0 and,for any v not in V 1,o(v)-i(v)  0 -Summing over all nodes in V 1, we have 2i-c  0 -Summing over all nodes not in V 1, we have 2o-c  0 -That is, i+o-c  0 -That is, m-2c  0 -That is, c  m/2

21. Relaxation and rounding technique -Polynomial-time 2-approximation algorithm for weighted version of MINIMUM VERTEX COVER -formulate the problem as linear integer programming -solve the relaxation -select nodes that have been chosen enough

22. ILP formulation

23. LP relaxation and rounding -Final solution: U = {i : x(i)  0.5}

24. Proof -The solution is feasible -Otherwise, one edge (i,j) is not filled (that is, x(i)+x(j)<1) -The solution has measure at most twice the optimum of LP relaxation -Each variable is at most doubled -The optimum of LP relaxationis at most equal to that of ILP -The set of feasible solutions is larger

25. Primal-dual technique -Classical approach for solving exactly combinatorial optimization problems -Weighted combinatorial problems are reduced to purely combinatorial, unweighted problems -Examples: Dijkstra (shortest path), Ford and Fulkerson (maximum flow), Edmonds (maximum matching) -Polynomial-time 2-approximation algorithm for weighted version of MINIMUM VERTEX COVER

26. LP formulation

27. Dual problem where N(i) denotes the neighborhood of i

28. begin y=0; U=Ø; while a not covered edge (i,j) exists increase y(i,j) until either i or j is filled if i (resp. j) is filled then put i (resp. j) in U end. 2-approximation algorithm -Simultaneously maintains a (possibly unfeasible) integer solution of LP formulation and a (not necessarily optimal) feasible solution of dual problem -At each step integer solution becomes more feasible and dual solution has better measure -Ends when integer solution becomes feasible

29. Proof -Feasibility: trivial -Performance ratio: -For any i  U, the ith constraint is tight. -Sum C of the weights of the nodes in U is equal to the sum P of the profit of the incident edges -P is at most twice the sum of the profit of all edges which is at most equal to the maximum profit -By duality, maximum profit is equal to minimum weight -Time complexity: -At most n iterations, where n is the number of nodes

30. MINIMUM PARTITION -INSTANCE: Finite set X of items, for each x i  X a positive integer weight a i -SOLUTION: A partition of X into disjoint sets Y 1 and Y 2 -MEASURE: Maximum between the sum of the weights of elements in V 1 and the sum of the weights of elements in V 2

31. Dynamic programming technique -Pseudo-polynomial time algorithm for MINIMUM PARTITION: -T: n x b-matrix (b=sum of the weights of all n elements) -T(i,j)=TRUE if a subset of {a 1,...,a i } exists whose sum is j -Construction of T: T(i+1,j)=T(i,j) or T(i,j-a i +1) -Final answer to the evaluation problem: -select true element of nth row of T that minimizes max(j,b-j) -Complexity: O(nb)=O(n 2 a max ), where a max is the maximum weight -Can be modified to obtain a feasible solution

32. The approximation algorithm -Ignore the last t digits of the numbers -Apply the pseudo-polynomial time algorithm -Return the corresponding solution in the original instance

33. Proof -Performance ratio: -m(x,y*(x’))-m*(x)  10 t n where y*(x’) denotes an optimal solution for scaled instance x’ -Performance ratio is at most 1+10 t n/m*(x) -m*(x)  a max -For any r, if we choose t=log 10 (a max (r-1) /n), then the performance ratio is at most r -Time complexity: -O(n 2 a’ max ) = O(n 3 /(r-1))

34. Class APX -NPO problems P that admit a polynomial-time r– approximation algorithm, for given constant r  1 -P is said to be r-approximable -Examples: MINIMUM BIN PACKING, MAXIMUM SAT, MAXIMUM CUT, MINIMUM VERTEX COVER

35. Class PTAS -NPO problems P that admit a polynomial-time r– approximation algorithm, for any r > 1 -Time must be polynomial in the length of the instance but not necessarily in 1/(r-1) -Time complexity O(n 1/(r-1) ) or O(2 1/(r-1) n 3 ) -P is said to admit a polynomial-time approximation scheme -Example: MINIMUM PARTITION

36. The NPO world NPO APX MINIMUM BIN PACKING MAXIMUM SAT MAXIMUM CUT MINIMUM VERTEX COVER PTAS MINIMUM PARTITION PO MINIMUM PATH