Local Search Based Approximation Algorithms. Vinayaka Pandit. IBM India Research Laboratory

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1 Local Search Based Approximation Algorithms The k-median problem Vinayaka Pandit IBM India Research Laboratory joint work with Naveen Garg, Rohit Khandekar, and Vijay Arya The 2011 School on Approximability, Bangalore p.1/51

2 Outline Local Search Technique Simple Example Using Max-CUT Analysis of a popular heuristic for the k-median problem Some interesting questions The 2011 School on Approximability, Bangalore p.2/51

3 Local Search Technique Let F denote the set of all feasible solutions for a problem instance I. Define a function N : F 2 F which associates for each solution, a set of neighboring solutions. Start with some feasible solution and iteratively perform local operations. Suppose S C F is the current solution. We move to any solution S N N(S C ) which is strictly better than S C. Output S L, a locally optimal solution for which no solution in N(S L ) is strictly better than S L itself. The 2011 School on Approximability, Bangalore p.3/51

4 Example: MAX-CUT Given a graph G = (V,E), Partition V into A,B s.t. #edges between A and B is maximized. Note that MAX-CUT is E. The 2011 School on Approximability, Bangalore p.4/51

5 Local Search for MAX-CUT Algorithm Local Search for MAX-CUT. 1. A,B any partition of V ; 2. While u V such that in-degree(u) > outdegree(u), do if(u A), Move u to B else, Move u to A done 3. return A, B The 2011 School on Approximability, Bangalore p.5/51

6 Neighborhood Function Solution Space: the set of all partitions. Neighborhood Function: Neighbors of a partition (A,B) are all the partitions (A,B ) obtained by interchanging the side of a single vertex. The 2011 School on Approximability, Bangalore p.6/51

7 Analysis for MAX-CUT u A B 1. in-d(u) out-d(u) (Apply Conditions for Local Optimality) 2. u V in-d(u) u V of local operations) out-d(u) (Consider suitable set 3. #Internal Edges #Cut Edges #Cut-edges E 2 2-approximation (Infer) The 2011 School on Approximability, Bangalore p.7/51

8 Local Search for Approximation Folklore: the 2-approx for MAX-CUT Fürer and Raghavachari: Additive Approx for Min. Degree Spanning Tree. Lu and Ravi: Constant Factor approx for Spanning Trees with maximum leaves. Könemann and Ravi: Bi-criteria approximation for Bounded Degree MST. Quite successful as a technique for Facility Location and Clustering problems. Started with Korupolu et. al. The 2011 School on Approximability, Bangalore p.8/51

9 The k-median problem We are given n points in a metric space. The 2011 School on Approximability, Bangalore p.9/51

10 The k-median problem We are given n points in a metric space. u d(u, w) + d(w, v) v d(u, w) d(w, v) w d(u,v) 0, d(u,u) = 0, d(u,v) = d(v,u) The 2011 School on Approximability, Bangalore p.10/51

11 The k-median problem We are given n points in a metric space. The 2011 School on Approximability, Bangalore p.11/51

12 The k-median problem We are given n points in a metric space. We want to identify k medians such that the sum of distances of all the points to their nearest medians is mini- The 2011 School on Approximability, Bangalore p.12/51

13 The k-median problem We are given n points in a metric space. We want to identify k medians such that the sum of lengths of all the red segments is minimized. The 2011 School on Approximability, Bangalore p.13/51

14 A local search algorithm Identify a median and a point that is not a median. The 2011 School on Approximability, Bangalore p.14/51

15 A local search algorithm And SWAP tentatively! The 2011 School on Approximability, Bangalore p.15/51

16 A local search algorithm Perform the swap, only if the new solution is better (has less cost) than the previous solution. The 2011 School on Approximability, Bangalore p.16/51

17 A local search algorithm Perform the swap, only if the new solution is better (has less cost) than the previous solution. Stop, if there is no swap that improves the solution. The 2011 School on Approximability, Bangalore p.16/51

18 The algorithm Algorithm Local Search. 1. S any k medians 2. While s S and s S such that, cost(s s + s ) < cost(s), do S S s + s 3. return S The 2011 School on Approximability, Bangalore p.17/51

19 The algorithm Algorithm Local Search. 1. S any k medians 2. While s S and s S such that, cost(s s + s ) < (1 ǫ)cost(s), do S S s + s 3. return S The 2011 School on Approximability, Bangalore p.18/51

20 Main theorem The local search algorithm described above computes a solution with cost (the sum of distances) at most 5 times the minimum cost. The 2011 School on Approximability, Bangalore p.19/51

21 Main theorem The local search algorithm described above computes a solution with cost (the sum of distances) at most 5 times the minimum cost. Korupolu, Plaxton, and Rajaraman (1998) analyzed a variant in which they permitted adding, deleting, and swapping medians and got (3 + 5/ǫ) approximation by taking k(1 + ǫ) medians. The 2011 School on Approximability, Bangalore p.19/51

22 Some notation j s S j S = { } S = k N S (s) cost(s) = the sum of lengths of all the red segments The 2011 School on Approximability, Bangalore p.20/51

23 Some more notation o N O (o) O = { } O = k The 2011 School on Approximability, Bangalore p.21/51

24 Some more notation s 1 s 2 N o s 1 o N o s 2 N o s 3 s 4 s 3 N O (o) N o s 4 N o s = N O (o) N S (s) The 2011 School on Approximability, Bangalore p.22/51

25 Local optimality of S Since S is a local optimum solution, The 2011 School on Approximability, Bangalore p.23/51

26 Local optimality of S Since S is a local optimum solution, We have, cost(s s + o) cost(s) for all s S,o O. The 2011 School on Approximability, Bangalore p.23/51

27 Local optimality of S Since S is a local optimum solution, We have, cost(s s + o) cost(s) for all s S,o O. We shall add k of these inequalities (chosen carefully) to show that, cost(s) 5 cost(o) > > The 2011 School on Approximability, Bangalore p.23/51

28 What happens when we swap <s,o>? o o i s N o i s N o s N S (s) All the points in N S (s) have to be rerouted to one of the facilities in S {s} + {o}. We are interested two types of clients: those belonging to N o s and those not belonging to N o s. The 2011 School on Approximability, Bangalore p.24/51

29 Rerouting j N o s o O j o i j S j s? N o i s N o s N S (s) Rerouting is easy. Send it to o. Change in cost = O j S j. The 2011 School on Approximability, Bangalore p.25/51

30 Rerouting j N o s o s? N o i s o i s j O j j S j s N o i s O j o i Look at N O (o i ). Map j to a unique j N O (o i ) outside N o i s and route via j. Change in cost = O j + O j + S j S j. Ensure that every client is involved in exactly one reroute. Therefore, the mapping need to be one-to-one and onto. The 2011 School on Approximability, Bangalore p.26/51

31 Desired mapping of clients inside N O (o) s 1 s 2 N o s 1 o N o s 2 N o s 3 N o s 4 s 4 s 3 N O (o) We desire a permutation π : N O (o) N O (o) that satisfies the following property: Client j N o s should get mapped to j N O (o), but outside N o s. The 2011 School on Approximability, Bangalore p.27/51

32 Notion of Capture s 1 s 2 N o s 1 o N o s 2 N o s 3 N o s 4 s 4 s 3 N O (o) We say that s S captures o O if Ns o > N O(o). 2 Note: A facility o O is captured precisely when a mapping as we described is not feasible. Capture graph The 2011 School on Approximability, Bangalore p.28/51

33 A mapping π s 1 s 2 N o s 1 o N o s 2 N o s 3 N o s 4 s 4 s 3 N O (o) We consider a permutation π : N O (o) N O (o) that satisfies the following property: if s does not capture o then a point j N o s should get mapped outside N o s. The 2011 School on Approximability, Bangalore p.29/51

34 A mapping π s 1 s 2 N o s 1 o N o s 2 N o s 3 N o s 4 s 4 s 3 N O (o) We consider a permutation π : N O (o) N O (o) that satisfies the following property: if s does not capture o then a point j N o s should get mapped outside N o s. The 2011 School on Approximability, Bangalore p.30/51

35 A mapping π s 1 s 2 N o s 1 o N o s 2 N o s 3 s 4 s 3 N O (o) N o s i N O (o) = l i + l/2 l π The 2011 School on Approximability, Bangalore p.31/51

36 Capture graph l O S l/2 Construct a bipartite graph G = (O,S,E) where there is an edge (o,s) if and only if s S captures o O. Capture The 2011 School on Approximability, Bangalore p.32/51

37 Swaps considered l O S l/2 The 2011 School on Approximability, Bangalore p.33/51

38 Swaps considered l O S l/2 Why consider the swaps? The 2011 School on Approximability, Bangalore p.33/51

39 Properties of the swaps considered l O S l/2 If s,o is considered, then s does not capture any o o. The 2011 School on Approximability, Bangalore p.34/51

40 Properties of the swaps considered l O S l/2 If s,o is considered, then s does not capture any o o. Any o O is considered in exactly one swap. The 2011 School on Approximability, Bangalore p.34/51

41 Properties of the swaps considered l O S l/2 If s,o is considered, then s does not capture any o o. Any o O is considered in exactly one swap. Any s S is considered in at most 2 swaps. The 2011 School on Approximability, Bangalore p.34/51

42 Focus on a swap s,o o s Consider a swap s,o that is one of the k swaps defined above. We know cost(s s + o) cost(s). The 2011 School on Approximability, Bangalore p.35/51

43 Upper bound on cost(s s + o) In the solution S s + o, each point is connected to the closest median in S s + o. The 2011 School on Approximability, Bangalore p.36/51

44 Upper bound on cost(s s + o) In the solution S s + o, each point is connected to the closest median in S s + o. cost(s s + o) is the sum of distances of all the points to their nearest medians. The 2011 School on Approximability, Bangalore p.36/51

45 Upper bound on cost(s s + o) In the solution S s + o, each point is connected to the closest median in S s + o. cost(s s + o) is the sum of distances of all the points to their nearest medians. We are going to demonstrate a possible way of connecting each client to a median in S s + o to get an upper bound on cost(s s + o). The 2011 School on Approximability, Bangalore p.36/51

46 Upper bound on cost(s s + o) o N O (o) s Points in N O (o) are now connected to the new median o. The 2011 School on Approximability, Bangalore p.37/51

47 Upper bound on cost(s s + o) o N O (o) s Thus, the increase in the distance for j N O (o) is at most O j S j. The 2011 School on Approximability, Bangalore p.38/51

48 Upper bound on cost(s s + o) o N O (o) s j Consider a point j N S (s) \ N O (o). The 2011 School on Approximability, Bangalore p.39/51

49 Upper bound on cost(s s + o) π(j) s o N O (o) s j Consider a point j N S (s) \ N O (o). Suppose π(j) N S (s ). (Note that s s.) The 2011 School on Approximability, Bangalore p.40/51

50 Upper bound on cost(s s + o) π(j) s o N O (o) s j Consider a point j N S (s) \ N O (o). Suppose π(j) N S (s ). (Note that s s.) Connect j to s now. The 2011 School on Approximability, Bangalore p.41/51

51 Upper bound on cost(s s + o) π(j) O π(j) S π(j) o s O j j New distance of j is at most O j + O π(j) + S π(j). The 2011 School on Approximability, Bangalore p.42/51

52 Upper bound on cost(s s + o) π(j) O π(j) S π(j) o s O j j New distance of j is at most O j + O π(j) + S π(j). Therefore, the increase in the distance for j N S (s) \ N O (o) is at most O j + O π(j) + S π(j) S j. The 2011 School on Approximability, Bangalore p.42/51

53 Upper bound on the increase in the cost Lets try to count the total increase in the cost. The 2011 School on Approximability, Bangalore p.43/51

54 Upper bound on the increase in the cost Lets try to count the total increase in the cost. Points j N O (o) contribute at most (O j S j ). The 2011 School on Approximability, Bangalore p.43/51

55 Upper bound on the increase in the cost Lets try to count the total increase in the cost. Points j N O (o) contribute at most (O j S j ). Points j N S (s) \ N O (o) contribute at most (O j + O π(j) + S π(j) S j ). The 2011 School on Approximability, Bangalore p.43/51

56 Upper bound on the increase in the cost Lets try to count the total increase in the cost. Points j N O (o) contribute at most (O j S j ). Points j N S (s) \ N O (o) contribute at most (O j + O π(j) + S π(j) S j ). Thus, the total increase is at most, (O j S j ) + (O j + O π(j) + S π(j) S j ). j N O (o) j N S (s)\n O (o) The 2011 School on Approximability, Bangalore p.43/51

57 Upper bound on the increase in the cost j N O (o) (O j S j ) + j N S (s)\n O (o) (O j + O π(j) + S π(j) S j ) The 2011 School on Approximability, Bangalore p.44/51

58 Upper bound on the increase in the cost j N O (o) (O j S j ) + j N S (s)\n O (o) (O j + O π(j) + S π(j) S j ) cost(s s + o) cost(s) The 2011 School on Approximability, Bangalore p.44/51

59 Upper bound on the increase in the cost j N O (o) (O j S j ) + j N S (s)\n O (o) (O j + O π(j) + S π(j) S j ) cost(s s + o) cost(s) 0 The 2011 School on Approximability, Bangalore p.44/51

60 Plan We have one such inequality for each swap s,o. (O j S j )+ (O j +O π(j) +S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) The 2011 School on Approximability, Bangalore p.45/51

61 Plan We have one such inequality for each swap s,o. (O j S j )+ (O j +O π(j) +S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) There are k swaps that we have defined. The 2011 School on Approximability, Bangalore p.45/51

62 Plan We have one such inequality for each swap s,o. (O j S j )+ (O j +O π(j) +S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) There are k swaps that we have defined. l O l/2 S The 2011 School on Approximability, Bangalore p.45/51

63 Plan We have one such inequality for each swap s,o. (O j S j )+ (O j +O π(j) +S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) There are k swaps that we have defined. l O l/2 Lets add the inequalities for all the k swaps and see what we get! S The 2011 School on Approximability, Bangalore p.45/51

64 The first term... (O j S j ) + (O j +O π(j) +S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) The 2011 School on Approximability, Bangalore p.46/51

65 The first term... (O j S j ) + (O j +O π(j) +S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) Note that each o O is considered in exactly one swap. The 2011 School on Approximability, Bangalore p.46/51

66 The first term... j N O (o) (O j S j ) + j N S (s)\n O (o) (O j +O π(j) +S π(j) S j ) 0. Note that each o O is considered in exactly one swap. Thus, the first term added over all the swaps is (O j S j ) o O j N O (o) The 2011 School on Approximability, Bangalore p.46/51

67 The first term... j N O (o) (O j S j ) + j N S (s)\n O (o) (O j +O π(j) +S π(j) S j ) 0. Note that each o O is considered in exactly one swap. Thus, the first term added over all the swaps is (O j S j ) o O j N O (o) = j (O j S j ) The 2011 School on Approximability, Bangalore p.46/51

68 The first term... j N O (o) (O j S j ) + j N S (s)\n O (o) (O j +O π(j) +S π(j) S j ) 0. Note that each o O is considered in exactly one swap. Thus, the first term added over all the swaps is (O j S j ) o O j N O (o) = j (O j S j ) = cost(o) cost(s). The 2011 School on Approximability, Bangalore p.46/51

69 The second term... (O j S j )+ (O j + O π(j) + S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) The 2011 School on Approximability, Bangalore p.47/51

70 The second term... (O j S j )+ (O j + O π(j) + S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) Note that O j + O π(j) + S π(j) S j. The 2011 School on Approximability, Bangalore p.47/51

71 The second term... (O j S j )+ (O j + O π(j) + S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) Note that Thus O j + O π(j) + S π(j) S j. O j + O π(j) + S π(j) S j 0. The 2011 School on Approximability, Bangalore p.47/51

72 The second term... (O j S j )+ (O j + O π(j) + S π(j) S j ) 0. j N O (o) j N S (s)\n O (o) Note that Thus O j + O π(j) + S π(j) S j. O j + O π(j) + S π(j) S j 0. Thus the second term is at most (O j + O π(j) + S π(j) S j ). j N S (s) The 2011 School on Approximability, Bangalore p.47/51

73 The second term... Note that each s S is considered in at most two swaps. The 2011 School on Approximability, Bangalore p.48/51

74 The second term... Note that each s S is considered in at most two swaps. Thus, the second term added over all the swaps is at most 2 (O j + O π(j) + S π(j) S j ) s S j N S (s) The 2011 School on Approximability, Bangalore p.48/51

75 The second term... Note that each s S is considered in at most two swaps. Thus, the second term added over all the swaps is at most 2 (O j + O π(j) + S π(j) S j ) s S j N S (s) = 2 j (O j + O π(j) + S π(j) S j ) The 2011 School on Approximability, Bangalore p.48/51

76 The second term... Note that each s S is considered in at most two swaps. Thus, the second term added over all the swaps is at most 2 (O j + O π(j) + S π(j) S j ) s S j N S (s) = 2 j (O j + O π(j) + S π(j) S j ) = 2 [ O j + j O π(j) + j S π(j) j S j ] j The 2011 School on Approximability, Bangalore p.48/51

77 The second term... Note that each s S is considered in at most two swaps. Thus, the second term added over all the swaps is at most 2 (O j + O π(j) + S π(j) S j ) s S j N S (s) = 2 j (O j + O π(j) + S π(j) S j ) = 2 [ O j + j O π(j) + j S π(j) j S j ] j = 4 cost(o). The 2011 School on Approximability, Bangalore p.48/51

78 Putting things together 0 s,o (O j S j ) + j N O (o) j N S (s)\n O (o) (O j + O π(j) + S π(j) The 2011 School on Approximability, Bangalore p.49/51

79 Putting things together 0 s,o (O j S j ) + j N O (o) j N S (s)\n O (o) (O j + O π(j) + S π(j) [cost(o) cost(s)] + [4 cost(o)] The 2011 School on Approximability, Bangalore p.49/51

80 Putting things together 0 s,o (O j S j ) + j N O (o) j N S (s)\n O (o) (O j + O π(j) + S π(j) [cost(o) cost(s)] + [4 cost(o)] = 5 cost(o) cost(s). The 2011 School on Approximability, Bangalore p.49/51

81 Putting things together 0 s,o (O j S j ) + j N O (o) j N S (s)\n O (o) (O j + O π(j) + S π(j) [cost(o) cost(s)] + [4 cost(o)] = 5 cost(o) cost(s). Therefore, cost(s) 5 cost(o). The 2011 School on Approximability, Bangalore p.49/51

82 A tight example (k 1) O S (k 1)/2 (k + 1)/2 The 2011 School on Approximability, Bangalore p.50/51

83 A tight example (k 1) O S (k 1)/2 (k + 1)/2 cost(s) = 4 (k 1)/2 + (k + 1)/2 = (5k 3)/2 The 2011 School on Approximability, Bangalore p.50/51

84 A tight example (k 1) O S (k 1)/2 (k + 1)/2 cost(s) = 4 (k 1)/2 + (k + 1)/2 = (5k 3)/2 cost(o) = 0 + (k + 1)/2 = (k + 1)/2 The 2011 School on Approximability, Bangalore p.50/51

85 Future directions We do not have a good understanding of the structure of problems for which local search can yield approximation algorithms. Starting point could be an understanding of the success of local search techniques for the curious capacitated facility location (CFL) problems. For CFL problems, we know good local search algorithms. But, no non-trivial approximations known using other techniques like greedy, LP rounding etc. The 2011 School on Approximability, Bangalore p.51/51

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