Local Search Heuristics for Facility Location Problems Rohit - PowerPoint PPT Presentation

Local Search Heuristics for Facility Location Problems Rohit Khandekar Dept. of Computer Science and Engineering, Indian Institute of Technology Delhi Joint work with: Vijay Arya, Naveen Garg, Vinayaka Pandit Kamesh Munagala, Adam Meyerson Departmental Seminar – p.1/43

Outline Define the k -median problem Simple local search algorithm Analysis Generalization Departmental Seminar – p.2/43

The k -median problem We are given n points in a metric space . Departmental Seminar – p.3/43

✆ ✟ � ✁ ✂ ✆ ✞ ✝ ✝ ✞ ✆ ✂ ✝ ✠ ✆ ✞ ✝ ✝ ✞ ✠ ✝ ☎ � � � ✁ ✄ ✂ ✂ ✁ ✁ The k -median problem We are given n points in a metric space . u d u w d w v v d w v d u w w d u v 0 d u u 0 d u v d v u Departmental Seminar – p.4/43

The k -median problem We are given n points in a metric space . Departmental Seminar – p.5/43

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 minimized. Departmental Seminar – p.6/43

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. Departmental Seminar – p.7/43

A brief bio-sketch of the k -median problem NP-hard Departmental Seminar – p.8/43

A brief bio-sketch of the k -median problem NP-hard Known to OR community since 60’s. Departmental Seminar – p.8/43

A brief bio-sketch of the k -median problem NP-hard Known to OR community since 60’s. Used for locating warehouses, manufacturing plants, etc. Departmental Seminar – p.8/43

A brief bio-sketch of the k -median problem NP-hard Known to OR community since 60’s. Used for locating warehouses, manufacturing plants, etc. Used also for clustering, data mining. Departmental Seminar – p.8/43

A brief bio-sketch of the k -median problem NP-hard Known to OR community since 60’s. Used for locating warehouses, manufacturing plants, etc. Used also for clustering, data mining. Received the attention of the Approximation algorithms community in early 90’s. Departmental Seminar – p.8/43

A brief bio-sketch of the k -median problem NP-hard Known to OR community since 60’s. Used for locating warehouses, manufacturing plants, etc. Used also for clustering, data mining. Received the attention of the Approximation algorithms community in early 90’s. Various algorithms via LP-relaxation, primal-dual scheme, etc. Departmental Seminar – p.8/43

A local search algorithm Departmental Seminar – p.9/43

A local search algorithm Start with any set of k medians. Departmental Seminar – p.9/43

A local search algorithm Identify a median and a point that is not a median. Departmental Seminar – p.10/43

A local search algorithm And SWAP tentatively! Departmental Seminar – p.11/43

A local search algorithm Perform the swap, only if the new solution is “better” (has less cost) than the previous solution. Departmental Seminar – p.12/43

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. Departmental Seminar – p.12/43

✞ ✍ ✎ ✆ ✏ ✌ ✌ ✏ ✎ ✆ ✡ ☞ ✌ ☛ ☞ ✡ The algorithm Algorithm Local Search. S any k medians 1. 2. While s S and s S such that, ✞✒✑ cost S s s cost S , do S S s s 3. return S Departmental Seminar – p.13/43

✌ ☞ ✆ ✆ ✎ ☞ ✍ ✌ ✞ ✆ ✎ ☛ ✞ ✡ ✎ ✡ ✏ ✌ ✏ The algorithm Algorithm Local Search. S any k medians 1. 2. While s S and s S such that, ε ✞✒✑ cost S s s 1 cost S , do S S s s 3. return S Departmental Seminar – p.14/43

Main theorem The local search algorithm described above computes a solution with cost (the sum of distances) at most 5 times the minimum cost. Departmental Seminar – p.15/43

✓ ✆ ✞ ✏ ✆ ✞ ✏ 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 ε swapping medians and got 3 5 approximation by ε taking k 1 medians. Departmental Seminar – p.15/43

� ✂ ✔ ✂ ✗ � ✔ ✖ ✖ ✕ ✔ Some notation s j S j S S k N S s cost S the sum of lengths of all the red segments Departmental Seminar – p.16/43

✗ ✖ ✖ ✔ � ✂ ✔ ✕ Some more notation o N O o O O k Departmental Seminar – p.17/43

� ✞ ✂ ✆ ✆ ✠ Some more notation s 2 s 1 N o s 2 o N o s 1 N o s 3 s 4 s 3 N O o N o s 4 N o ✞✙✘ N O o N S s s Departmental Seminar – p.18/43

Local optimality of S Since S is a local optimum solution, Departmental Seminar – p.19/43

☞ ✟ ☞ ✞ ✆ ✚ ✎ ✆ ✏ ✝ ✞ 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 Departmental Seminar – p.19/43

✝ ✞ ✞ ✚ ✆ ✆ ✟ ✞ ✏ ☞ ✑ ✎ ✛ ✆ ✆ ✞ ☞ 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 > > Departmental Seminar – p.19/43

✆ ✜ � ✞ ✂ ✜ ✜ ☞ ✚ ☞ Capture s 2 s 1 N o s 2 o N o s 1 N o s 3 s 4 s 3 N O o N o s 4 We say that s S captures o O if N O o N o ✜✣✢ s 2 Capture graph Departmental Seminar – p.20/43

✞ ✆ � ✤ ✂ ✞ ☞ ✆ A mapping π s 2 s 1 N o s 2 o N o s 1 N o s 3 s 4 s 3 N O o N o s 4 We consider a permutation π : N O o N O o that satisfies the following property: N o if s does not capture o then a point j s should get mapped outside N o s . Departmental Seminar – p.21/43

✞ ✆ � ✤ ✂ ✞ ☞ ✆ A mapping π s 2 s 1 N o s 2 o N o s 1 N o s 3 s 4 s 3 N O o N o s 4 We consider a permutation π : N O o N O o that satisfies the following property: N o if s does not capture o then a point j s should get mapped outside N o s . Departmental Seminar – p.22/43

✂ ☎ � ✔ ✂ ✥ ✖ � ✖ A mapping π s 2 s 1 N o s 2 o N o s 1 N o s 3 s 4 s 3 N O o N o s 4 N O o l 1 2 i i l 2 l π Departmental Seminar – p.23/43

✝ ✞ ☞ ✆ ☞ ✞ ✝ ✝ ✆ ✠ ✥ ✦ 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 Departmental Seminar – p.24/43

✥ ✦ Swaps considered l O S l 2 Departmental Seminar – p.25/43

✥ ✦ Swaps considered l O S l 2 “Why consider the swaps?” Departmental Seminar – p.25/43

✌ ✦ ✥ ✧ ✝ ✠ ★ ✍ Properties of the swaps considered l O S l 2 If s o is considered, then s does not capture any o o . Departmental Seminar – p.26/43

✝ ✧ ☞ ✠ ✦ ✍ ✥ ✌ ★ 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. Departmental Seminar – p.26/43

★ ✝ ☞ ☞ ✦ ✠ ✥ ✍ ✌ ✧ 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. Departmental Seminar – p.26/43

✟ ✞ ✏ ✆ ✎ ✞ ✆ ✩ ★ ✝ ✧ ✫ ✪ s o Focus on a swap o s Consider a swap s o that is one of the k swaps defined above. We know cost S s o cost S . Departmental Seminar – p.27/43

✎ ✭ ✮ ✎ ✬ ✏ ✏ 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 . Departmental Seminar – p.28/43

✆ ✎ ✏ ✭ ✞ ✏ ✮ ✬ ✎ ✎ ✏ 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 . cost S s o is the sum of distances of all the points to their nearest medians. Departmental Seminar – p.28/43

✏ ✞ ✏ ✎ ✎ ✬ ✏ ✏ ✎ ✆ ✞ ✮ ✏ ✎ ✭ ✆ ✎ 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 . 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 . Departmental Seminar – p.28/43

� ✬ ✞ ✭ ✆ ✂ ✮ Upper bound on cost S s o N O o s o Points in N O o are now connected to the new median o . Departmental Seminar – p.29/43

✆ ✂ ✎ ✭ ☞ ✚ ✮ ✬ ✞ � Upper bound on cost S s o N O o s o Thus, the increase in the distance for j N O o is at most O j S j Departmental Seminar – p.30/43