1. LP-Based Algorithms for Capacitated Facility Location Hyung-Chan An EPFL July 29, 2013 Joint work with Mohit Singh and Ola Svensson

2. Capacitated facility location problem  Given a metric cost c on  D: set of clients  F: set of facilities 10 2 3 12

3. Capacitated facility location problem  Given a metric cost c on  D: set of clients  F: set of facilities  U i : capacity of i ∈ F  o i : opening cost of i ∈ F 5 ≤3 1 ≤2 20 ≤3

4. Capacitated facility location problem  Given a metric cost c on  D: set of clients  F: set of facilities  U i : capacity of i ∈ F  o i : opening cost of i ∈ F  Want:  Choose S ⊆ F to open 5 ≤3 1 ≤2 20 ≤3

5. Capacitated facility location problem  Given a metric cost c on  D: set of clients  F: set of facilities  U i : capacity of i ∈ F  o i : opening cost of i ∈ F  Want:  Choose S ⊆ F to open  Assign every client to an open facility f : D → S  Capacities satisfied 5 ≤3 1 ≤2 20 ≤3

6. Capacitated facility location problem  Given a metric cost c on  D: set of clients  F: set of facilities  U i : capacity of i ∈ F  o i : opening cost of i ∈ F  Want:  Choose S ⊆ F to open  Assign every client to an open facility f : D → S  Capacities satisfied  Minimize Σ i ∈ S o i + Σ j ∈ D c j,f(j) = (20 + 5) + (2 + 2 + 2 + 5 + 2 + 3) 5 20 2 3 2 5 22

7. Successful special case  Uncapacitated facility location problem  U i = ∞ ∀ i  NP-hard to approximate better than 1.463 [Guha & Khuller 1999] [Sviridenko]  1.488-approximation algorithm [Li 2011]

8. Successful special case  Uncapacitated facility location problem  U i = ∞ ∀ i  NP-hard to approximate better than 1.463 [Guha & Khuller 1999] [Sviridenko]  1.488-approximation algorithm [Li 2011]  Combining a linear program(LP)-rounding algorithm with a primal-dual algorithm  LP-rounding 3.16-approximation [Shmoys, Tardos, Aardal 1997], LP-rounding & greedy 2.41- approximation [Guha & Khuller 1999], LP-rounding 1.74-approximation [Chudak & Shmoys 1999], primal-dual 3-approximation [Jain & Vazirani 2001], combining 1.73-approximation [Charikar & Guha 1999], primal-dual 1.61-approximation [Jain, Mahdian, Markakis, Saberi, Vazirani 2003], LP-rounding 1.59- approximation [Sviridenko 2002], primal-dual 1.52-approximation [Mahdian, Ye, Zhang 2006], combining 1.5-approximation algorithm [Byrka & Aardal 2010]

9. The Question  Can we use these LP-based techniques to solve the capacitated facility location problem?

10. The Question  Can we use these LP-based techniques to solve the capacitated facility location problem?  All known approximation algorithms based on local search [Bansal, Garg, Gupta 2012], [Pal, Tardos, Wexler 2001], [Korupolu, Plaxton, Rajaraman 2000]  5-approximation

11. The Question  Can we use these LP-based techniques to solve the capacitated facility location problem?  Rich toolkit of algorithmic techniques

12. The Question  Can we use these LP-based techniques to solve the capacitated facility location problem?  Rich toolkit of algorithmic techniques  Per-instance performance guarantee  Application to related problems  One of the ten Open Problems selected by the textbook of Williamson and Shmoys

13. The Question  Can we use these LP-based techniques to solve the capacitated facility location problem?  Why is this hard?  Standard LP relaxation fails to bound the optimum within a reasonable factor  Relaxed problems: uncapacitated problem, capacities can be violated [Abrams, Meyerson, Munagala, Plotkin 2002], facilities can be opened multiple times [Shmoys, Tardos, Aardal 1997], [Jain, Vazirani 2001], opening costs are uniform [Levi, Shmoys, Swamy 2012]

14. The Question  Can we use these LP-based techniques to solve the capacitated facility location problem?  Why is this hard?  Standard LP relaxation fails to bound the optimum within a reasonable factor  No LP relaxation known that is algorithmically amenable

15. Main result There is a good LP: its optimum is within a constant factor of the true optimum. In particular, there is a poly-time algorithm that finds a solution whose cost is within a constant factor of the LP optimum. Theorem

16. Our relaxation  Standard LP rewritten  ∀ i ∈ F y i = 1 if open, 0 if not  ∀ i ∈ F, j ∈ D x ij = 1 if j is assigned to i, 0 if not  Consider a multicommodity flow network:  arc (j, i) of capacity x ij  j ∈ D is a source of commodity j with demand 1  i ∈ F is a sink of  commodity-oblivious capacity y i ∙U i  commodity-specific capacity y i ∙ 1 ≤3 ≤2 ≤3 y=1 y=0 y=1 All shown arcs are of capacity 1; others 0.

17. Our relaxation  Standard LP rewritten  ∀ i ∈ F y i = 1 if open, 0 if not  ∀ i ∈ F, j ∈ D x ij = 1 if j is assigned to i, 0 if not  Consider a multicommodity flow network:  arc (j, i) of capacity x ij  j ∈ D is a source of commodity j with demand 1  i ∈ F is a sink of  commodity-oblivious capacity y i ∙U i  commodity-specific capacity y i ∙ 1 ≤3 ≤2 ≤3 y=1 y=0 y=1 All shown arcs are of capacity 1; others 0. Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject tothe flow network defined by (x, y) is feasible x ij, y i ∈ {0, 1 }

18. Our relaxation  Standard LP rewritten  ∀ i ∈ F y i = 1 if open, 0 if not  ∀ i ∈ F, j ∈ D x ij = 1 if j is assigned to i, 0 if not  Consider a multicommodity flow network:  arc (j, i) of capacity x ij  j ∈ D is a source of commodity j with demand 1  i ∈ F is a sink of  commodity-oblivious capacity y i ∙U i  commodity-specific capacity y i ∙ 1 ≤3 ≤2 ≤3 y=1 y=0 y=1 All shown arcs are of capacity 1; others 0. Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject tothe flow network defined by (x, y) is feasible x ij, y i ∈ [0, 1]

19. Our relaxation  Standard LP rewritten  ∀ i ∈ F y i = 1 if open, 0 if not  ∀ i ∈ F, j ∈ D x ij = 1 if j is assigned to i, 0 if not  Consider a multicommodity flow network:  arc (j, i) of capacity x ij  j ∈ D is a source of commodity j with demand 1  i ∈ F is a sink of  commodity-oblivious capacity y i ∙U i  commodity-specific capacity y i ∙ 1 Instance with one client and one facility with capacity ≤10 ≤3 ≤2 ≤3 y=1 y=0 y=1 All shown arcs are of capacity 1; others 0.

20. Our relaxation  Consider arbitrary partial assignment g : D ↛ F  Suppose that clients are assigned according to g  (x, y) should still give a feasible solution for the remaining clients: i.e.,  the flow network should still be feasible when  only the remaining clients have demand of 1  commodity-oblivious capacity of i is y i ∙(U i -|g - 1 (i)|)  In similar spirit as knapsack-cover inequalities [Wolsey 1975] [Carr, Fleischer, Leung, Phillips 2000]  LP constraint: the flow network defined by (g, x, y) is feasible for all g  Is this really a relaxation? ≤32 ≤21 ≤3 y=1 y=0 y=1 All shown arcs are of capacity 1; others 0. ≤3 ≤2 ≤3 x x

21. Our relaxation  Is this really a relaxation? No.  The only facility adjacent from remaining clients has 1 ∙0 = 0 commodity-oblivious capacity  Introduce backward edges corresponding to g  Now flows can be routed along alternating paths ≤3 ≤2 ≤30 y=1 y=0 y=1 All shown arcs are of capacity 1; others 0. ≤3 ≤2 ≤3 x

22. Our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ partial assignment g where MFN(g, x, y) is a multicommodity flow network with  arc (j, i) of capacity x ij,  arc (i, j) of capacity 1 if g assigns j to i,  j ∈ D is a source of commodity j with demand 1 if not assigned by g,  i ∈ F is a sink of  commodity-oblivious capacity y i ∙(U i -|g - 1 (i)|) and  commodity-specific capacity y i ∙ 1.

23. Our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ partial assignment g  Automatically embraces  Standard LP when g is the empty partial function  Knapsack-cover inequalities  Can be separated with respect to any given g  Algorithm uses feasibility of MFN(g, x, y) for a single g  Invoke standard techniques [Carr, Fleischer, Leung, Phillips 2000], [Levi, Lodi, Sviridenko 2007]

24. LP-rounding without exact separation  Relaxed separation oracle  Either finds a violated inequality or rounds the given point  Our rounding algorithm gives one  It does not rely on optimality  We can separate with respect to a given g

25. LP-rounding without exact separation  Relaxed separation oracle  Either finds a violated inequality or rounds the given point  (Optimization) ellipsoid method  Queries (exact) separation oracle with x*, which either  Finds an inequality violated by x*  Determines x* is feasible – c(x) ≤ c(x*) added to system  Can run with a relaxed separation oracle  Finds an inequality violated by x*  Rounds x* – c(x) ≤ c(x*) added to system

26. LP-rounding algorithm

27.  I ← {i ∈ F | y * i > ½}, S ← {i ∈ F | y * i ≤ ½}; open I  Can afford it  Choose a partial assignment g : D ↛ I  For each client j assigned by g,  assign j in the same way  Remaining clients are to be assigned to S Lemma We can choose a partial assignment g s.t.  g is cheap  MFN(g, x *, y * ) has a feasible flow where all flow is drained at S.

28. LP-rounding algorithm  Remaining clients are to be assigned to S  MFN(g, x *, y * ) remains feasible even when “restricted” to S  MFN( ∅, x *, y * ) is feasible when restricted to remaining clients i.e., it is feasible for the standard LP  Commodity-oblivious capacity of i ∈ S is y i ∙U i ≤ U i / 2  Capacity constraints are not tight  Can use Abrams et al.’s algorithm based on the standard LP  Finds 18-approx soln where capacities are violated by 2 Lemma We can choose a partial assignment g s.t.  g is cheap  MFN(g, x *, y * ) has a feasible flow where all flow is drained at S.

29. LP-rounding algorithm  Can use Abrams et al.’s algorithm based on the standard LP  Finds 18-approx soln where capacities are violated by 2  A 288-approximation algorithm  Obvious improvements, but perhaps not leading to ≤5  Determination of integrality gap remains open Lemma We can choose a fractional partial assignment g s.t.  g is cheap  MFN(g, x *, y * ) has a feasible flow where at least half of each commodity’s flow is drained at S.

30. Lemma We can choose a partial assignment g s.t.  g is cheap  MFN(g, x *, y * ) has a feasible flow where all flow is drained at S. LP-rounding algorithm

31.  Simplifying assumption  For each client j, all its incident edges in the support have equal costs  Support of x * := { (i, j) | x * ij > 0 } ≤2 ≤1 ≤2 x-values shown on edges 1/2 1/3 1 1 y=1 y=1/2 Lemma We can choose a partial assignment g s.t.  g is cheap  MFN(g, x *, y * ) has a feasible flow where all flow is drained at S. LP-rounding algorithm

32.  Simplifying assumption  For each client j, all its incident edges in the support have equal costs  Support of x * := { (i, j) | x * ij > 0 } ≤2 ≤1 ≤2 Lemma We can choose a partial assignment g s.t.  g is cheap  MFN(g, x *, y * ) has a feasible flow where all flow is drained at S. LP-rounding algorithm

33.  I ← {i ∈ F | y * i > ½}, S ← {i ∈ F | y * i ≤ ½}; open I  Find a maximum bipartite matching g on the support of x *  j ∈ D is matched at most once  i ∈ I is matched up to U i times  i ∈ S is not matched ≤2 ≤1 ≤2 I S

34. LP-rounding algorithm  I ← {i ∈ F | y * i > ½}, S ← {i ∈ F | y * i ≤ ½}; open I  Find a maximum bipartite matching g on the support of x *  j ∈ D is matched at most once  i ∈ I is matched up to U i times  i ∈ S is not matched ≤2 ≤1 ≤2 I S

35. LP-rounding algorithm  I ← {i ∈ F | y * i > ½}, S ← {i ∈ F | y * i ≤ ½}; open I  Find a maximum bipartite matching g on the support of x *  j ∈ D is matched at most once  i ∈ I is matched up to U i times  i ∈ S is not matched  Now we observe MFN(g, x *, y * )  There exists no path from a remaining client to a facility in I that is “undermatched” ≤2 ≤1 ≤2 I S ≤1 ≤2 I S

36. LP-rounding algorithm  I ← {i ∈ F | y * i > ½}, S ← {i ∈ F | y * i ≤ ½}; open I  Find a maximum bipartite matching g on the support of x *  j ∈ D is matched at most once  i ∈ I is matched up to U i times  i ∈ S is not matched  Now we observe MFN(g, x *, y * )  There exists no path from a remaining client to a facility in I that is “undermatched”  “Fully matched” facility has zero capacity  All flow drained at S ≤2 ≤1 ≤2 I S ≤21 ≤10 ≤2 I S x x x

37. Lifting the assumption

38.  Problem  g may become expensive compared to c(x*)  Solution  Find a fractional partial assignment g ≤ x *

39. Lifting the assumption  Problem  g may become expensive compared to c(x*)  Solution  Find a fractional partial assignment g ≤ x * Σ i ∈ I g ij ≤ 1 ∀ j ∈ D Σ j ∈ D g ij ≤ U i ∀ i ∈ I g ij = 0 ∀ i ∈ S

40. Lifting the assumption  Problem  g may become expensive compared to c(x*)  Solution  Find a fractional partial assignment g ≤ x * 1/2 1/3 1 1 S s t 1 1 1 1 1 1 1 2

41. Lifting the assumption  Problem  g may become expensive compared to c(x*)  Solution  Find a fractional partial assignment g ≤ x *  But we defined MFN(g, x *, y * ) only for integral g…

42. (Extending) our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ partial assignment g where MFN(g, x, y) is a multicommodity flow network with  arc (j, i) of capacity x ij,  arc (i, j) of capacity 1 if g assigns j to i,  j ∈ D is a source of commodity j with demand 1 if not assigned by g,  i ∈ F is a sink of  commodity-oblivious capacity y i ∙(U i -|g - 1 (i)|) and  commodity-specific capacity y i ∙ 1.

43. (Extending) our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ fractional part. asgn. g where MFN(g, x, y) is a multicommodity flow network with  arc (j, i) of capacity x ij,  arc (i, j) of capacity 1 if g assigns j to i,  j ∈ D is a source of commodity j with demand 1 if not assigned by g,  i ∈ F is a sink of  commodity-oblivious capacity y i ∙(U i -|g - 1 (i)|) and  commodity-specific capacity y i ∙ 1.

44. (Extending) our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ fractional part. asgn. g where MFN(g, x, y) is a multicommodity flow network with  arc (j, i) of capacity x ij,  arc (i, j) of capacity g ij,  j ∈ D is a source of commodity j with demand 1 if not assigned by g,  i ∈ F is a sink of  commodity-oblivious capacity y i ∙(U i -|g - 1 (i)|) and  commodity-specific capacity y i ∙ 1.

45. (Extending) our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ fractional part. asgn. g where MFN(g, x, y) is a multicommodity flow network with  arc (j, i) of capacity x ij,  arc (i, j) of capacity g ij,  j ∈ D is a source of commodity j with demand d j := 1 - Σ i ∈ I g ij,  i ∈ F is a sink of  commodity-oblivious capacity y i ∙(U i -|g - 1 (i)|) and  commodity-specific capacity y i ∙ 1.

46. (Extending) our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ fractional part. asgn. g where MFN(g, x, y) is a multicommodity flow network with  arc (j, i) of capacity x ij,  arc (i, j) of capacity g ij,  j ∈ D is a source of commodity j with demand d j := 1 - Σ i ∈ I g ij,  i ∈ F is a sink of  commodity-oblivious capacity y i ∙(U i - Σ j ∈ D g ij ) and  commodity-specific capacity y i ∙ 1.

47. (Extending) our relaxation Minimize Σ i ∈ F o i y i + Σ i ∈ F, j ∈ D c ij x ij subject toMFN(g, x, y) is feasible ∀ fractional part. asgn. g where MFN(g, x, y) is a multicommodity flow network with  arc (j, i) of capacity x ij,  arc (i, j) of capacity g ij,  j ∈ D is a source of commodity j with demand d j := 1 - Σ i ∈ I g ij,  i ∈ F is a sink of  commodity-oblivious capacity y i ∙(U i - Σ j ∈ D g ij ) and  commodity-specific capacity y i ∙d j.

48. Lifting the assumption Lemma We can choose a fractional partial assignment g s.t.  g ≤ x *  MFN(g, x *, y * ) has a feasible flow where at least half of each commodity’s flow is drained at S.

49. Lifting the assumption Lemma We can choose a fractional partial assignment g s.t.  g ≤ x *  MFN(g, x *, y * ) has a feasible flow where at least half of each commodity’s flow is drained at S.  Hoping too much: y=.01 ≤1 y=.99.01.99.01d j = 0.0001.01.99

50. Lifting the assumption Lemma We can choose a fractional partial assignment g s.t.  g ≤ 2x *  MFN(g, x *, y * ) has a feasible flow where at least half of each commodity’s flow is drained at S.  What’s the goal?  Goal is not in making MFN(g, x *, y * ) feasible – the ellipsoid method guarantees this  Will find an assignment that leaves on clients “too much” demand to be served by remaining capacities in I

51. Lifting the assumption  Two different type of “paths”  In MFN(g, x *, y * )  In the residual graph of the partial assignment  Previous argument  A path from a client with positive demand to a facility in I with positive capacity corresponds to a path from an undermatched client to an undermatched facility and therefore does not exist.

52. LP-rounding algorithm  I ← {i ∈ F | y * i > ½}, S ← {i ∈ F | y * i ≤ ½}; open I  Find a maximum bipartite matching g on the support of x *  j ∈ D is matched at most once  i ∈ I is matched up to U i times  i ∈ S is not matched  Now we observe MFN(g, x *, y * )  There exists no path from a remaining client to a facility in I that is “undermatched” ≤2 ≤1 ≤2 I S ≤1 ≤2 I S

53. Lifting the assumption  Two different type of “paths”  In MFN(g, x *, y * )  In the residual graph of the partial assignment ≤1 All weights on edges 1/2 x*x*g ≤1 Residual Graph ≤1 MFN(g, x *, y * )

54. Lifting the assumption  Consider a maximum fractional matching g and its residual graph s t

55. Lifting the assumption  Consider a maximum fractional matching g and its residual graph  R : reachable from an undermatched client  N : not reachable s t DRDR DNDN IRIR ININ S

56. Lifting the assumption  Consider a maximum fractional matching g and its residual graph  R : reachable from an undermatched client  N : not reachable  Consider MFN(g, x *, y * ) s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S

57. Lifting the assumption  No facility in I R is undermatched a  R : reachable from an undermatched client  N : not reachable s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S

58. Lifting the assumption  No facility in I R is a sink s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinks

59. Lifting the assumption  No facility in I R is a sink  Every client with positive demand is in D R s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources

60. Lifting the assumption  No facility in I R is a sink  Every client with positive demand is in D R  Consider j ∈ D R and i ∈ I N  (j, i) is saturated: g ij = 2x* ij s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j j i i ×

61. Lifting the assumption  Consider j ∈ D R and i ∈ I N  (j, i) is saturated: g ij = 2x* ij  Set g ij ← 0  Increases d j by Σ i ∈ I N 2x* ij, which is twice the total capacity from j to I N s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j j

62. Lifting the assumption  For every j ∈ D R, its demand is at least twice the total capacity of arcs from j to I N in MFN(g, x *, y * ) s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j j

63. Lifting the assumption  For every j ∈ D R, its demand is at least twice the total capacity of arcs from j to I N in MFN(g, x *, y * ) Compare to:  For every j ∈ D R, there does not exist a path from j to I N in MFN(g, x *, y * ) s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j j

64. Lifting the assumption  Say: arcs from j to I N is labeled as commodity j s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources

65. Lifting the assumption  Say: arcs from j to I N is labeled as commodity j  Want: there exists a feasible flow where, for every j ∈ D R, every arc from j to I N is used by commodity j s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources

66. Lifting the assumption  Say: arcs from j to I N is labeled as commodity j  Recall g i’j’ ← 0 for j’ ∈ D R and i’ ∈ I N s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources ×

67. Lifting the assumption  Say: arcs from j to I N is labeled as commodity j  Recall g i’j’ ← 0 for j’ ∈ D R and i’ ∈ I N  Suppose g i’j’ > 0 for some j’ ∈ D N and i’ ∈ I R s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j’ × i’

68. Lifting the assumption  Say: arcs from j to I N is labeled as commodity j  Recall g i’j’ ← 0 for j’ ∈ D R and i’ ∈ I N  Suppose g i’j’ > 0 for some j’ ∈ D N and i’ ∈ I R s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources ×

69. Lifting the assumption  Say: arcs from j to I N is labeled as commodity j  Recall g i’j’ ← 0 for j’ ∈ D R and i’ ∈ I N  Suppose g i’j’ > 0 for some j’ ∈ D N and i’ ∈ I R  Any flow path ending at I N contains an arc from D R to I N s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources

70. Lifting the assumption  Any flow path ending at I N contains an arc from D R to I N s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j

71. Lifting the assumption  Any flow path ending at I N contains an arc from D R to I N  Suppose j ∈ D R drains the highest fraction of its flow at I N and this fraction is >1/2  Want: there exists a feasible flow where, for every j ∈ D R, every arc from j to I N is used by commodity j s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j

72.  Want: there exists a feasible flow where, for every j ∈ D R, every arc from j to I N is used by commodity j  Any flow path ending at I N contains an arc from D R to I N  Suppose j ∈ D R drains the highest fraction(>½) of its flow at I N  There is a flow path of j that “steals” an arc of k ∈ D R Lifting the assumption s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j k

73.  Want: there exists a feasible flow where, for every j ∈ D R, every arc from j to I N is used by commodity j  Any flow path ending at I N contains an arc from D R to I N  Suppose j ∈ D R drains the highest fraction(>½) of its flow at I N  There is a flow path of j that “steals” an arc of k ∈ D R Lifting the assumption s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j k

74.  Want: there exists a feasible flow where, for every j ∈ D R, every arc from j to I N is used by commodity j  k ∈ D R recovered its stolen arc Lifting the assumption s t DRDR DNDN IRIR ININ S DRDR DNDN IRIR ININ S possible sinkspossible sources j k

75. Main result There is a good LP: its optimum is within a constant factor of the true optimum. In particular, there is a poly-time algorithm that finds a solution whose cost is within a constant factor of the LP optimum. Theorem

76. The Question  Can we use LP-based techniques to solve the capacitated facility location problem? Yes!  Can we use other LP-based techniques with our new relaxation?  Can we apply these results to related problems?  Can we improve our integrality gap bounds?

77. Thank you.