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

Capacitated facility location problem  Given a metric cost c on  D: set of clients  F: set of facilities

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

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

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

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) + ( )

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

Successful special case  Uncapacitated facility location problem  U i = ∞ ∀ i  NP-hard to approximate better than [Guha & Khuller 1999] [Sviridenko]  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 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 approximation [Sviridenko 2002], primal-dual 1.52-approximation [Mahdian, Ye, Zhang 2006], combining 1.5-approximation algorithm [Byrka & Aardal 2010]

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

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

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

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

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]

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

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

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.

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 }

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]

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

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

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.

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]

LP-rounding algorithm

 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.

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.

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.

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

 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

 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

 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

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

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

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

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

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?

Thank you.