Recent advances in facility location optimization

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Recent advances in facility location optimization

• Binay K Bhattacharya
• School of Computing Science
• Simon Fraser University

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What is facility locating?
Client needs this kind of service.
Facility provides some kind of service.
GOOD SERVICE
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Location problems in most general form can be
stated as follows

• a set of clients originates demands for some
  kind of goods or services.
• the demands of the customers must be supplied by
  one or more facilities.
• the decision process must establish where to
  locate the facilities.
• issues like cost reduction, demand capture,
  equitable service supply, fast response time etc
  drive the selection of facility placement.

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The basic elements of location models are

• a universe, U, from which a set C of client
  input positions is selected,
• a distance metric, d U x U ? R, defined over
  the universe R,
• an integer, p 1, denoting the number of
  facilities to be located, and
• an optimization function g that takes as input a
  set of client positions and p facility positions
  and returns a function of their distances as
  measured by the metric d.

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Problem statement

• Select a set F of p facility positions in
  universe U that minimizes g(F,C).

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Models of the universe of clients and facilities

• Continuous space
• Universe is defined as a region, such that
  clients and facilities may be placed anywhere
  within the continuum, and the number of
  possible locations is uncountably infinite.

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• Discrete space
• Universe is defined by a discrete set of
  predefined positions.
• Network space
• Universe is defined by an undirected weighted
  graph. Client positions are given by the
  vertices. Facilities may be located anywhere on
  the graph.

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Where should the radio tower be located?

• Tower may be positioned anywhere
  (continuous)
• Tower may be positioned in five available slots
  (discrete)
• Tower may be located on the roadside
  (network)

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Distance metric

• Distance metric between two elements of the
  universe further differentiates between specific
  problems
• Minkowski distance (Euclidean, Manhattan etc)
• Network distance

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Optimization function sum or maximum

• The objective function is the most significant
  characterization of a facility location problem.
• p-center problem
• Given a universe U, a set of points C, a metric
  d, and a positive integer p, a p-center of C is a
  set of p points F of the universe U that
  minimizes
• maxjeC minieF dij.

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p-median problem Given a universe U, a set of
points C, a metric d, and a positive integer p, a
p-median of C is a set of p points F of the
universe U that minimizes SjeC minieF dij.
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Euclidean p-center
3-center
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Euclidean p-center

• (Megiddo and Supowit, 1984) p-center is
  NP-hard.
• (Feder and Green 1988) e-approximation remains
  NP- hard for any e lt (1v7)/2 1.8229.
• (Hwang, Lee and Chang 1993) p-center problem in
  R2 can be solved in O(nvp) time.

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Summary
Time complexities of algorithmic solutions to the
Euclidean p-center problem

• Drezner 1984, Hoffmann 2005 (arbitrary p and d
  1)
• Megiddo 1983 (p1, d2)
• Agarwal et al 1993, Chazelle et al. 1995 (p 1
  and d fixed)
• Chan 1999 (p2, d2)
• Agarwal and Sharir 1998 (p2, d arbitrary)
• Agarwal and Procopuic 1998 (p fixed, d arbitrary)

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Euclidean p-median
1-median
Minimize the average distance to the facility
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Euclidean p-median

• 1-median is not unique if the points are
  linearly dependent.
• 1-median is unique if the points are linearly
  independent (Kupitz and Martini, 1997)
• Difficult to compute 1-median in the plane.
  Bajaj(1987, p 177) states there exist no
  exact algorithm under models of computation
  where the root of an algebraic equation is
  obtained using arithmetic operations and the
  extraction of kth roots
• Chandrasekhar and Tamir (1990) a polynomial
  time algorithm for an e-approx of the 1-median.
• Indyk (1999) and Bose et al (2003) linear in n
  and polynomial in 1/ e (1-median)

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Euclidean p-median

• Hassin and Tamir(1991) O(pn) time solution for a
  line.
• Suppowit and Meggido (1984) Problem is NP-hard
  in the plane, and can not be approximated to
  within 3/2.

Summary of results
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Three problems

• Mobile facility location problem.
• p-median problem in tree networks
• Approximation algorithms for network facility
  location problems.

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Mobile Facility Location
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Collaborators Sergey Bereg, Univ. of Texas at
Dallas Binay Bhattacharya, SFU Stephane
Durocher, UBC David Kirkpatrick, UBC Michael
Segal, Ben-Gurion Univ.
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(Static) Facility Location
sites/clients
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(Static) Facility Location
sites/clients
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Static Facility Location
sites/clients
facilities/servers
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Center problems
1-center
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Center problems
1-center
Minimize the maximum distance to the facility
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Center problems
p-center
Minimize maximum distance to the closest facility
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Median problems
1-median
Minimize the average distance to the facility
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Dynamic Facility Location
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Dynamic Facility Location
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Dynamic Facility Location
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Dynamic Facility Location
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• discrete changes (insert/delete)
• focus on data structures to avoid
• (full) recomputation

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Mobile Facility Location
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Mobile Facility Location
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Mobile Facility Location
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• clients change position continuously
• goal update facility location(s) in
• a well-behaved fashion

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• New constraints/criteria
• facility locations should change
• continuously (unlike exact 2-center)

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• New constraints/criteria
• facility locations should change
• continuously (unlike exact 2-center)
• facilities may be subject to bounds
• on velocity (unlike exact 1-center)

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p1
p3
p2
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p1
c
c
p2
p3
p2
p3
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p1
c
c
p2
p3
p2
p3
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• New constraints/criteria
• facility locations should change
• continuously (unlike exact 2-center)
• facilities may be subject to bounds
• on velocity (unlike exact 1-center)
• quality of approximation

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• New constraints/criteria
• facility locations should change
• continuously (unlike exact 2-centre)
• facilities may be subject to bounds
• on velocity (unlike exact 1-centre)
• quality of approximation
• simplicity of facility motion

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Mobile 1-center results

• 1-dimensional Exact 1-center can be
  maintained (need appropriate data structures)

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Mobile 1-center results

• 2-dimensional
• Theorem For any velocity v 0, there exist
  three sites such that a unit velocity motion of
  the two of the sites induces an instantaneous
  velocity gt v of the Euclidean 1-center.

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Mobile 1-center results

• 2-dimensional (approximate)
• Observation Let a1, a2,.., an be fixed
  nonnegative numbers such that a1 a2.. an 1.
  If all the sites s1, s2, , sn move with velocity
  at most 1, a1s1 a2s2.. ansn also moves with
  velocity at most 1.

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Mobile 1-center results

• 2-dimensional (approximate)
• Lemma The center of mass of a set of n sites
  provides (2-2/n)-approximation of the Euclidean
  1-center. The facility moves with velocity at
  most 1.

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Mobile 1-center results

• 2-dimensional (approximate)
• Lemma The center of mass of a set of n sites
  provides (2-2/n)-approximation of the Euclidean
  1-center. The facility moves with velocity at
  most 1.
• Surprisingly, the above approximation factor is
  asymptotically optimal.

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Mobile 1-center results

• Higher velocity approximation.
• Bounding box center

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Mobile 1-center results

• Higher velocity approximation.
• Bounding box center

Bounding box center moves with velocity at most
v2. The center is (1 v2)/2 approxi- mation of
the Euclidean 1-center.
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Mobile 1-center results

• Summary
• 2-approximation is realizable with velocity 1.
• (1v2)/2-approximation is realizable with
  velocity v2.

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Mobile 1-center results

• Summary
• 2-approximation is realizable with velocity 1.
• (1v2)/2-approximation is realizable with
  velocity v2.

Question What approximation factor can be
achieved if the velocity of the facility f is
restricted to some constant between 1 and v2 ?
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Mobile 1-center results

• Summary
• 2-approximation is realizable with velocity 1.
• (1v2)/2-approximation is realizable with
  velocity v2.

Question What approximation factor can be
achieved if the velocity of the facility f is
restricted to some constant between 1 and v2 ?
A combination of the center of mass and the
center of the bounding box.
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Mobile 1-center results

• (f1, v1) (location, velocity) of center of
  mass
• (f2, v2) (location, velocity) of bounding box
  center
• Mixing strategy
• (f, v) (location, velocity) of the mixing
  center where
• f a f1 (1- a) f2, and
• v a v1 (1-a) v2, 1 a 0.

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Mobile 1-center results
Upper Bound Lemma For any e gt 0, there is a
strategy such that (a) the approximation factor
is (1e), and (b) the velocity never exceeds
(2e)(1e)/v(2e e2).
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Mobile 1-center results
Lower Bound Lemma For any e gt 0, any
(1e)-approximate mobile 1-center has velocity at
least 1/(8ve).
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Approximations to the Euclidean 1-center

• Strategy Approximations vmax

Euclidean center 1 8 Center of
mass 2 1 Bounding box (1v2)/2 1.2071 v2
1.4142 Gaussian center 1.1153 4/p
1.2732
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Euclidean 1-median

• The Euclidean 1-median moves with arbitrarily
  high velocity.
• The center of mass provides a (2-2/n)-approximati
  on using unit velocity
• There are examples where the approximation ratio
  of the center of mass is arbitrarily close to 2.
• An approximation ratio better than 2/v3 to the
  Euclidean 1-median is impossible where the
  facility is constrained to move no faster than
  the clients.

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Open Problems

• Provide tighter bounds for mobile 1-center and
  1-median problems.
• k-center and k-median, k 2?
• 3-dimensions?
• Clusterings?

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p-median problem in trees
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Vertex optimality of p-median

• Theorem (Hakimi, 1965) There always exists an
  optimal p-median solution where the facilities
  are placed only at the vertices of the network.

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Vertex optimality of p-median

• Theorem (Hakimi, 1965) There always exists an
  optimal p-median solution where the facilities
  are placed only at the vertices of the network.

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Open problems

• The conjecture of Chrobak et al gj(Tx) e
  O(Tx). If true, this will have
  significant impact on the complexity.
• Can we remove the requirement that p is fixed?
• able to do when the tree is balanced
• currently, not so when the tree is not
  unbalanced.
• Extension of these to partial k-trees?

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Summary of problems solved for tree networks
Problem Vertex Current Prev. weights
best best p-median (const p)
O(nlogp2n) O(pn2) 3-median
O(nlog3n) O(n2) 2-median (MWD)
/- O(nlogn) O(nlog2n) 2-median (WMD)
/- O(nhlog2n) O(n3) p-center (const p)
O(n) O(nlog2n) 1-center
/- O(nlog3n) O(n2) Collection
depots 1-median O(nlogn) O(n2) 1-center
O(n) O(n2)
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Collaborators

• Boaz Ben Moshe
• Robert Benkoczi
• David Breton
• Binay Bhattacharya
• Qiaosheng Shi

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Papers

• Discrete Mathematics (to appear)
• MFCS 2003,
• ESA 2005
• ISAAC 2005
• LATIN 2006
• unpublished

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Approximation algorithms
A ?-approximation algorithm for an optimization
problem is a polynomial-time algorithm that is
guaranteed to find a feasible solution of the
objective function value within a factor of ? of
the optimal. The performance guarantee of the
algorithm is ?.
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Three algorithmic techniques

• LP Rounding Rounding fractional optimal
  solution to nearbyoptimal integral solution.

• Primal-Dual Use LP implicitly to find a
  solution

• Local Search Iteratively improving the
  integer solution searching nearby solutions.

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Overview of LP rounding

• Formulate the problem as an IP problem.
• Solve the corresponding LP-relaxation. LP-relaxed
  solution is a lower bound on IP.
• Round the relaxed optimal solution.
• Show that the rounding does not increase the cost
  too much.

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Uncapacitated facility location problem

• The universe is a network which is a complete
  graph G.
• The client set (C) and the facility set (F) are
  subset of vertices of G.
• Each facility i of F has an opening cost fi if it
  is open.
• Problem is to select a subset of the facilities
  such that the total cost to serve all the
  clients of C is minimized.

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Mathematical models

• Notations used
• C set of customers/clients.
• F set of candidate facility locations.
• wj service demand of customer j.
• fi fixed cost of establishing a facility at
  location i.
• dij per unit cost of servicing (distance of)
  customer j from facility i.
• yi decision variable which takes value 1 if
  facility i is opened, otherwise it is 0.
• xij customer js demand is supplied from
  facility i.

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Integer programming formulation
Xij 1
j
i
facilities clients
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Integer programming formulation
Xij 1
j
i
Xij 0
i
facilities clients
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Integer programming formulation
Xij 1
j
i
facilities clients
yi 1
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Integer programming formulation
Corresponding LP-relaxed formulation (Primal)
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Dual LP formulation
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Interplay between the primal and dual variables

• Let (x,y) and (v,ß) be the optimal solution
  of LP-Primal and LP-Dual respectively.
• SjeC vj LP-Opt (strong duality theorem)
• SjeC vj LP-Opt for any feasible v (weak
  duality theorem)
• If xij gt 0 implies cij ? vj (closeness
  property)

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Assume that (x,y) and (v,ß) are known. We
can visualize the fractional solution of LP-R as
follows
?i xij 1 ? j
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Shmoys, Tardos and Aardal (1997)
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One iteration

• Select the remaining client j with minimum vj.
• Construct a cluster which includes

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One iteration

• Select the remaining client j with minimum vj.
• Construct a cluster which includes
• client j

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One iteration

• Select the remaining client j with minimum vj.
• Construct a cluster which includes
• client j
• all facilities used fractionally by j

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One iteration

• Select the remaining client j with minimum vj.
• Construct a cluster which includes
• client j
• all facilities used fractionally by j
• all clients that use this facility
  (fractionally)

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One iteration
action

• Open the facility in the cluster with the
  smallest opening cost.
• Assign all the clients of the cluster to the
  opened facility.

j
i
Observe that the unopened facilities of this
cluster are never going to be in any other
cluster.
facilities clients
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Assignment cost of a client k in the cluster
j
i
xij gt 0
k
facilities clients
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Assignment cost of a client k in the cluster
i(j)
j
i
xij gt 0
k
facilities clients
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Assignment cost of a client k in the cluster
k is directly connected to facility (i(j))
i(j)
j
i
xi(j)k gt 0
Due to the closeness property the assignment cost
of client k is at most vk i.e. ci(j)k vk
k
facilities clients
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Assignment cost of a client k in the cluster
i(j)
j
i
From triangle inequality ci(j)k ? cik cij
ci(j)j ? vk vj vj ? 3vk
i
k
facilities clients
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Assignment cost of a client k in the cluster
i(j)
From triangle inequality ci(j)k ? cik cij
ci(j)j ? vk vj vj ? 3vk
j
i
i
k
Total assignment cost is at most 3?? j ? C vj,
which is at most 3LP-opt ? 3IP-opt.
facilities clients
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opening cost of i(j) fi(j)
fi(j) min fi
i(j)
facilities used by j
? ? fi xij
facilities used by j
? ? fi yi
facilities used by j
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opening cost of i(j) fi(j)
fi(j) min fi
i(j)
facilities used by j
? ? fi yi
facilities used by j
Total opening cost is ?
fiyi , which is at most LP-opt ? IP-opt.
open facilities
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Theorem The LP-rounding algorithm is a
4-approximation algorithm for the uncapacitated
facility location problem.
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Clustered randomized rounding(Chudak and Shmoys,
2003)

• Choose the cluster center in increasing vj Si
  e F cijxij, (let j be the center)
• In cluster centered at j, open a facility i at
  random with probabilty xij.
• Open independently each facility i that is not
  contained in the neighborhood of any cluster with
  probability yi.
• Assign each client to its nearest open facility.

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Clustered randomized rounding(Chudak and Shmoys,
2003)

• Choose cluster center in increasing vj Si e F
  , say j.
• In cluster centered at j, open a facility i at
  random with probabilty xij.
• Open independently each facility i that is
  contained in the neighborhood of any cluster with
  probability yi.
• Assign each client to its nearest open facility.

• Every client will have an open facility in its
  neighbor-hood (directly connected) with large
  probability (gt 1-1/e ).
• Expected cost of the solution is (12/e)IP-opt.

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Primal-Dual method
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Integer programming formulation
Corresponding LP-relaxed formulation
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Dual LP formulation
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Algorithm of Jain and Vazirani, 1999

• LP-R or LP-Dual solution is not required.
• First constructs a feasible dual solution (v,ß)
  and then using the dual solution constructs an
  integer feasible solution (x,y) of LP-R, and
  hence for IP.
• (x,y) and (v,ß) satisfy the Dual CS condition
  and partially satisfy the Primal CS condition.
• The algorithm is a 3-approximation algorithm.

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Interpretation of the dual variables

• Suppose I F and f C ? I is an optimal
  integral solution. i.e. yi 1 iff i e I and xij
  1 iff i f(j).
• Suppose (v,ß) denotes an optimal dual solution.
• If i e I, Sj e C ßij fi .
• Each open facility is fully paid for by the
  clients using the facility.
• If i f(j), vj ßij cij.
• We can think of vj as the total price paid by
  client j of this cij goes towards the use of
  edge (i,j) and ßij is the contribution of j
  towards the opening of facility i.

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Some terminologies

• A facility is fully paid when Sj e C ßij fi .
• A client j has reached a facility i if vj cij.
  
• If, in addition, i is fully paid, j gets
  connected to i.

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Algorithm to compute a feasible (v,ß)(phase 1)

• Set vj ? 0 ßij ? 0 for all i and j.
• UNTIL all j e C are connected DO
• -- increase vj for all unconnected j .
• -- increase ßij for all i and j satisfying
• j has reached i
• j is not yet connected
• i is not fully paid.

4
c
3
a
1
b
2
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Algorithm to compute a feasible (v,ß)(phase 1)

• Set vj ? 0 ßij ? 0 for all i and j.
• UNTIL all j e C are connected DO
• -- increase vj for all unconnected j.
• -- increase ßij for all i and j satisfying
• j has reached i
• j is not yet connected
• i is not fully paid.

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Algorithm to compute a feasible (v,ß)(phase 1)

• Set vj ? 0 ßij ? 0 for all i and j.
• UNTIL all j e C are connected DO
• -- increase vj for all unconnected j .
• -- increase ßij for all i and j satisfying
• j has reached i
• j is not yet connected
• i is not fully paid.

1
a
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Algorithm to compute a feasible (v,ß)(phase 1)

• Set vj ? 0 ßij ? 0 for all i and j.
• UNTIL all j e C are connected DO
• -- increase vj for all unconnected.
• -- increase ßij for all i and j satisfying
• j has reached i
• j is not yet connected
• i is not fully paid.

4
c
3
a
1
b
2
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Algorithm to compute a feasible (v,ß)(phase 1)

• Set vj ? 0 ßij ? 0 for all i and j.
• UNTIL all j e C are connected DO
• -- increase vj for all unconnected.
• -- increase ßij for all i and j satisfying
• j has reached i
• j is not yet connected
• i is not fully paid.

4
c
3
a
1
b
2
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Algorithm to compute a feasible (v,ß)(phase 1)

• Set vj ? 0 ßij ? 0 for all i and j.
• UNTIL all j e C are connected DO
• -- increase vj for all unconnected.
• -- increase ßij for all i and j satisfying
• j has reached i
• j is not yet connected
• i is not fully paid.

4
c
3
a
1
b
2
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Algorithm to compute a feasible (v,ß)(phase 1)

• Set vj ? 0 ßij ? 0 for all i and j.
• UNTIL all j e C are connected DO
• -- increase vj for all unconnected.
• -- increase ßij for all i and j satisfying
• j has reached i
• j is not yet connected
• i is not fully paid.

4
c
3
1
a
b
2
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Algorithm to compute an integral (x,y)(phase 2)
j
i
ßij gt 0
4
c
3
1
a
b
2
facilities clients
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Algorithm to compute an integral (x,y)(phase 2)
j
i
ßij gt 0
4
c
j
3
i
1
a
b
2
facilities clients
i is fully paid before i.
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Algorithm to compute an integral (x,y)(phase 2)
j
Case j is directly connected to i, i.e. ßij
gt0 From Phase 1 algorithm vj - ßij cij
i.e. vi cij
i
ßij gt 0
j
i
facilities clients
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Algorithm to compute an integral (x,y)(phase 2)
j
Case j is indirectly connected to i. i.e.
ßij 0. From triangle inequality cij cij
cij cij. Since the facility i is fully paid
before the facility i, vj vj. Therefore
cii 3vj.
i
ßij gt 0
j
i
facilities clients
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One iteration

• Select the earliest fully paid facility i not
  opened yet (set yi 1).
• Remove all clients j with ßij gt0 (set xij 1)

j
i
ßij gt 0
facilities clients
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One iteration

• Select the cheapest fully paid facility i not
  opened yet (set yi 1).
• Remove all clients j with ßij gt0 (set xij 1)
• Remove all clients j where j is indirectly
  connected to i. (set xij 1)

j
i
ßij gt 0
j
facilities clients
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Relaxed complementary slackness (CS) property
Let (x,y) and (v,ß) be the solution determined
during Phase 1 and 2 respectively. Relaxed
Primal CS xij gt 0 implies 1/3cij vj ßij
cij for all i, j. yi gt 0 implies Sj e C ßij
fi for all i Dual CS vj gt 0 implies Si e F
xij 1 for j ßij gt 0 implies xij yi for all
i, j
Theorem The primal-dual algorithm of Jain and
Vazirani is a 3-approximation
algorithm. Charikar and Guha(1999) showed that
this algorithm alone can not improved the
solution any further.
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Summary of UFLP approximation algorithms
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Generalizations

• p-median
• Charikar, Guha, Tardos and Shmoys
• Arya, Garg, Khandekar, Pandit, Myerson and
  Munagala
• Jain and Vazirani
• Charikar and Guha
• Jain, Mahadian and Saberi
• ((1 2/e) lower bound)

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Generalizations

• Capacitated facility location
• Soft
• Shmoys, Tardos, Aardal
• Chudak and Shmoys
• Jain and Vazirani
• Korupolu, Rajaraman and Plaxton
• Chudak and Williamson

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Generalizations

• Capacitated facility location
• Hard
• Pal, Tardos and Wexler

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Generalizations

• Fault tolerant facility location
• Jain and Vazirani
• Guha, Myerson and Munagala
• Swamy and Shmoys
• Jain, Mahdian, Markakis, Saberi, Vazirani

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Generalizations

• Facility location with penalties
• Charikar, Khullar, Mount and Narasimhan
• Jain, Mahdian, Marakis, Saberi and Vazirani

• Minimum sum of cluster diameters
• ..

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Conclusions

• Rich source of problems
• requiring exact solution
• requiring approximate solution
• requiring fast solution
• Developed tools have wider applications
• scheduling
• clustering
• Most of the problems in a general network is
  NP-hard
• partial k-trees where k is small?

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Thank you