The Matroid Median Problem
Viswanath Nagarajan
IBM Research
Joint with R. Krishnaswamy, A. Kumar,
Y. Sabharwal, B. Saha
k-Median Problem
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Set of locations in a metric space (V,d)
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Symmetric, triangle inequality
Place k facilities such that sum of connection
costs (to nearest facility) is minimized:
minFµV, |F|·k u2V d(u,F)
k-Median Results
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poly(log n) approx via tree embeddings [B ’96]
LP rounding O(1)-approx [CGST ’99]
Lagrangian relaxation + primal dual [JV ’01]
Local search with p-exchanges [AGKMMP ’04]
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best known ratio 3+²
Hardness of approximation ¼ 1.46
[GK ’98]
Red-Blue Median
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Facilities are of two different types
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Separate bounds kr and kb on facilities
Recently introduced
[HKK ’10]
Motivated by Content Distribution Networks
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Partition V into red and blue sets
T facility-types (RB Median is T=2)
O(1)-approximation ratio via Local Search
kr=3
kb=2
Matroid Median
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Given matroid M on ground-set V
Locate facilities F that are independent in M
Minimize connection cost
Recap matroid M=(V, Iµ2V)
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A,B 2 I and |A|<|B| ) 9 e 2 BnA : A[{e} 2 I
Substantial generalization of RB Median
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The CDN application with T facility-types reduces
to partition matroid constraint
A
e
B
k1=2 k2=3 k3=1 k4=2
Talk Outline
Thm: 16-approximation for Matroid Median
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Bad example for Local Search
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LP relaxation
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Phase I : sparsification
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Phase II: reformulation
Local Search?
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Partition matroid with T parts
T-1 exchange local search
 Swap up to T-1 facilities in each step
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Unlikely to work beyond T=O(1)
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m
1
m
m
m
m
Eg. T=5
Uniform metric on T+1
Clients n=mT+1
OPT = 1 (small fac.)
LOPT = m (big fac.)
locality gap (n/T)
LP relaxation
min u v d(u,v) ¢ xuv
s.t. v xuv = 1
8u2V
xuv · yv
8 u,v 2 V
v2S yv · r(S)
8 Sµ V
x, y ¸ 0.
u
xuv
v
y2M
clients
facilities
connection
constraints
matroid rank
constraints
Solving the LP
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Exponential number of rank constraints
Use separation oracle:
minSµV r(S) - v2S yv
An instance of submodular minimization
Also more efficient algorithms to separate
over the matroid polytope
[C ’84]
Solvable in poly-time via Ellipsoid algorithm
Idea for approach(1)
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Problem non-trivial even if metric is a tree 
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Even O(log n)-approximation not obvious
What’s easier than a tree?
Suppose input is special star-like instance
One root facility (can help any client)
client 2
Others are private facilities (help only 1 client)
client 1
client 3
root facility
Idea for approach(2)
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Recall LP variables
 yj : facility opening (in matroid polytope)
 xij : connection
For any client i, private j 2 P(i) WMA xij = yj
Connection constraint j xij = 1
So xir = 1 - j2P(i) xij = 1 - j2P(i) yj
Can eliminate all connection variables !
client i
private facilities P(i)
r
Idea for approach(3)
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Reformulate the LP
xij
xir
min i [ j2P(i) dij ¢ yj + dir¢(1- j2P(i) yj) ]
s.t.
j2P(i) yj · 1, 8 clients i
To ensure xir ¸ 0
y2M
matroid constraint
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This is just an instance of intersection of M
with partition matroid from P(i)s 
Idea for approach(4)
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Start with LP optimum (x,y) of arbitrary
matroid median instance
Phase I: Use (x,y) to form clusters of disjoint
star-like instances
Phase II: Resolve the new star-LP
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(x,y) itself restricted to the stars not integral
Show that new LP is integral
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¼ matroid intersection
Phase I: sparsify LP solution
Outline
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Modify LP connections x in four steps
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Key: no change in facility variables y
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Similar to [CGST ’99]
Need to ensure y remains in matroid polytope
Not true in [CGST ’99]
Require some more (technical) work
Step 1: cluster clients
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Lu = v duv¢xuv, contribution of u to LP obj.
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B(u) is local ball of u
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vertices within distance 2¢Lu from u
Order clients u in increasing Lu
Pick maximal disjoint set of local balls
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3
1
2
T are the chosen clients
Move each client to T-client close to it
5
3
4
6
1
4
5
2
Loss in obj · 4¢ LP*
(additive)
6
Obs on step 1
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Local balls of T clients are disjoint
y-value inside any local ball ¸ ½
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Markov inequality
Restrict to clients T (now weighted)
For any p,q2T : d(p,q) ¸ 2¢(LPp + LPq)
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well separated clients
y¸½
separated
T balls
More obs on step 1
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Suppose y-value in each T’s local ball ¸ 1
Then instance of matroid intersection:
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Matroid M and partition from local-ball(T)
Resolving suitable LP ) integral soln
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Will need intersection with `laminar’
constraints, not just partition matroid
Step 2: private facilities
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Ensure that each facility in some T-ball or
helps at most one client (ie. private)
Break connections from all except closest
client 1 to facility j
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Reconnect to facilities in B(1), y-value ¸ ½
Total reconnection for any client · ½
j
3
2
1
Constant factor
loss in obj
Step 3: uniform objective
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Each connection from client p to any facility
in B(q) will pay same objective d(p,q)
Since p,q well separated d(p,q) · O(1)¢ d(p,j)
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For any j 2 B(q)
Constant factor loss in obj
p
q
Step 4: building stars
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WMA each client i 2 T connected to
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Set of `outer’ connections ¼ directed tree
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Its private facilities P(i), OR
Its closest other client k2T, ie. facility in B(k)
Unique out-edge from each client
Lem: Can modify outer connection to `star’
Constant factor
loss in obj
The star structure
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One pseudo-root { r, r’ }
Every other client connected to either r or r’
All LP-connections x are from client i to:
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private facility j2P(i), obj d(i,j) OR
facility in B(k) with k2{ r, r’ }, uniform obj d(i,k)
r
r’
i
Phase II: using star
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Will drop all the connection x-variables
WMA xij = yj for j2P(i) private facilities
Total outer connection=1 - j2P(i) xij =1 - j2P(i) yj
Each outer-connection pays same obj d(i,r)
Want property (in integral soln) that P(i)=; )
there is a recourse connection to r
Do not quite ensure this, but…
Phase II contd.
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Add constraint that y(P(r)) + y(P(r’)) ¸ 1
Indeed feasible for (x,y) since each local ball
has y-value ¸ ½
This ensures (in integral soln) that P(i)=; )
there is a recourse connection to r or r’
Lose another constant factor in obj
Phase II: new LP
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Apply constraints for each star to get LP
min i [ j2P(i) dij ¢ yj + d(i,r(i)) ¢(1- j2P(i) yj)]
s.t. j2P(i) yj · 1, 8 clients i
laminar
constraints
y(P(r)) + y(P(r’)) ¸ 1, 8 p-root {r, r’}
y2M
matroid
constraint
Lem: Integral polytope (via proof similar to
matroid intersection)
Summarize
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Using LP solution and metric properties
reduce to star-like instances
Formulate new LP for star-like instances, with
only facility variables
New LP is integral
Other Results
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O(1)-approximation for prize-collecting
version of matroid median
Knapsack Median problem (knapsack
constraint on open facilities)
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Give bi-criteria approx, violate budget by wmax
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Can we get true O(1)-approx?
Handle other constraints in k-median?
Thank You