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The Matroid Median Problem. Viswanath Nagarajan IBM Research Joint with R. Krishnaswamy, A. Kumar, Y. Sabharwal, B. Saha. k-Median Problem. Set of locations in a metric space (V,d) Symmetric, triangle inequality

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The Matroid Median Problem


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the matroid median problem

The Matroid Median Problem

Viswanath Nagarajan

IBM Research

Joint with R. Krishnaswamy, A. Kumar, Y. Sabharwal, B. Saha

k median problem
k-Median Problem
  • Set of locations in a metric space (V,d)
    • 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
k-Median Results
  • 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]
    • best known ratio 3+²
  • Hardness of approximation ¼ 1.46 [GK ’98]
red blue median
Red-Blue Median
  • Facilities are of two different types
    • Partition V into red and blue sets
  • Separate bounds kr and kb on facilities
  • Recently introduced [HKK ’10]
  • Motivated by Content Distribution Networks
    • T facility-types (RB Median is T=2)
  • O(1)-approximation ratio via Local Search

kr=3kb=2

matroid median
Matroid Median
  • Given matroid M on ground-set V
  • Locate facilities F that are independent in M
  • Minimize connection cost
  • Recap matroid M=(V, Iµ2V)
    • A,B2 I and |A|<|B| )9 e 2 BnA : A[{e} 2 I
  • Substantial generalization of RB Median
    • The CDN application with T facility-types reduces to partition matroid constraint

B

A

e

k3=1

k4=2

k1=2

k2=3

talk outline
Talk Outline

Thm: 16-approximation for Matroid Median

  • Bad example for Local Search
  • LP relaxation
  • Phase I : sparsification
  • Phase II: reformulation
local search
Local Search?
  • Partition matroid with T parts
  • T-1 exchange local search
    • Swap up to T-1 facilities in each step
  • Unlikely to work beyond T=O(1)

Eg. T=5

Uniform metric on T+1

Clients n=mT+1

OPT = 1 (small fac.)

LOPT = m (big fac.)

m

1

m

m

m

m

locality gap (n/T)

lp relaxation
LP relaxation

min uv d(u,v) ¢xuv

s.t. vxuv = 1 8 u 2 V

xuv·yv8 u,v 2 V

v2Syv· r(S) 8 Sµ V

x, y¸ 0.

connection

constraints

matroid rank

constraints

xuv

u

v

y 2 M

facilities

clients

solving the lp
Solving the LP
  • 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
Idea for approach(1)
  • Problem non-trivial even if metric is a tree 
    • 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)

Others are private facilities (help only 1 client)

client 2

client 1

client 3

root facility

idea for approach 2
Idea for approach(2)
  • Recall LP variables
    • yj : facility opening (in matroid polytope)
    • xij : connection
  • For any client i, private j2 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

r

private facilities P(i)

idea for approach 3
Idea for approach(3)
  • Reformulate the LP

min i[j2P(i) dij¢ yj + dir¢(1- j2P(i) yj) ]

s.t. j2P(i) yj· 1, 8 clients i

y 2 M

  • This is just an instance of intersection of M with partition matroid from P(i)s 

xij

xir

To ensure xir¸ 0

matroid constraint

idea for approach 4
Idea for approach(4)
  • 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
    • (x,y) itself restricted to the stars not integral
  • Show that new LP is integral
    • ¼ matroid intersection
outline
Outline
  • Modify LP connections x in four steps
    • Similar to [CGST ’99]
  • Key: no change in facility variables y
    • Need to ensure y remains in matroid polytope
    • Not true in [CGST ’99]
  • Require some more (technical) work
step 1 cluster clients
Step 1: cluster clients
  • Lu = v duv¢xuv, contribution of u to LP obj.
  • B(u) is local ball of u
    • vertices within distance 2¢Lufrom u
  • Order clients u in increasing Lu
  • Pick maximal disjoint set of local balls
    • T are the chosen clients
    • Move each client to T-client close to it

Loss in obj · 4¢ LP*

(additive)

3

4

3

4

5

5

1

1

6

6

2

2

obs on step 1
Obs on step 1
  • Local balls of T clients are disjoint
  • y-value inside any local ball ¸ ½
    • Markov inequality
  • Restrict to clients T (now weighted)
  • For any p,q2T : d(p,q) ¸ 2¢(LPp + LPq)
    • well separated clients

y¸½

separated

T balls

more obs on step 1
More obs on step 1
  • Supposey-value in each T’s local ball ¸ 1
  • Then instance of matroid intersection:
    • Matroid M and partition from local-ball(T)
  • Resolving suitable LP ) integral soln
  • Will need intersection with `laminar’ constraints, not just partition matroid
step 2 private facilities
Step 2: private facilities
  • 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
    • Reconnect to facilities in B(1), y-value ¸ ½
    • Total reconnection for any client · ½

j

3

1

Constant factor

loss in obj

2

step 3 uniform objective
Step 3: uniform objective
  • 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)
    • For any j 2 B(q)
  • Constant factor loss in obj

p

q

step 4 building stars
Step 4: building stars
  • WMA each client i 2 T connected to
    • Its private facilities P(i), OR
    • Its closest other client k2T, ie. facility in B(k)
  • Set of `outer’ connections ¼ directed tree
    • Unique out-edge from each client
  • Lem: Can modify outer connection to `star’

Constant factor

loss in obj

the star structure
The star structure
  • One pseudo-root { r, r’ }
  • Every other client connected to either r or r’
  • All LP-connections x are from client i to:
    • 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
Phase II: using star
  • 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
Phase II contd.
  • 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
Phase II: new LP
  • 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

y(P(r)) + y(P(r’)) ¸ 1, 8 p-root {r, r’}

y 2 M

Lem: Integral polytope (via proof similar to matroid intersection)

laminar

constraints

matroid

constraint

summarize
Summarize
  • 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
Other Results
  • O(1)-approximation for prize-collecting version of matroid median
  • Knapsack Median problem (knapsack constraint on open facilities)
    • Give bi-criteria approx, violate budget by wmax
    • Can we get true O(1)-approx?
  • Handle other constraints in k-median?