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## The Matroid Median Problem

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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
- Place k facilities such that sum of connection costs (to nearest facility) is minimized:

minFµV, |F|·ku2V d(u,F)

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

- 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

- 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

Thm: 16-approximation for Matroid Median

- Bad example for Local Search
- LP relaxation
- Phase I : sparsification
- Phase II: reformulation

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

min uv 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

- 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)

- 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)

- Recall LP variables
- yj : facility opening (in matroid polytope)
- xij : connection
- For any client i, private j2 P(i) WMA xij = yj
- Connection constraintj 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)

- 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)

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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.

- 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

- 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

- 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

- 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?

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