The Matroid Median Problem

Viswanath NagarajanIBM 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|·k u2V 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=3 kb=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,B 2 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

A Be k1=2 k4=2k2=3 k3=1

Talk OutlineThm: 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)

m

m

mm

m

1Eg. T=5

Uniform metric on T+1Clients n=mT+1

OPT = 1 (small fac.)LOPT = m (big fac.)

locality gap (n/T)

LP relaxationmin u v d(u,v) ¢ xuv

s.t. v xuv = 1 8 u 2 V

xuv · yv 8 u,v 2 V

v2S yv · r(S) 8 Sµ V

x, y ¸ 0.

y 2 M

facilitiesclients

u vxuv

connectionconstraints

matroid rankconstraints

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

root facility

client 1

client 2

client 3

One root facility (can help any client)

Others are private facilities (help only 1 client)

Idea for approach(2)

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 !

r

client i

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

To ensure xir ¸ 0

matroid constraint

xirxij

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

Phase I: sparsify LP solution

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¢Lu from 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

12

3 45

61

2

43

56

Loss in obj · 4¢ LP*(additive)

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

T balls

y¸½

separated

More obs on step 1 Suppose y-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

j1

2

3 Constant factorloss in obj

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

qp

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 factorloss 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)

matroid constraint

laminar constraints

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?

Thank You