Download - The Matroid Median Problem
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
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?