The Matroid Median Problem Viswanath Nagarajan IBM Research Joint with R. Krishnaswamy, A. Kumar, Y. Sabharwal, B. Saha
Jan 03, 2016
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?