Dependent Randomized Rounding in Matroid Polytopes (& Related Results) Chandra Chekuri Jan...
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Transcript of Dependent Randomized Rounding in Matroid Polytopes (& Related Results) Chandra Chekuri Jan...
Dependent Randomized Rounding in Matroid Polytopes (& Related
Results)
Chandra Chekuri Jan Vondrak Rico Zenklusen
Univ. of Illinois IBM Research MIT
Example: Congestion Minimization
s3
s2
s1
t1
t2
t3
Choose a path for each pair
Minimize max number of paths using any edge (congestion)
Special case: Edge-Disjoint Paths
G
Example: Congestion Minimization
s3
s2
s1
t1
t2
t3
Choose a path for each pair
Minimize max number of paths using any edge (congestion)
Special case: Edge-Disjoint Paths
[Raghavan-Thompson’87]•Solve mc-flow relaxation (LP)•Randomly pick a path according to fractional solution•Chernoff bounds to show approx ratio of O(log n/log log n)
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Chernoff-Hoeffding Concentration Bounds
• X1, X2, ..., Xn independent {0,1} random variables
• E[Xi] = Pr[Xi = 1] = xi
• a1, a2, ..., an numbers in [0,1]
• μ = E[Σi ai Xi] = Σi ai xi
Theorem: • Pr[Σi ai Xi > (1+δ) μ] ≤ ( e δ / (1+δ) δ ) μ
• Pr[Σi ai Xi < (1 - δ) μ] ≤ exp(- μ δ2/2)
Example: Multipath Routing
s3
s2
s1
t1
t2
t3
Choose ki paths for pair (si, ti)(assume paths for pair disjoint)
Minimize max number of paths using any edge (congestion)
k2 = 1
k1 = 2
k3 = 2
G
Example: Multipath Routing
s3
s2
s1
t1
t2
t3
Choose ki paths for pair (si, ti)(assume paths for pair disjoint)
Minimize max number of paths using any edge (congestion)
[Srinivasan’99]•Solve mc-flow relaxation (LP)•Randomized pipage rounding• O(log n/log log n) approx via negative correlation
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Dependent Randomized Rounding
Randomized rounding while maintaining some dependency/correlation between variables
Dependent Randomized Rounding
Randomized rounding while maintaining some dependency/correlation between variables
Several variants in literature
This talk: dependent randomized rounding to satisfy a matroid base constraint while retaining concentration bounds similar to independent rounding
Briefly, related work on matroid intersection and non-bipartite graph matchings
Crossing Spanning Trees and ATSP
Undir graph G=(V,E)
Cuts S1, S2, …, Sm
Find spanning tree T that minimizes max # of edges crossing a given cut
[Bilo-Goyal-Ravi-Singh-’04]
[Fekete-Lubbecke-Meijer’04]
Crossing Spanning Trees and ATSP
Undir graph G=(V,E)
Cuts S1, S2, …, Sm
Find spanning tree T that minimizes max # of edges crossing a given cut
[Asadpour etal]
• Solve LP: x point in spanning tree polytope of G
• Dependent rounding via maximum entropy sampling
• O(log m/log log m) approx
• Also O(log n/log log n) for ATSP (several other ideas)
1
0.4 0.6
0.9
0.4
0.3
1
0.7
0.7
Tool: Negative Correlation
• X1, X2 two binary ({0,1}) random variables
• X1, X2 are negatively correlated if E[X1 X2] ≤ E[X1] E[X2]
• That is,
Pr[X1 = 1 | X2 = 1] ≤ Pr[X1 = 1] and
Pr[X2 = 1 | X1 = 1] ≤ Pr[X2 = 1]
Tool: Negative Correlation
• X1, X2 two binary random variables
• X1, X2 are negatively correlated if E[X1 X2] ≤ E[X1] E[X2]
• That is,
Pr[X1 = 1 | X2 = 1] ≤ Pr[X1 = 1] and
Pr[X2 = 1 | X1 = 1] ≤ Pr[X2 = 1]
• Also implies (1-X1), (1-X2) are negatively correlated
Negative Correlation
• X1, X2, ..., Xn binary random variables
• X1, X2, ..., Xn are negatively correlated if for any index set J {1,2, ..., n}
• E[ i J Xi ] ≤ i J E[Xi ] and
• E[ i J (1-Xi)] ≤ i J E[(1-Xi)]
Negative Correlation and Concentration
• X1, X2, ..., Xn binary random variables that are negatively correlated (can be dependent)
• E[Xi] = Pr[Xi = 1] = xi
• a1, a2, ..., an numbers in [0,1]
• μ = E[Σi ai Xi] = Σi ai xi
Theorem: [Panconesi-Srinivasan’ 97]• Pr[Σi ai Xi > (1+δ) μ] ≤ ( e δ / (1+δ) δ ) μ
• Pr[Σi ai Xi < (1 - δ) μ] ≤ exp(- μ δ2/2)
Connecting the dots ...
What is common between the two applications?
Integer Program:
min λs.tA x ≤ λbx is a base in a matroid
A non-neg matrix, packing constraints
Multipath: x corresponds to choosing ki paths for pair siti
from Pi Crossing tree: x induces a spanning tree
congestion
Matroids
M=(N, I ) where N is a finite ground set and I 2N is a set of independent sets such that
• I is not empty
• I is downward closed: B I and A B A I
• A, B I and |A| < |B| implies there is i B\A such that A+i I
Matroid Examples
• Uniform matroid: I = { S : |S| ≤ k }
• Partition matroid: I = { S : |S Nj| ≤ kj, 1 ≤ i ≤ h } where N1, ..., Nh partition N, and kj are integers
• Graphic matroid: G = (V, E) is a graph and M=(E, I) where I = { S E : S induces a forest }
Bases in Matroid
• B I is a base of a matroid M=(N, I) if B is a maximal independent set
• All bases have same cardinality
• Matroids can also be defined via bases
• Example: spanning trees in a graph
Base Exchange Theorem
B’ and B’’ are distinct bases in a matroid M=(N, I)
Strong Base Exchange Theorem: There are elements i B’\B’’ and j B’’\B’ such that B’-i+j and B’’-j+i are both bases.
B’ B’’
B’B’’ B’B’’
i j B’-i+j and B’’-j+i are both bases
Dependent Rounding in Matroids
• M = (N, I ) is a matroid with |N| = n
• B(M) is the base polytope: conv{1B : B is a base}
• x is a fractional point in B(M)
• Round x to a random base B such that • Pr[i B] = xi for each i N
• Xi (indicator for i B ) variables are negatively correlated
Our Work
Two methods for arbitrary matroids:
1. Randomized pipage rounding for matroids [Calinescu-C-Pal-Vondrak’07,’09]
2. Randomized swap rounding [C-Vondrak-Zenklusen’09]
This talk: randomized swap rounding
Randomized Swap Rounding
• Express x = mj=1 βi Bi (convex comb. of
bases)
• C1 = B1 , β = β1
• For k = 1 to m-1 do • Randomly Merge β Ck & βk+1 Bk+1 into (β+βk+1)
Ck+1
• Output Cm
Swap Rounding
0.2 C1 + 0.1 B2 + 0.5 B3 + 0.15 B4 + 0.05 B5
0.3 C2 + 0.5 B3 + 0.15 B4 + 0.05 B5
0.8 C3 + 0.15 B4 + 0.05 B5
0.95 C4 + 0.05 B5
C5
x = 0.2 B1 + 0.1 B2 + 0.5 B3 + 0.15 B4 + 0.05 B5
Merging two Bases
Merge B’ and B’’ into a random B that looks like B’ with probability p and like B’’ with probability (1-p)
Merging two Bases
Merge B’ and B’’ into a random B that looks like B’ with probability p and like B’’ with probability (1-p)
Option: Pick B’ with prob. p and B’’ with prob. (1-p) ?
Will not have negative correlation properties!
Merging two Bases
B’ B’’
B’B’’ B’B’’
i j
prob p
prob 1-p
B’ B’’
B’B’’ B’B’’
i i
B’ B’’
B’B’’ B’B’’
j j
p
p
1-p
1-p
Swap Rounding for Matroids
Theorem: Randomized-Swap-Rounding with x B(M) outputs a random base B such that• Pr[i B] = xi for each i N
• Xi (indicator for i B ) variables are negatively correlated
Negative correlation gives concentration bounds for linear functions of the Xi s
Swap Rounding for Matroids
Theorem: Randomized-Swap-Rounding with x B(M) outputs a random base B such that• Pr[i B] = xi for each i N
• Xi (indicator for i B ) variables are negatively correlated
Additional properties for submodular functions:• E[f(B)] ≥ F(x) where F is multilinear extension of f• Pr[ f(B) < (1-δ) F(x)] ≤ exp(- F(x) δ2/8)
(concentration for lower tail of submod functions)
Several Applications
Can handle matroid constraint plus packing constraints
x B(M) and Ax ≤ b
• (1-1/e) approximation for submodular functions subject to a matroid plus O(1) knapsack/packing constraints (or many “loose” packing constraints)
• Simpler rounding and proof for “thin” spanning trees in ATSP application ([Asadpour etal’10])
• ...
Proof idea for Negative Correlation
Process is a vector-valued martingale:
• each iteration merges two bases
• merging bases involves swapping elements in each step
In each step only two elements i and j involved
Proof idea for Negative Correlation
In each step only two elements i and j involved
Xi, Xj before swap step and X’i, X’j after swap step
1. E[X’i | Xi, Xj ] = Xi and E[X’j | Xi, Xj ] = Xj
2. X’i + X’j = Xi + Xj
Proof idea for Negative Correlation
In each step only two elements i and j involved
Xi, Xj before swap step and X’i, X’j after swap step
1. E[X’i | Xi, Xj ] = Xi and E[X’j | Xi, Xj ] = Xj
2. X’i + X’j = Xi + XjE[X’iX’j|Xi,Xj ] = ¼ E[(X’i+X’j)2| Xi,Xj ] − ¼ E[(X’i - X’j)2| Xi,Xj ]
= ¼ (Xi+Xj)2 − ¼ E[(X’i - X’j)2| Xi, Xj ]
≤ ¼ (Xi+Xj)2 − ¼ (Xi - Xj)2
≤ Xi Xj
Beyond matroids?
Question: Can we obtain negative correlation for other combinatorial structures/polytopes?
Beyond matroids?
Question: Can we obtain negative correlation for other combinatorial structures/polytopes?
Answer: No.
Negative correlation implies the polytope is “essentially” a matroid base polytope
Other Comments
• Swap rounding advantage: • identifies exchange property as the key• Idea generalizes/inspires work for other
structures such as matroid intersection, and b-matchings with some restrictions
• Lower tail for submodular functions uses martingale analysis (does not follow from negative correlation)
• Negative correlation not needed for concentration
Do we need negative correlation for concentration?
No.
• Lower tail for submodular functions shown via martingale method
• Also can show concentration for linear functions in the matroid intersection polytope and non-bipartite matching (a the loss of a bit in expectation)
Example: Rounding in bipartite-matching polytope
xe = ½ on each edge
Can we round x to a matching?
If we want to preserve expectation of x only choice is to pick one of two perfect matchings, each with prob ½
Large positive correlation!
Informal Statements
For any point x in the bipartite matching polytope
• Can round x to a matching preserving expectation and negative correlation holds for edge variables incident to any vertex [Srinivasan’99]
• Can round x to a matching x’ s.t E[x’] = (1-γ) x and concentration holds for any linear functions of x (the exponent in tail bound depends on γ) [CVZ]
• Above results generalize to matroid intersection and non-bipartite matchings [CVZ]
Submodular Functions
• Non-negative submodular set functions
f(A) ≥ 0 for all A
• Monotone submodular set functions
f(ϕ) = 0 and f(A) ≤ f(B) for all A B
• Symmetric submodular set functionsf(A) = f(N\A) for all A
Multilinear Extension of f
[CCPV’07] inspired by [Ageev-Sviridenko]
For f: 2N R+ define F:[0,1]N R+ as
x = (x1, x2, ..., xn) [0,1]|N|
F(x) = Expect[ f(x) ] = S N f(S) px(S)
= S N f(S) i S xi i N\S (1-xi)
Multilinear Extension of f
For f: 2N R+ define F:[0,1]N R+ as
F(x) = S N f(S) i S xi i N\S (1-xi)
F is smooth submodular ([Vondrak’08])
• F/xi ≥ 0 for all i (monotonicity)
• 2F/xixj ≤ 0 for all i,j (submodularity)