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Report
Algorithms for submodular objectives:
continuous extensions &
dependent randomized rounding
Chandra Chekuri
Univ. of Illinois, Urbana-Champaign
Modern aspects
of Submodularity
Georgi a Institute
of Technology
Modern Aspects of
Submodularity
March 19-22, 2012
Location:TBD
ORGANIZING COMMITTEE: Shabbir Ahmed (School of ISyE, GaTech), Nina
Balcan (School of CS, GaTech), Satoru Iwata (RIMS, Kyoto University), and
Prasad Tetali (School of Math and School of CS, GaTech)
Wor k sh op T h em e:
Submodular functions are discrete analo gues of convex functions, arising in various fields of
computer science and o perations research. Since the seminal work of Jack Edmonds (1970),
submodularity has long been recognized as a common structure of many efficient ly solvable
combinatorial optimization problems. Recent algo rithmic developments in the past decade
include combinatorial strongly polynomial algorithm for minimization, constant factor
approximation algorithms for maximization, and efficient methods for learning submodular
functions. In addition, submodular functions find no vel applications in combinatorial auctions,
machine learning, and so cial networks. This workshop aims at providing a forum for researchers
from a variety of backgrounds for exchanging result s, ideas, and problems on submo dular
optimization and its applicatio ns. The first day will be devo ted to tutorial-style lectures!
Con f i r m ed Sp eak er s : Jef f Bi l m es (University of Washington), Ch an d r a Ch ek u r i (UIUC),
Li sa Fl ei sc h er (Dartmouth), Sat or u Fu j i sh i ge (Kyoto University), Gagan Goel (Google
Research), M i ch el Goem an s (MIT), Car l os Gu est r i n (CMU), Ni ck H ar v ey (UBC), Sat or u Iw at a
(Kyoto University), K am al Jai n (Microsoft Research), A n d r ea K r au se (ETH), Jon Lee (U.
Michigan), T om M cCor m i ck (UBC), A r an y ak M eh t a (Google Research), V ah ab M i r r ok n i
(Google Research), K az u o M u r ot a (University of Tokyo), K i y oh i t o N agan o (University of
Tokyo), M au r i ce Qu ey r an n e (UBC), A m i n Sab er i (Stanford), A k i y osh i Sh i ou r a (Tohoku
University), M ax i m Sv i r i d en k o (IBM), Zoy a Sv i t k i n a (Google), Jan V on d r ak (IBM), Lasz l o
V egh (Georgia Tech).
For additional information, please visit: http://arc.gatech.edu/upco ming_events.php
Sponsors: Algorithms & Randomness Center , ACO Ph.D. Program (Georgia Tech),
Yandex Corporation (Russia), ...
Combinatorial Optimization
• N a finite ground set
• w : N ! R weights on N
max/min w(S)
s.t S µ N satisfies constraints
Combinatorial Optimization
• N a finite ground set
• w : N ! R weights on N
•
S µ 2N feasible solutions to problem
max/min w(S)
s.t S 2 S
Examples: poly-time solvable
• max weight matching
• s-t shortest path in a graph
• s-t minimum cut in a graph
• max weight independent set in a matroid and
intersection of two matroids
• ...
Examples: NP-Hard
• max cut
• min-cost multiway/multiterminal cut
• min-cost (metric) labeling
• max weight independent set in a graph
• ...
Approximation Algorithms
A is an approx. alg. for a problem:
• A runs in polynomial time
• maximization problem: for all instances I of the
problem A(I) ¸ ® OPT(I)
• minimization problem: for all instances I of the
problem A(I) · ® OPT(I)
• ® is the worst-case approximation ratio of A
This talk
min/max f(S)
s.t. S 2 S
f is a non-negative submodular set function on N
Motivation:
• several applications
• mathematical interest
• modeling power and new results
Submodular Set Functions
A function f : 2N ! R+ is submodular if
f(A+j) – f(A) ¸ f(B+j) – f(B) for all A ½ B, j 2 N\B
A
B
j
f(A+j) – f(A) ≥ f(A+i+j) – f(A+i) for all A  N , i, j  N\A
Submodular Set Functions
A function f : 2N ! R+ is submodular if
f(A+j) – f(A) ¸ f(B+j) – f(B) for all A ½ B, i 2 N\B
A
B
j
f(A+j) – f(A) ≥ f(A+i+j) – f(A+i) for all A  N , i, j  N\A
Equivalently: f(A) + f(B) ≥ f(AB) + f(AB) 8 A,B  N
Cut functions in graphs
• G=(V,E) undirected graph
• f : 2V ! R+ where f(S) = |δ(S)|
S
Coverage in Set Systems
• X1, X2, ..., Xn subsets of set U
• f : 2{1,2, ..., n} ! R+ where f(A) = |[ i
X1
X5
X4
X2
X3
in A
Xi |
X1
X5
X4
X2
X3
Submodular Set Functions
• Non-negative submodular set functions
f(A) ≥ 0 8 A ) f(A) + f(B) ¸ f(A[ B) (sub-additive)
• Monotone submodular set functions
f(ϕ) = 0 and f(A) ≤ f(B) for all A  B
• Symmetric submodular set functions
f(A) = f(N\A) for all A
Other examples
• Cut functions in hypergraphs (symmetric non-negative)
• Cut functions in directed graphs (non-negative)
• Rank functions of matroids (monotone)
• Generalizations of coverage in set systems (monotone)
• Entropy/mutual information of a set of random variables
• ...
Max-Cut
max f(S)
s.t S 2 S
• f is cut function of a given graph G=(V,E)
•
S = 2V : unconstrained
• NP-Hard!
Unconstrained problem
min/max f(S)
• minimization poly-time solvable assuming value
oracle for f
• Ellipsoid method [GLS’79]
• Strongly-polynomial time combinatorial algorithms
[Schrijver, Iwata-Fleischer-Fujishige ’00]
• maximization NP-Hard even for explicit cut-function
Techniques
min/max f(S)
s.t. S 2 S
f is a non-negative submodular set function on N
• Greedy
• Local Search
• Mathematical Programming Relaxation + Rounding
Math. Programming approach
min/max
w(S)
s.t S 2 S
min/max
w¢x
s.t x 2 P(S)
xi 2 [0,1]
indicator
variable for i
Exact algorithm: P(S) = convexhull( {1S : S 2 S})
Math. Programming approach
min/max
w(S)
s.t S 2 S
min/max
w¢x
s.t x 2 P(S)
Round
x* 2 P(S)
to S* 2 S
Exact algorithm: P(S) = convexhull( {1S : S 2 S})
Approx. algorithm: P(S) ¾ convexhull( {1S : S 2 S})
P(S) solvable: can do linear optimization over it
Math. Programming approach
min/max f(S)
min/max g(x)
s.t S 2 S
s.t x 2 P(S)
P(S) ¶ convexhull( {1S : S 2 S}) and solvable
Round
x* 2 P(S)
to S* 2 S
Math. Programming approach
min/max f(S)
min/max g(x)
s.t S 2 S
s.t x 2 P(S)
• What is the continuous extension g ?
• How to optimize with objective g ?
• How do we round ?
Round
x* 2 P(S)
to S* 2 S
Continuous extensions of f
For f : 2N ! R+ define g : [0,1]N ! R+ s.t
• for any S µ N want f(S) = g(1S)
• given x = (x1, x2, ..., xn)  [0,1]N want polynomial
time algorithm to evaluate g(x)
• for minimization want g to be convex and for
maximization want g to be concave
Canonical extensions: convex
and concave closure
x = (x1, x2, ..., xn)  [0,1]N
min/max  S ®S f(S)
S ®S = 1
S ®S = xi for all i
®S ¸ 0 for all S
-
f (x) for minimization and f+(x) for maximization: convex
and concave respectively for any f
Submodular f
• For minimization f-(x) can be evaluated in poly-time
via submodular function minimization
• Equivalent to the Lovasz-extension
• For maximization f+(x) is NP-Hard to evaluate even
when f is monotone submodular
• Rely on the multi-linear-extension
Lovasz-extension of f
f»(x) = Eµ 2 [0,1][ f(xµ) ] where xµ = { i | xi ¸ µ }
Example: x = (0.3, 0, 0.7, 0.1)
xµ = {1,3} for µ = 0.2 and xµ = {3} for µ = 0.6
f»(x) = (1-0.7) f(;) + (0.7-0.3)f({3}) + (0.3-0.1) f({1,3})
+ (0.1-0) f({1,3,4}) + (0-0) f({1,2,3,4})
Properties of
»
f
• f» is convex iff f is submodular
• f»(x) = f-(x) for all x when f is submodular
• Easy to evaluate f»
• For submod f : solve relax. via convex optimization
min
f»(x)
s.t x 2 P(S)
Multilinear extension of f
[Calinescu-C-Pal-Vondrak’07] inspired by [Ageev-Svir.]
For f : 2N ! R+ define F : [0,1]N ! R+ as
x = (x1, x2, ..., xn)  [0,1]N
R: random set, include i independently with prob. xi
F(x) = E[ f(R) ] =  S  N f(S) i  S xi  i  N\S (1-xi)
Properties of F
• F(x) can be evaluated by random sampling
• F is a smooth submodular function
• 2F/xixj ≤ 0 for all i,j.
Recall f(A+j) – f(A) ≥ f(A+i+j) – f(A+i) for all A, i, j
• F is concave along any non-negative direction vector
• F/xi ≥ 0 for all i if f is monotone
Maximizing F
max { F(x) | xi 2 [0,1] for all i} is NP-Hard
equivalent to unconstrained maximization of f
When f is monotone
max { F(x) | i xi · k, xi 2 [0,1] for all i} is NP-Hard
Approximately maximizing F
[Vondrak’08]
Theorem: For any monotone f, there is a (1-1/e)
approximation for the problem max { F(x) | x  P }
where P [0,1]N is any solvable polytope.
Algorithm: Continuous-Greedy
Approximately maximizing F
[C-Vondrak-Zenklusen’11]
Theorem: For any non-negative f, there is a ¼
approximation for the problem max { F(x) | x  P }
where P [0,1]n is any down-closed solvable polytope.
Remark: 0.325-approximation can be obtained
Remark: Current best 1/e ' 0.3678 [Feldman-NaorSchwartz’11]
Algorithms: variants of local-search and continuous-greedy
Math. Programming approach
min/max f(S)
min/max g(x)
s.t S 2 S
s.t x 2 P(S)
✔
Round
x* 2 P(S)
to S* 2 S
• What is the continuous extension g ?
• Lovasz-extension for min and multilinear ext. for max
✔
• How to optimize with objective g ?
• Convex optimization for min and O(1)-approx. alg for max
• How do we round ?
Rounding
Rounding and approximation depend on S and P(S)
Two competing issues:
• Obtain feasible solution S* from fractional x*
• Want f(S*) to be close to g(x*)
Rounding approach
Viewpoint: objective function is complex
• round x* to S* to approximately preserve objective
• fix/alter S* to satisfy constraints
• analyze loss in fixing/altering
Rounding to preserve objective
x* : fractional solution to relaxation
Minimization: f»(x) = Eµ 2 [0,1][ f(xµ) ]
Pick µ uniformly at random in [0,1] (or [a, b])
S* = { i | x*i ¸ µ }
Maximization: F(x) = E[f(R)]
S* = pick each i 2 N independently with probability ® x*i
(® · 1)
Maximization
max f(S)
s.t S 2 I
I µ 2N is a downward closed family
A2I&B½A)B2I
Captures “packing” problems
Maximization
High-level results:
• optimal rounding in matroid polytopes [CalinescuC-Vondrak-Pal’07,C-Vondrak-Zeklusen’09]]
• contention resolution scheme based rounding
framework [C-Vondrak-Zenklusen’11]
Max k-Coverage
max f(S)
s.t S 2 I
• X1,X2,...,Xn subsets of U and integer k
• N = {1,2,...,n}
• f is the set coverage function (monotone)
•
I = { A µ N : |A| · k } (cardinality constraint)
• NP-Hard
Greedy
[Nemhauser-Wolsey-Fisher’78, FNW’78]
• Greedy gives (1-1/e)-approximation for the problem
max { f(S) | |S| · k } when f is monotone Obtaining
a (1-1/e + ²)-approximation requires exponentially
many value queries to f
• Greedy give ½ for maximizing monotone f over a
matroid constraint
• Unless P=NP no (1-1/e +²)-approximation for
special case of Max k-Coverage [Feige’98]
Matroid Rounding
[Calinescu-C-Pal-Vondrak’07]+[Vondrak’08]=[CCPV’09]
Theorem: There is a randomized (1-1/e) ' 0.632
approximation for maximizing a monotone f subject
to any matroid constraint.
[C-Vondrak-Zenklusen’09]
Theorem: (1-1/e-²)-approximation for monotone f
subject to a matroid and a constant number of
packing/knapsack constraints.
Rounding in Matroids
[Calinescu-C-Pal-Vondrak’07]
Theorem: Given any point x in P(M), there is a randomized
polynomial time algorithm to round x to a vertex
X(hence an indep set of M) such that
• E[X] = x
• f(X) = F(X) ≥ F(x)
[C-Vondrak-Zenklusen’09]
Different rounding with additional properties and apps.
Contention Resolution Schemes
•
I an independence family on N
• P(I) a relaxation for I and x 2 P(I)
• R: random set from independent rounding of x
CR scheme for P(I): given x, R outputs R’ µ R s.t.
1. R’ 2 I
2. and for all i, Pr[i 2 R’ | i 2 R] ¸ c
Rounding and CR schemes
max F(x)
s.t x 2 P(I)
Round
x* 2 P(I)
to S* 2 I
Theorem: A monotone CR scheme for P(I) can be used
to round s.t.
E[f(S*)] ¸ c F(x*)
Via FKG inequality
Summary for maximization
• Optimal results in some cases
• Several new technical ideas and results
• Questions led to results even for modular case
• Similar results for modular and submodular (with in
constant factors) for most known problems
Minimization
• Landscape is more complex
• Many problems that are “easy” in modular case are
hard in submodular case: shortest paths, spanning
trees, sparse cuts ...
• Some successes via Lovasz-extension
• Future: need to understand special families of
submodular functions and applications
Submodular-cost Vertex Cover
• Input: G=(V,E) and f : 2V ! R+
• Goal: min f(S) s.t S is a vertex cover in G
• 2-approx for modular case well-known
• 2-approx for sub-modular costs [KoufogiannakisYoung’ 99, Iwata-Nagano’99, Goel etal’99]
Submodular-cost Vertex Cover
• Input: G=(V,E) and f : 2V ! R+
• Goal: min f(S) s.t S is a vertex cover in G
min f»(x)
xi + xj ¸ 1
xi ¸ 0
for all ij 2 E
for all i 2 V
Rounding
min f»(x)
xi + xj ¸ 1 for all ij 2 E
xi ¸ 0 for all i 2 V
Pick µ 2 [0, 1/2] uniformly at random
Output S = { i | xi ¸ µ }
Rounding Analysis
min f»(x)
xi + xj ¸ 1 for all ij 2 E
xi ¸ 0 for all i 2 V
Pick µ 2 [0, 1/2] uniformly at random
Output S = { i | x*i ¸ µ }
Claim 1: S is a vertex cover with probability 1
Claim 2: E[ f(S) ] · 2 f»(x*)
Proof: 2f»(x) = 2s10 f(xµ) dµ ¸ 2s1/20 f(xµ) = E[ f(S) ]
Submodular-cost Set Cover
• Input: Subsets X1,...,Xn of U, f : 2N ! R+
• Goal: min f(S) s.t [i 2 S Xi = U
• Rounding according to objective gives only k-approx
where k is max-element frequency. Also integrality
gap of (k)
• [Iwata-Nagano’99] (k/log k)-hardness
A Labeling Problem
Labels
G=(V,E)
Assign label to each vertex to minimize cost:
• to label v with i cost c(v,i)
• if edge uv get assigned different labels pay cost w(uv)
A Labeling Problem
Labels
G=(V,E)
Assign label to each vertex to minimize cost:
• to label v with i cost c(v,i)
• if edge uv get assigned different labels pay cost w(uv)
A Labeling Problem
• Labeling problem has many applications in computer
vision [Boykov-Veksler-Zabih, ...]
• Generalized to metric labeling by [Kleinberg-Tardos’99]
• 2-approx for uniform case by interesting relaxation and
rounding
• O(log k)-approximation for general metric building on
uniform case
• LP relaxation and O(log k) gap [C-Khanna-Naor-Zosin’01]
• Multiway-cut is special case
Submodular Cost Allocation
[C-Ene’11] A new model
• V a finite ground set
• L = {1, 2, ..., k} set of labels
• for each i a submodular cost fn fi : 2V ! R+
• Goal: assign labels to V to minimize  i fi(Ai) where
Ai µ V assigned label i
Labeling Problem as
Submodular Cost Allocation
Labels
G=(V,E)
For each label i define function fi : 2V ! R
fi(S) = v 2 S c(v,i)
+ w(±(S))/2
Submodular Cost Allocation:
Relaxation
x(u,i) 2 {0,1} whether u allocated label i
xi = (x(u1,i), x(u2,i),....,x(un,i))
min i fi»(xi)
i x(u,i) = 1 for all u 2 V
x(u,i) ¸ 0 for all u 2 V, i 2 L
Relaxation
• [Calinescu-Karloff-Rabani’98] relaxation for
multiway cut is a special case
• [Kleinberg-Tardos’99] relaxation for uniform metric
labeling is a special case
Rounding
min i fi»(xi)
i x(u,i) = 1 for all u 2 V
x(u,i) ¸ 0 for all u 2 V, i 2 L
Pick µ 2 [0,1] at random
for each i let Ai = { u | x(u,i) ¸ µ }
Claim: E[ i f(Ai) ] = i fi»(xi)
Rounding
min i fi»(xi)
i x(u,i) = 1 for all u 2 V
x(u,i) ¸ 0 for all u 2 V, i 2 L
Pick µ 2 [0,1] at random
for each i let Ai = { u | x(u,i) ¸ µ }
Claim: E[ i f(Ai) ] = i fi»(xi)
The Ai overlap & not all vertices assigned labels!
Rounding
(0,0,1)
(0,1,0)
(1,0,0)
Rounding
(0,0,1)
(0,1,0)
(1,0,0)
Fixing
Overlap
• If fi are monotone overlap does not matter
• If fi arise from single symmetric function uncross using
posi-modularity
• Ensure no overlap by picking µ 2 (1/2, 1]
Not all vertices labeled
• Repeat rounding until all vertices labeled
• Assign all unlabeled vertices to some label
Results
[C-Ene’11a,C-Ene’11b]
• Rounding “explains” [CKR’98,KT’99,...]
• Submodular multiway partition: 1.5-approximation
for symmetric, 2-approximation for general.
Improve 2 and (k-1) respectively
• Recover 2-approx for uniform metric labeling
• Obtain O(log k) for generalization of labeling, hublocation, non-metric facility location ...
Concluding Remarks
• Substantial progress on constrained submodular
function optimization in the last few years
• New tools and connections including a general
framework via continuous extensions and dependent
randomized rounding
• Increased awareness and applications
• Much work to be done, especially on minimization
problems
Thanks!

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