algorithms for submodular objectives: continuous extensions & dependent randomized rounding

Algorithms for submodular objectives: continuous extensions &

dependent randomized rounding

Chandra ChekuriUniv. of Illinois, Urbana-Champaign

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

f is a non-negative submodular set function on NMotivation:• several applications• mathematical interest• modeling power and new results

min/max f(S)

s.t. S 2 S

Submodular Set Functions

A function f : 2N ! R+ is submodular if

A Bj

f(A+j) – f(A) ¸ f(B+j) – f(B) for all A ½ B, j 2 N\B

f(A+j) – f(A) ≥ f(A+i+j) – f(A+i) for all A N , i, j N\A


A function f : 2N ! R+ is submodular if

A Bj

f(A+j) – f(A) ¸ f(B+j) – f(B) for all A ½ B, i 2 N\B

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(AB) + f(AB) 8 A,B N

• G=(V,E) undirected graph• f : 2V ! R+ where f(S) = |δ(S)|

Cut functions in graphs

S

Coverage in Set Systems

• X1, X2, ..., Xn subsets of set U

• f : 2{1,2, ..., n} ! R+ where f(A) = |[ i in A Xi |

X1

X2 X3

X4X5

X1

X2 X3

X4X5


• Non-negative submodular set functionsf(A) ≥ 0 8 A ) f(A) + f(B) ¸ f(A[ B) (sub-additive)

• Monotone submodular set functionsf(ϕ) = 0 and f(A) ≤ f(B) for all A B

• Symmetric submodular set functionsf(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

• f is cut function of a given graph G=(V,E)• S = 2V : unconstrained • NP-Hard!

max f(S)

s.t S 2 S

Unconstrained problem

• 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

min/max f(S)

Techniques

f is a non-negative submodular set function on N• Greedy • Local Search• Mathematical Programming Relaxation +

Rounding

min/max f(S)

s.t. 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)Exact algorithm: P(S) = convexhull( {1S : S 2

S})

xi 2 [0,1] indicator variable for i


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


P(S) ¶ convexhull( {1S : S 2 S}) and solvable

min/max f(S)

s.t S 2 S

min/max g(x)

s.t x 2 P(S)



• What is the continuous extension g ?• How to optimize with objective g ?• How do we round ?

min/max f(S)

s.t S 2 S

min/max g(x)

s.t x 2 P(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 = 0.6f»(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/xixj ≤ 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 monotonemax { 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 obtainedRemark: Current best 1/e ' 0.3678 [Feldman-Naor-

Schwartz’11]Algorithms: variants of local-search and continuous-

greedy


• 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 ?

min/max f(S)

s.t S 2 S

min/max g(x)

s.t x 2 P(S)


✔

✔

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

I µ 2N is a downward closed family A 2 I & B ½ A ) B 2 I

Captures “packing” problems

max f(S)

s.t S 2 I

Maximization

High-level results:• optimal rounding in matroid polytopes

[Calinescu-C-Vondrak-Pal’07,C-Vondrak-Zeklusen’09]]

• contention resolution scheme based rounding framework [C-Vondrak-Zenklusen’11]

Max k-Coverage

• 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

max f(S)

s.t S 2 I

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 xCR 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

Theorem: A monotone CR scheme for P(I) can be used to round s.t.

E[f(S*)] ¸ c F(x*)Via FKG inequality

max F(x)

s.t x 2 P(I)

Round x* 2 P(I)to S* 2 I

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

[Koufogiannakis-Young’ 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 for all ij 2 E xi ¸ 0 for all i 2 V

Rounding

Pick µ 2 [0, 1/2] uniformly at randomOutput S = { i | xi ¸ µ }


Rounding Analysis

Pick µ 2 [0, 1/2] uniformly at randomOutput S = { i | x*i ¸ µ }

Claim 1: S is a vertex cover with probability 1Claim 2: E[ f(S) ] · 2 f»(x*)Proof: 2f»(x) = 2s1

0 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

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)

G=(V,E)

Labels

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

For each label i define function fi : 2V ! R

fi(S) = v 2 S c(v,i) + w(±(S))/2

G=(V,E)

Labels

Submodular Cost Allocation: Relaxation

x(u,i) 2 {0,1} whether u allocated label ixi = (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

Pick µ 2 [0,1] at randomfor each i let Ai = { u | x(u,i) ¸ µ }

Claim: E[ i f(Ai) ] = i fi»(xi)

min i fi»(xi)


Rounding

Pick µ 2 [0,1] at randomfor 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!

min i fi»(xi)


Rounding

(1,0,0)

(0,1,0)

(0,0,1)

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,

hub-location, 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!

algorithms for submodular objectives: continuous extensions & dependent randomized rounding

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