Submodular Set Function Maximizationvia the Multilinear Relaxation & Dependent

Rounding

Chandra ChekuriUniv. of Illinois, Urbana-Champaign

Max weight independent set

• N a finite ground set• w : N ! R+ weights on N• I µ 2N is an independence family of subsets• I is downward closed: A 2 I and B ½ A ) B 2 I

max w(S)

s.t S 2 I

Independence families

• stable sets in graphs• matchings in graphs and hypergraphs• matroids and intersection of matroids• packing problems: feasible {0,1} solutions

to A x · b where A is a non-negative matrix

Max weight independent set

• max weight stable set in graphs• max weight matchings • max weight independent set in a matroid• max weight independent set in intersection of two matroids• max profit knapsack • etc

max w(S)

s.t S 2 I

This talk

f is a non-negative submodular set function on NMotivation:• several applications• mathematical interest

max f(S)

s.t. S 2 I

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, i 2 N\B

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

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

Submodular Set Functions

• 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• ...

Example: Max-Cut

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

max f(S)

s.t S 2 I

Example: 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

Approximation Algorithms

A is an approx. alg. for a maximization problem:• A runs in polynomial time• for all instances I of the problem A(I) ¸ ®

OPT(I) ® (· 1) is the worst-case approximation ratio of A

Techniques

f is a non-negative submodular set function on N• Greedy • Local Search• Multilinear relaxation and rounding

max f(S)

s.t. S 2 I

Greedy and Local-Search

[Nemhauser-Wolsey-Fisher’78, Fisher-Nemhauser-Wolsey’78]

• Work well for “combinatorial” constraints: matroids, intersection of matroids and generalizations

• Recent work shows applicability to non-monotone functions [Feige-Mirrokni-Vondrak’07] [Lee-Mirrokni-Nagarajan-Sviridenko’08] [Lee-Sviridenko-Vondrak’09] [Gupta etal, 2010]

Motivation for mathematical programming approach

• Quest for optimal results• Greedy/local search not so easy to adapt for

packing constraints of the form Ax · b• Known advantages of geometric and continuous

optimization methods and the polyhedral approach

Math. Programming approach

max w(S)

s.t S 2 I

max w¢x

s.t x 2 P(I)

Exact algorithm: P(I) = convexhull( {1S : S 2 I})

xi 2 [0,1] indicator variable for i

Math. Programming approach

max w(S)

s.t S 2 I

max w¢x

s.t x 2 P(I)

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

Exact algorithm: P(I) = convexhull( {1S : S 2 I})

Approx. algorithm: P(I) ¾ convexhull( {1S : S 2 I})

P(I) solvable: can do linear optimization over it

Math. Programming approach

max f(S)

s.t S 2 I

max F(x)

s.t x 2 P(I)

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

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

Math. Programming approach

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

max f(S)

s.t S 2 I

max F(x)

s.t x 2 P(I)

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

Some results[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.

What is special about 1-1/e?

Greedy gives (1-1/e)-approximation for the problem max { f(S) | |S| · k } when f is monotone [NWF’78]• Obtaining a (1-1/e + ²)-approximation requires

exponentially many value queries to f [FNW’78]

• Unless P=NP no (1-1/e +²)-approximation for special case of Max k-Coverage [Feige’98]

New results give (1-1/e) for any matroid constraint improving ½ . Moreover, algorithm is interesting and techniques have been quite useful.

Submodular Welfare Problem

• n items/goods (N) to be allocated to k players

• each player has a submodular utility function fi(Ai) is the utility to i if Ai is allocation to i)

• Goal: maximize welfare of allocation i fi(Ai)

Can be reduced to a single f and a (partition) matroid constraint and hence (1-1/e) approximation

Some more results[C-Vondrak-Zenklusen’11]

• Extend approach to non-monotone f• Rounding framework via contention resolution

schemes

• Several results from framework including the ability to handle intersection of different types of constraints

Math. Programming approach

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

max f(S)

s.t S 2 I

max F(x)

s.t x 2 P(I)

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

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

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)

Why multilinear extension?

• Ideally a concave extension to maximize• Could choose (“standard”) concave closure

f+ of f• Evaluating f+(x) is NP-Hard!

Properties of F• F(x) can be evaluated (approximately) 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

Math. Programming approach

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

max f(S)

s.t S 2 I

max F(x)

s.t x 2 P(I)

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

Maximizing F

max { F(x) | xi · k, xi 2 [0,1] } 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

Algorithm: Local-Search variants

Local-Search based algorithm

Problem: max { F(x) | x 2 P }, P is down-monotone

x* = a local optimum of F in PQ = { z 2 P | z · 1-x* }y* = a local optimum of F in QOutput better of x* and y*

Local-Search based algorithm

Problem: max { F(x) | x 2 P }, P is down-monotone

x* = a local optimum of F in PQ = { z 2 P | z · 1-x* }y* = a local optimum of F in QOutput better of x* and y*

Theorem: Above algorithm gives a ¼ approximation.

Math. Programming approach

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

max f(S)

s.t S 2 I

max F(x)

s.t x 2 P(I)

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

Rounding

Rounding and approximation depend on I and P(I)Two results:• For matroid polytope a special rounding• A general approach via contention

resolution schemes

Rounding in Matroids

Matroid M = (N, I)Independence polytope: P(M) = convhull({1S |

S 2 I}) given by following system [Edmonds] i 2 S xi · rankM(S) 8 S µ N

x 2 [0,1]N

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)

[C-Vondrak-Zenklusen’09] Different rounding with additional properties and apps.

Rounding

F(x*) = E[f(R)] where R is obtained by independently rounding each i with probability x*

i

R unlikely to be in I

max F(x)

s.t x 2 P(I)

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

Rounding

F(x*) = E[f(R)] where R is obtained by independently rounding each i with probability x*

i

R unlikely to be in I Obtain R’ µ R s.t. R’ 2 I and E[f(R’)] ¸ c f(R)

max F(x)

s.t x 2 P(I)

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

A simple question?

0.9

1

0.4

0.6

0.4

10.7

0.7

0.3

0.6

0.1

• x is a convex combination of spanning trees• R: pick each e 2 E independently with

probability xe

Question: what is the expected size of a maximal forest in R? (n - # of connected components)

A simple question?

• x is a convex combination of spanning trees of G

• R: pick each e 2 E independently with probability xe

Question: what is the expected size of a maximal forest in R? (n - # of connected components)Answer: ¸ (1-1/e) (n-1)

Related question

• x is a convex combination of spanning trees of G

• R: pick each e 2 E independently with probability xe

Want a (random) forest R’ µ R s.t. for every edge e

Pr[e 2 R’ | e 2 R] ¸ c

Related question

• x is a convex combination of spanning trees of G

• R: pick each e 2 E independently with probability xe

Want a (random) forest R’ µ R s.t. for every edge e

Pr[e 2 R’ | e 2 R] ¸ c ) there is a forest of size e c xe = c (n-1) in R

Related question• x is a convex combination of spanning trees of G• R: pick each e 2 E independently with probability

xe

Want a (random) forest R’ µ R s.t. for every edge ePr[e 2 R’ | e 2 R] ¸ c

Theorem: c = (1-1/e) is achievable & optimal [CVZ’11]

(true for any matroid)

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

Remarks

[CVZ’11]• Several existing rounding schemes are CR

schemes• CR schemes for different constraints can be

combined for their intersection• CR schemes through correlation gap and LP

duality

Math. Programming approach

Problem reduced to finding a good relaxation P(I) and a contention resolution scheme for P(I)

max f(S)

s.t S 2 I

max F(x)

s.t x 2 P(I)

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

Concluding Remarks

• Substantial progress on submodular function maximization problems in the last few years

• New tools and connections including a general framework via the multilinear relaxation

• Increased awareness and more applications• Several open problems still remain

Thanks!