Submodular Set Function Maximization via the Multilinear Relaxation

Submodular Set Function Maximization via the Multilinear Relaxation & Dependent Rounding Chandra Chekuri Univ. of Illinois, Urbana-Champaign

Max weight independent set • N a finite ground set • w : N ! R+ weights on N • I µ 2 N 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 w(S) s. t S 2 I • 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

This talk max f(S) s. t. S 2 I f is a non-negative submodular set function on N Motivation: • several applications • mathematical interest

Submodular Set Functions A function f : 2 N ! R+ is submodular if f(A+j) – f(A) ¸ f(B+j) – f(B) for all A ½ B, i 2 NB A B j f(A+j) – f(A) ≥ f(A+i+j) – f(A+i) for all A N , i, j NA

Submodular Set Functions A function f : 2 N ! R+ is submodular if f(A+j) – f(A) ¸ f(B+j) – f(B) for all A ½ B, i 2 NB A B j f(A+j) – f(A) ≥ f(A+i+j) – f(A+i) for all A N , i, j NA 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 : 2 V ! R+ where f(S) = |δ(S)| S

Coverage in Set Systems • X 1, X 2, . . . , Xn subsets of set U • f : 2{1, 2, . . . , n} ! R+ where f(A) = |[ i X 1 X 5 X 4 X 2 X 3 in A Xi | X 1 X 5 X 4 X 2 X 3

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(NA) 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 max f(S) s. t S 2 I • f is cut function of a given graph G=(V, E) • I = 2 V : unconstrained • NP-Hard

Example: Max k-Coverage max f(S) s. t S 2 I • X 1, X 2, . . . , 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

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 max f(S) s. t. S 2 I f is a non-negative submodular set function on N • Greedy • Local Search • Multilinear relaxation and rounding

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) max w¢x s. t x 2 P(I) S 2 I xi 2 [0, 1] indicator variable for i Exact algorithm: P(I) = convexhull( {1 S : S 2 I})

Math. Programming approach max w(S) max w¢x s. t x 2 P(I) S 2 I Round x* 2 P(I) to S* 2 I Exact algorithm: P(I) = convexhull( {1 S : S 2 I}) Approx. algorithm: P(I) ¾ convexhull( {1 S : S 2 I}) P(I) solvable: can do linear optimization over it

Math. Programming approach max f(S) max F(x) s. t x 2 P(I) S 2 I P(I) ¶ convexhull( {1 S : S 2 I}) and solvable Round x* 2 P(I) to S* 2 I

Math. Programming approach max f(S) max F(x) s. t x 2 P(I) S 2 I • What is the continuous extension F ? • How to optimize with objective F ? • How do we round ? 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 max f(S) max F(x) s. t x 2 P(I) S 2 I • What is the continuous extension F ? • How to optimize with objective F ? • How do we round ? Round x* 2 P(I) to S* 2 I

Multilinear extension of f [CCPV’ 07] inspired by [Ageev-Sviridenko] For f : 2 N ! R+ define F : [0, 1]N ! R+ as x = (x 1, x 2, . . . , 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 NS (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 • 2 F/ 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

Math. Programming approach max f(S) max F(x) s. t x 2 P(I) S 2 I Round x* 2 P(I) to S* 2 I • What is the continuous extension F ? ✔ • How to optimize with objective F ? • How do we round ?

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 P Q = { z 2 P | z · 1 -x* } y* = a local optimum of F in Q Output 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 P Q = { z 2 P | z · 1 -x* } y* = a local optimum of F in Q Output better of x* and y* Theorem: Above algorithm gives a ¼ approximation.

Math. Programming approach max f(S) max F(x) s. t x 2 P(I) S 2 I Round x* 2 P(I) to S* 2 I • What is the continuous extension F ? ✔ • How to optimize with objective F ? ✔ • How do we round ?

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({1 S | S 2 I}) given by following system [Edmonds] i 2 S xi · rank. M(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 max F(x) s. t x 2 P(I) Round x* 2 P(I) to S* 2 I F(x*) = E[f(R)] where R is obtained by independently rounding each i with probability x*i R unlikely to be in I

Rounding max F(x) s. t x 2 P(I) Round x* 2 P(I) to S* 2 I 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)

A simple question? 0. 6 0. 4 0. 7 0. 6 0. 1 1 0. 7 1 0. 3 0. 9 0. 4 • 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 e Pr[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 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

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 max f(S) max F(x) s. t x 2 P(I) S 2 I Round x* 2 P(I) to S* 2 I Problem reduced to finding a good relaxation P(I) and a contention resolution scheme for P(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!