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

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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

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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

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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

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

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Submodular Set Functions A function f : 2 N ! R + is submodular if AB j 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

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Submodular Set Functions A function f : 2 N ! R + is submodular if AB j 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(A B) + f(A B) 8 A,B N

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G=(V,E) undirected graph f : 2 V ! R + where f(S) = | δ (S)| Cut functions in graphs S

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Coverage in Set Systems X 1, X 2,..., X n subsets of set U f : 2 {1,2,..., n} ! R + where f(A) = | [ i in A X i | X1X1 X2X2 X3X3 X4X4 X5X5 X1X1 X2X2 X3X3 X4X4 X5X5

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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

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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...

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

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Example: Max k-Coverage X 1,X 2,...,X n 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

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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

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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

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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]

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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

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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( { 1 S : S 2 I }) x i 2 [0,1] indicator variable for i

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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( { 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

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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( { 1 S : S 2 I }) and solvable

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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

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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.

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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.

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Submodular Welfare Problem n items/goods (N) to be allocated to k players each player has a submodular utility function f i (A i ) is the utility to i if A i is allocation to i) Goal: maximize welfare of allocation i f i (A i ) Can be reduced to a single f and a (partition) matroid constraint and hence (1-1/e) approximation

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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

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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

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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,..., x n ) [0,1] N R: random set, include i independently with prob. x i F( x ) = E [ f(R) ] = S N f(S) i S x i i N\S (1-x i )

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Why multilinear extension? Ideally a concave extension to maximize Could choose (“standard”) concave closure f + of f Evaluating f + ( x ) is NP-Hard!

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Properties of F F( x ) can be evaluated (approximately) by random sampling F is a smooth submodular function 2 F/ x i x j ≤ 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/ x i ≥ 0 for all i if f is monotone

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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

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Maximizing F max { F(x) | x i · k, x i 2 [0,1] } is NP-Hard

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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

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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

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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 *

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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.

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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

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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

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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 x i · rank M (S) 8 S µ N x 2 [0,1] N

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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.

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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

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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

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A simple question? 0.9 1 0.4 0.6 0.4 1 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 x e Question: what is the expected size of a maximal forest in R? (n - # of connected components)

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A simple question? x is a convex combination of spanning trees of G R: pick each e 2 E independently with probability x e Question: what is the expected size of a maximal forest in R? (n - # of connected components) Answer: ¸ (1-1/e) (n-1)

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Related question x is a convex combination of spanning trees of G R: pick each e 2 E independently with probability x e Want a (random) forest R’ µ R s.t. for every edge e Pr[e 2 R’ | e 2 R] ¸ c

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Related question x is a convex combination of spanning trees of G R: pick each e 2 E independently with probability x e 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 x e = c (n-1) in R

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

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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

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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

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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

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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

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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

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Thanks!

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