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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 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 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. S2 I f is a non-negative submodular set function on N Motivation: • several applications • mathematical interest 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 j A 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 f(A+j) – f(A) ¸ f(B+j) – f(B) for all A ½ B, i 2 N\B j A 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 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 in A Xi | X1 X1 X5 X5 X4 X4 X2 X2 X3 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 • ... Example: Max-Cut max f(S) s.t S 2 I • f is cut function of a given graph G=(V,E) • I = 2V : unconstrained • NP-Hard Example: 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 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. S2 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 xi 2 [0,1] indicator variable for s.t S 2 I s.t x 2 P(I) i Exact algorithm: P(I) = convexhull( {1S : S 2 I}) Math. Programming approach max w(S) max w¢x Round x* 2 P(I) to s.t S 2 I s.t x 2 P(I) 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) max F(x) Round x* 2 P(I) to s.t S 2 I s.t x 2 P(I) S* 2 I P(I) ¶ convexhull( {1S : S 2 I}) and solvable Math. Programming approach max f(S) max F(x) Round x* 2 P(I) to s.t S 2 I 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 ? 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) Round x* 2 P(I) to s.t S 2 I 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 ? 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 max f(S) max F(x) Round x* 2 P(I) to s.t S 2 I 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 ? 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) Round x* 2 P(I) to s.t S 2 I 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 ? 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 max F(x) Round x* 2 P(I) s.t 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) Round x* 2 P(I) s.t 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? 0.4 0.6 0.3 0.6 0.1 0.7 1 0.7 0.9 1 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) Round x* 2 P(I) s.t 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) Round x* 2 P(I) to s.t S 2 I s.t x 2 P(I) 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!

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posted: | 10/9/2013 |

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