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Algorithms for submodular objectives: continuous extensions & dependent randomized rounding

Algorithms for submodular objectives: continuous extensions & dependent randomized rounding. Chandra Chekuri Univ. 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.

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Algorithms for submodular objectives: continuous extensions & dependent randomized rounding

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  1. Algorithms for submodularobjectives: continuous extensions &dependent randomized rounding Chandra Chekuri Univ. of Illinois, Urbana-Champaign

  2. Combinatorial Optimization • N a finite ground set • w : N !Rweights on N max/minw(S) s.tS µ N satisfies constraints

  3. Combinatorial Optimization • N a finite ground set • w : N !Rweights on N • Sµ2Nfeasible solutions to problem max/minw(S) s.tS 2S

  4. 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 • ...

  5. Examples: NP-Hard • max cut • min-cost multiway/multiterminal cut • min-cost (metric) labeling • max weight independent set in a graph • ...

  6. 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

  7. This talk min/maxf(S) s.t.S 2S f is a non-negative submodular set function on N Motivation: • several applications • mathematical interest • modeling power and new results

  8. 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, j 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

  9. 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, i2 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(AB) + f(AB) 8A,B  N

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

  11. Coverage in Set Systems • X1, X2, ..., Xnsubsets of set U • f : 2{1,2, ..., n} !R+where f(A) = |[i in AXi | X1 X1 X5 X5 X4 X4 X2 X2 X3 X3

  12. Submodular Set Functions • Non-negative submodular set functions f(A) ≥ 0 8A )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

  13. 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 • ...

  14. Max-Cut maxf(S) s.tS 2S • f is cut function of a given graph G=(V,E) • S = 2V: unconstrained • NP-Hard!

  15. Unconstrained problem min/maxf(S) • 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

  16. Techniques min/maxf(S) s.t.S 2S f is a non-negative submodular set function on N • Greedy • Local Search • Mathematical Programming Relaxation + Rounding

  17. Math. Programming approach min/maxw(S) s.tS 2S min/maxw¢x s.tx2 P(S) xi2[0,1] indicator variable for i Exact algorithm: P(S) = convexhull( {1S : S 2S})

  18. Math. Programming approach min/maxw(S) s.tS 2S min/maxw¢x s.tx2 P(S) Round x*2 P(S) to S*2S Exact algorithm: P(S) = convexhull( {1S : S 2S}) Approx. algorithm: P(S)¾convexhull( {1S : S 2S}) P(S) solvable: can do linear optimization over it

  19. Math. Programming approach min/maxf(S) s.tS 2S min/maxg(x) s.tx2 P(S) Round x*2 P(S) to S*2S P(S)¶convexhull( {1S : S 2S}) and solvable

  20. Math. Programming approach • What is the continuous extension g ? • How to optimize with objective g ? • How do we round ? min/maxf(S) s.tS 2S min/maxg(x) s.tx2 P(S) Round x*2 P(S) to S*2S

  21. 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

  22. Canonical extensions: convex and concave closure x= (x1, x2, ..., xn)  [0,1]N min/max S ®S f(S) S®S = 1 S®S = xifor all i ®S¸ 0 for all S f-(x) for minimization and f+(x) for maximization: convex and concave respectively for anyf

  23. Submodularf • 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

  24. Lovasz-extension of f f»(x) = Eµ2 [0,1][ f(xµ) ] wherexµ = { i | xi¸µ } Example:x = (0.3, 0, 0.7, 0.1) xµ = {1,3} forµ = 0.2 andxµ = {3} forµ = 0.6 f»(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})

  25. Properties of f» • f»is convex ifff is submodular • f»(x) = f-(x) for all x when f is submodular • Easy to evaluate f» • For submodf : solve relax. via convex optimization minf»(x) s.tx2 P(S)

  26. 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 iindependently with prob. xi F(x) =E[ f(R) ] =S N f(S) iS xi iN\S (1-xi)

  27. Properties of F • F(x) can be evaluated by random sampling • F is a smooth submodular function • 2F/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 • Fis concave along any non-negative direction vector • F/xi ≥ 0 for all iif f is monotone

  28. Maximizing F max { F(x) | xi2 [0,1] for all i} is NP-Hard equivalent to unconstrained maximization of f When f is monotone max { F(x) | ixi· k, xi2[0,1] for all i} is NP-Hard

  29. 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

  30. 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]nis any down-closed solvable polytope. Remark: 0.325-approximation can be obtained Remark: Current best 1/e ' 0.3678 [Feldman-Naor-Schwartz’11] Algorithms: variantsof local-search and continuous-greedy

  31. Math. Programming approach min/maxf(S) s.tS 2S min/maxg(x) s.tx2 P(S) Round x*2 P(S) to S*2S • 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 ? ✔ ✔

  32. Rounding Rounding and approximation depend on Sand P(S) Two competing issues: • Obtain feasible solution S* from fractional x* • Want f(S*) to be close to g(x*)

  33. 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

  34. Rounding to preserve objective x* : fractional solution to relaxation Minimization: f»(x) = Eµ2 [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 i2 N independently with probability ®x*i (®· 1)

  35. Maximization maxf(S) s.tS 2I Iµ2Nis a downward closed family A 2I& B ½ A)B 2I Captures “packing” problems

  36. Maximization High-level results: • optimal rounding in matroidpolytopes[Calinescu-C-Vondrak-Pal’07,C-Vondrak-Zeklusen’09]] • contention resolution scheme based rounding framework [C-Vondrak-Zenklusen’11]

  37. Max k-Coverage maxf(S) s.tS 2I • X1,X2,...,Xnsubsets 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

  38. 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]

  39. 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.

  40. 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.

  41. Contention Resolution Schemes • Ian independence family on N • P(I) a relaxation for I and x2 P(I) • R: random set from independent rounding of x CR scheme for P(I): given x, R outputs R’ µ R s.t. • R’ 2I • and for all i, Pr[i2 R’ | i2 R] ¸ c

  42. Rounding and CR schemes maxF(x) s.tx2 P(I) Round x*2 P(I) to S*2I Theorem: A monotone CR scheme for P(I) can be used to round s.t. E[f(S*)] ¸ c F(x*) Via FKG inequality

  43. 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

  44. 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

  45. Submodular-cost Vertex Cover • Input:G=(V,E) and f : 2V!R+ • Goal:min f(S) s.tS 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]

  46. Submodular-cost Vertex Cover • Input:G=(V,E) and f : 2V!R+ • Goal:min f(S) s.tS is a vertex cover in G min f»(x) xi + xj¸ 1 for all ij2 E xi¸ 0 for all i2 V

  47. Rounding min f»(x) xi + xj¸ 1 for all ij2 E xi¸ 0 for all i2 V Pick µ2 [0, 1/2] uniformly at random Output S = { i | xi¸µ }

  48. Rounding Analysis min f»(x) xi + xj¸ 1 for all ij2 E xi¸ 0 for all i2 V Pick µ2 [0, 1/2] uniformly at random Output S = { i | x*i¸µ } Claim 1: S is a vertex cover with probability 1 Claim 2: E[ f(S) ] · 2 f»(x*) Proof: 2f»(x) = 2s10 f(xµ) dµ¸ 2s1/20 f(xµ) = E[ f(S) ]

  49. Submodular-cost Set Cover • Input:SubsetsX1,...,XnofU,f : 2N!R+ • Goal:min f(S) s.t[i2 SXi = 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

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