Presentation is loading. Please wait.

Presentation is loading. Please wait.

Submodular Set Function Maximization A Mini-Survey Chandra Chekuri Univ. of Illinois, Urbana-Champaign.

Similar presentations


Presentation on theme: "Submodular Set Function Maximization A Mini-Survey Chandra Chekuri Univ. of Illinois, Urbana-Champaign."— Presentation transcript:

1 Submodular Set Function Maximization A Mini-Survey Chandra Chekuri Univ. of Illinois, Urbana-Champaign

2 Submodular Set Functions A function f : 2 N  R is submodular if f(A) + f(B) ≥ f(A  B) + f(A  B) for all A,B  N Equivalently, f(A+j) – f(A) ≥ f(A+i+j) – f(A+i) for all A  N and i, j  N\A

3 Submodular Functions Non-negative submodular set functions f(A) ≥ 0 for all A 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

4 Well-known Examples Cut functions in undirected graphs and hypergraphs (symmetric non-negative) Cut functions in directed graphs (non-negative) Rank functions of matroids (monotone) Coverage in set systems (monotone) many others...

5 Maximizing Submodular Set Functions Given f on a ground set N via a value oracle max S  N f(S) S satisfies some constraints

6 Maximizing Submodular Set Functions Given f on a ground set N via a value oracle max S  N f(S) S satisfies some constraints Motivation: Many non-trivial applications (easy to miss!) Generalize known results for modular functions

7 Unconstrained Problem max S  N f(S) Uninteresting for monotone f NP-Hard for non-negative f (Max-Cut is a special case) Very hard to approximate for arbitrary f (reduction from Set Packing)

8 Unconstrained Problem [Feige-Mirrokni-Vondrak’07] First O(1) approximation for non-negative f ! Easy O(1) algorithms, ½ for symmetric case Non-trivial 2/5 = 0.4 approximation (slight improvement to 0.41 [Vondrak])

9 Unconstrained Problem [Feige-Mirrokni-Vondrak’07] First O(1) approximation for non-negative f ! Easy O(1) algorithms, ½ for symmetric case Non-trivial 2/5 = 0.4 approximation (slight improvement to 0.41 [Vondrak]) Better than ½ requires exponential # of value queries Open Problem: Close gap between 0.41 and ½

10 Unconstrained Problem [Feige-Mirrokni-Vondrak’07] Random set algorithm: pick each i in N with prob ½, let R be random set E[f(R)] ≥ OPT/4 E [f(R)] ≥ OPT/2 for symmetric f Simple Local Search: Initialize S to best singleton S = local optimum for adding or deleting if improvement Output better of S and N\S 1/3 approx for non-negative f and ½ for symmetric f

11 Local Search Analysis Lemma: If S is a local optimum then for any I  S or S  I, f(I) ≤ f(S). Proof: Say S  I and f(I) > f(S) then by submodularity there exists i in I\S s.t f(S+i) > f(S). Corollary: Let S* be an optimum solution and S be a local opt. f(S  S*) ≤ f(S) and f(S  S*) ≤ f(S)

12 Local Search Analysis f(S  S*) ≤ f(S) and f(S  S*) ≤ f(S) f(S  S*) + f(N\S) ≥ f(S*\S) + f(N) ≥ f(S*\S) 2f(S) + f(N\S) ≥ f(S*\S) + f(S*  S) ≥ f(S*) implies max (f(S), f(N\S)) ≥ f(S*)/3 S N\S S*

13 Maximizing Submodular Set Functions with Constraints max S  N f(S) S satisfies some constraints Question: what constraints? For maximization probs, packing constraints natural S  I I is a downward-closed: A  I, B  A implies B  I

14 Matroid and Knapsack Constraints Combinatorial packing constraints I is the intersection of some p matroids on N Lemma: every downward-close family I on N is the intersection of p matroids on N (for some p) Knapsack or matrix packing constraints A is a m x n non-negative matrix, b is m x 1 vector I = { x  {0,1} n | A x ≤ b } Combination of matroid and knapsack constraints

15 Matroid Constraints Uniform matroid: I = { S : |S| ≤ k } Partition matroid: I = { S : |S  N i | ≤ k i, 1 ≤ i ≤ h } where N 1,..., N h partition N, and k i are integers Laminar matroid: I = { S : |S  U| ≤ k(U), U in F } for a laminar family of sets F Graphic matroid Matroid polytope is integral and hence one can hope to capture constraints via relaxation in polytope

16 Monotone Functions

17 Cardinality Constraint max f(S) such that |S| ≤ k Max k-Cover problem is special case Greedy gives (1-1/e) approximation [Nemhauser- Wolsey-Fisher’78] Unless P=NP no better approximation [Feige’98] Many applications, routinely used

18 Greedy Algorithm 1.S =  2.While |S| < k do i  argmax j f S (j) S  S+i 3.Output S

19 Greedy Analysis S j : first j elements picked by Greedy f(S) = δ 1 + δ δ k δ j ≥ (OPT – f(S j-1 ))/k (monotonicity and submod) f(S) ≥ (1-1/k) k OPT

20 A Different Analysis S = {i 1, i 2,..., i k } picked by Greedy in order S* = {i* 1, i* 2,..., i* k } an optimum solution Form perfect matching between S and S* s.t elements in S  S* are matched to themselves S S*

21 A Different Analysis Form perfect matching between S and S* Renumber S* s.t i j is matched to i* j For each j, because Greedy chose i* j instead of i* j δ j = f(S j-1 + i j ) – f(S j-1 ) ≥ f(S j-1 + i* j ) – f(S j-1 ) ≥ f S (i* j ) Summing up f(S) ≥ Σ j f S (i* j ) ≥ f S (S*) = f(S  S*) – f(S) hence f(S) ≥ f(S  S*)/2 ≥ f(S*)/2 if f is monotone

22 Why weaker analysis? Let M=(N, I) be any matroid on N Solve max f(S) such that S  I [Nemhauser-Wolsey-Fisher’78] Theorem: Greedy is ½ approximation, and analysis is tight even for partition matroids Many applications, unfortunately unknown, till recently, to approximation algorithms community!

23 Base Exchange Theorem B and B’ are distinct bases in a matroid M=(N, I) Strong Base Exchange Theorem: There are elements i  B\B’ and i’  B’\B such that B-i+i’ and B’-i’+i are both bases. Corollary: There is a perfect matching between B\B’ and B’\B such that for each matched pair (i,i’), B-i+i’ is a base.

24 Proof for Greedy B is the solution that Greedy outputs B* is an optimum solution B and B* are bases of M by monotonicity of f Same argument as before works by using perfect matching between B and B*

25 Multiple Matroids max f(S) such that S is in intersection of p matroids [Fisher-Nemhauser-Wolsey’78] Theorem: Greedy is 1/(p+1) approximation Generalize matching argument to match one element of Greedy to p elements of OPT Also works for p-systems

26 Non-negative functions? max f(S) such that S is in intersection of p matroids [Lee-Mirrokni-Nagarajan-Sviridenko’08] Theorem: For fixed p, local search based algorithm that achieves Ω (1/p) approximation. [Gupta-Roth-Schoenbeck-Talwar’09] For all p and with simple proof combining Greedy and unconstrained algorithm. Slightly worse constants.

27 Knapsack Constraints Similar ideas can be used with standard guessing large items etc

28 Can we do better? ½ for one matroid, hardness is (1-1/e) 1/(p+1) for p matroids, hardness is  (p/log p)

29 Can we do better? [Calinescu-C-Pal-Vondrak’07]+[Vondrak’08]=[CCPV’09] Theorem: There is a randomized (1-1/e) approximation for maximizing a monotone f subject to a matroid constraint.

30 Can we do better? [Calinescu-C-Pal-Vondrak’07]+[Vondrak’08]=[CCPV’09] Theorem: There is a randomized (1-1/e) approximation for maximizing a monotone f subject to a matroid constraint. [Lee-Sviridenko-Vondrak’09] Theorem: For fixed p≥2, there is a local-search based 1/(p+ ε ) approximation for intersection of p matroids New useful insight for (two) matroid intersection

31 Multilinear Extension of f Question: Is there a useful continuous relaxation of f such that it can be optimized? And can we round it effectively?

32 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| F(x) = Expect[ f(x) ] =  S  N f(S) p x (S) =  S  N f(S)  i  S x i  i  N\S (1-x i )

33 Multilinear Extension of f For f: 2 N  R + define F:[0,1] N  R + as F(x) =  S  N f(S)  i  S x i  i  N\S (1-x i ) F is smooth submodular ([Vondrak’08])  F/  x i ≥ 0 for all i (monotonicity)  2 F/  x i  x j ≤ 0 for all i,j (submodularity)

34 Optimizing F(x) [Vondrak’08] Theorem: For any down-monotone polytope P  [0,1] n max F(x) s.t x  P can be optimized to within a (1- 1/e) approximation if we can do linear optimization over P Algorithm: Continuous-Greedy

35 Generic Approach Want to solve: max f(S) s.t S  I Relaxation: max F(x) s.t x  P(I ) P(I) is a polytope that captures/relaxes S  I Can solve to within (1-1/e) with continuous greedy How to round? F is a non-linear function

36 Rounding in Matroids Let M=(S,I) be a matroid P(M) is the independent set polytope of M B(M) is the base polytope of M Algorithm: Run continuous greedy to obtain a point x  B(M) such that F(x) ≥ (1-1/e) OPT Round x to a vertex of B(M) (a base)

37 Rounding in Matroids [CCPV’07] Theorem: Given any point x in B(M), there is a polynomial time algorithm to round x to a vertex x* (hence a base of M) such that F(x*) ≥ F(x). “Pipage” rounding technique of [Ageev-Sviridenko] adapted to matroids

38 New Rounding Method [C-Vondrak-Zenklusen’09] Randomized Swap-Rounding: Express x =  m j=1 β i B i (convex comb. of bases) B = B 1, β = β 1 For k = 2 to m do Randomly Merge β B and β k B k into (β + β k ) B’ Set B = B’, β = (β + β k ) Output B

39 Merging two Bases Merge B’ and B’’ into a random B that looks “half” like B’ and “half” like B’’

40 Merging two Bases B’ B’’ B’  B’’ ij Base ExchangeTheorem: B’-i+j and B’’-j+i are both bases

41 Merging two Bases B’ B’’ B’  B’’ ij prob ½ B’ B’’ B’  B’’ ii B’ B’’ B’  B’’ jj

42 Swap Rounding for Matroids Theorem: Swap-Merging with input x in B(M) outputs a random base B such that 1.E[f(B)] ≥ F(x) and 2.Pr[ f(B) < (1- δ ) F(x)] ≤ exp(- F(x) δ 2 /8) (concentration for lower tail of submod functions) and 3.For any vector a  [0,1] n, let μ = ax then Pr[a 1 B < (1- δ ) μ ] ≤ exp(- μ δ 2 /2) Pr[a 1 B > (1+ δ ) μ ] ≤ ( e δ / (1+ δ ) δ ) μ (concentration for linear functions) Almost like independent random rounding of x

43 Applications Can handle matroid constraint plus packing constraints x  P(M) and Ax ≤ b (1-1/e) approximation for submodular functions subject to a matroid plus O(1) knapsack/packing constraints (or many “loose” packing constraints) Simpler rounding and proof for “thin” spanning trees in ATSP application ([Asadpour etal’10])...

44 Other Tools On-going work [C-Vondrak-Zenklusen] Extension of continuous greedy to handle multiple submodular functions simultaneously Depending rounding via swap method for polyhedra matroid intersection matchings and b-matchings in non-bipartite graphs... Many applications of dependent randomized rounding[Arora-Frize-Kaplan, Srinivasan’01,...., Asadpour etal,..]

45 Illustrative Application Maximum Bipartite Flow in Networks with Adaptive Channel Width [Azar-Madry-Moscibroda- Panigrahy-Srinivasan’09] Problem motivated by capacity allocation in wireless networks (1-1/e) approximation via a specialized LP and complicated analysis An easy O(log n) approximation

46 Problem Definition Base stations Clients θ (B,C) : threshold capacity between B and C α (C): max flow that C desires from base stns β (B): total capacity of B to serve clients For each base station B, decide an operating point τ (B) ≤ β (B) If τ (B) > θ (B,C) then u(B,C) capacity of link (B,C) is 0, otherwise u(B,C) = τ (B) Maximize flow from base stations to clients β (B) α (C) u(B,C)

47 Reduction Base stations Clients β (B) α (C) t B Copies of B one for each τ (B) u(τ (B),C) N : all copies on left f : 2 N  R + where f(S) = flow from S to t Partition Matroid constraint: can pick only one copy for each B

48 Summary Substantial progress on submodular function maximization problems Increased awareness and more applications (New) tools and connections: continuous greedy, dependent rounding, local search,... Several open problems still remain

49 Open Problems Unconstrained for non-neg f. Close gap bet 0.41 and ½ max f(S) s.t S in intersection of p matroids p=1 we have 1-1/e approx which is tight unless P=NP for fixed p ≥ 2, 1/(p+ ε ) approx, otherwise 1/(p+1) hardness is Ω (p/log p) for large p when f is modular close gaps, most interesting for small p How to round F(x) for more than 1 matroid? Don’t know integrality gap for p=2! max f(x) s.t x in P(M) and Ax ≤ b, A is k-column-sparse. Want Ω (1/k) approx. Known without matroid constraint [Bansal-Korula-Nagarajan-Srinivisan’10]


Download ppt "Submodular Set Function Maximization A Mini-Survey Chandra Chekuri Univ. of Illinois, Urbana-Champaign."

Similar presentations


Ads by Google