Submodular Optimization and Approximation Algorithms

Satoru Iwata (RIMS, Kyoto University)

Submodular Functions

•  Cut Capacity Functions •  Matroid Rank Functions •  Entropy Functions

Finite Set

Entropy Functions

Information Sources

Entropy of the Joint Distribution

Conditional Mutual Information

Positive Definite Symmetric Matrices

Ky Fan’s Inequality

Extension of the Hadamard Inequality

Discrete Concavity

Diminishing Returns

Discrete Convexity

Convex Function

Discrete Convexity Lovász (1983)

： Linear Interpolation

： Convex

： Submodular

Murota (2003)

Discrete Convex Analysis

Submodular Function Minimization

Minimization Algorithm

Evaluation Oracle

Minimizer

Assumption:

Ellipsoid Method Grötschel, Lovász, Schrijver (1981)

Submodular Function Minimization Grötschel, Lovász, Schrijver (1981, 1988)

Iwata, Fleischer, Fujishige (2000) Schrijver (2000)

Iwata (2003)

Fleischer, Iwata (2000)

Orlin (2007)

Iwata (2002)

Fully Combinatorial

Ellipsoid Method

Cunningham (1985)

Iwata, Orlin (2009)

Symmetric Submodular Functions

Symmetric

Symmetric Submodular Function Minimization

Minimum Cut Algorithm by MA-ordering Nagamochi & Ibaraki (1992) Minimum Degree Ordering Nagamochi (2007)

Queyranne (1998)

Application to Clustering

Minimize

subject to

Greedy Split

-Approximation

Monotone, Submodular

Zhao, Nagamochi, Ibaraki (2005)

Submodurlar Function Maximization

Approximation Algorithms Nemhauser, Wolsey, Fisher (1978) Monotone SF Cardinality Constraint (1-1/e)-Approximation

Vondrák (STOC 2008) Monotone SF Matroid Constraint (1-1/e)-Approximation

Submodurlar Function Maximization

Approximation Algorithms Feige, Mirrokni, Vondrák (FOCS 2007) Nonnegative SF 2/5-Approximation

Lee, Mirrokni, Nagarajan, Sviridenko (STOC 2009) Nonnegative SF 1/4-Approximation (Matroid Constraint) 1/5-Approximation (Knapsack Constraints)

Feature Selection Random Variables

Predict from subset “Sick”

“Fever” “Rash” “Cough” Maximize subject to Conditionally

Independent

Submodular

Krause & Guestrin (2005)

Submodular Welfare Problem

Monotone, Submodular

Utility Functions

Maximize

subject to

-Approximation Vondrák (2008)

Algorithm with for monotone submodular functions

Construct a set function such that

For what function is this possible?

Approximating Submodular Functions Goemans, Harvey, Iwata & Mirrokni (2009)

Question What Kind of Approximation Algorithms Can Be Extended to Optimization Problems

with Submodular Cost or Constraints ?

Cf. Submodular Flow (Edmonds & Giles, 1977)

Svitkina & Fleischer (FOCS 2008) Sampling Algorithms, Lower Bounds Submodular Sparsest Cut Submodular Load Balancing

Question How Can We Exploit Discrete Convexity

in Design of Approximation Algorithms?

Cf. Ellipsoid Method (Grötschel, Lovász & Schrijver, 1981)

Chudak & Nagano (SODA 2007) Facility Location with Submodular Penalty

Submodular Function Minimization under Covering Constraints

•  Submodular Vertex Cover Problem Rounding Algorithm •  Submodular Cost Set Cover Problem Rounding Algorithm Primal-Dual Algorithm •  Submodular Edge Cover Problem Inapproximability

Iwata & Nagano (2009)

The Vertex Cover Problem Graph

Vertex Cover

Find a Vertex Cover

Minimizing

The Vertex Cover Problem Integer Programming Formulation

Minimize

subject to

Linear Programming Relaxation (LPR)

The Vertex Cover Problem

Lemma (Nemhauser & Trotter, 1974)

LPR has a half-integral optimal solution.

Every nonsingular submatrix in the coefficient matrix has a half-integral inverse.

The Vertex Cover Problem

Optimal Solution of LPR

Vertex Cover

Vertex Cover

2-Approximation Solution

Submodular Vertex Cover Graph

Submodular

Find a Vertex Cover Minimizing

The Vertex Cover Problem

Relaxation Problem

Convex Programming Relaxation (CPR)

Minimize

subject to

Lemma

CPR has a half-integral optimal solution.

Proof of Half-Integrality

Every nonsingular submatrix in the coefficient matrix has a half-integral inverse.

Rounding Algorithm Half-Integral Optimal Solution of CPR

Vertex Cover

2-Approximation Solution

Vertex Cover

How to Solve CRP

Find a Vertex Cover

Minimizing

in

How to Solve CPR

Vertex Cover

Optimal Vertex Cover with

How to Solve CPR Vertex Cover with

Feasible to CPR

Lower Bounds Goel, Karande, Tripathi, Wang (FOCS 2009)

No polynomial algorithm can solve the submodular vertex cover problem approximately within a factor smaller than 2.

Rounding Algorithm Using Ellipsoid Method for Monotone Submodular Functions

Submodular Cost Set Cover Find Covering with Minimum

Rounding Algorithm -Approximation

Primal-Dual Algorithm

Greedy -Approximation Algorigm for Monotone Submodular Functions

Koufogiannakis & Young (ICALP 2009)

Relaxation Problem

Minimize

(SCP)

subject to

Ellipsoid Method

Rounding Algorithm

Optimal Solution of (SCP)

Set Cover

-Approximation Solution

Minimizer of among　 　 Supersets of

Rounding Algorithm

Set Cover

Set Cover

Monotone, Submodular

Primal-Dual Approximation

Maximize

subject to

Dual Problem of (SCP)

Submodular Polyhedron

Primal-Dual Approximation Initially,

Unique Maximal Set with

Primal-Dual Approximation

Set Cover

Set Cover

-Approximation

Submodular Cost Set Cover What if is not bounded?

For each Minimizer of Covering Set Cover

Set Cover

-Approximation

Greedy -Approx.?

Submodular Edge Cover Graph

Edge Cover

Submodular, Nonnegative,

Find an Edge Cover Minimizing

NP-hard MIN 2SAT

Inapproximability Result

Theorem No oracle-polynomial algorithm can solve the submodular edge cover problem approximately within a factor of

Summary Submodular Vertex Cover

CP-Rounding

2-Approx.

No Oracle-Polynomial Algorithm with an Approximation Ratio.

Submodular Cost Set Cover

CP-Rounding

Primal-Dual -Approx.

Submodular Edge Cover

-Approx.