A Min-Max Theorem for Transversal Submodular Functions and Its - PDF document

A Min-Max Theorem for Transversal Submodular Functions and Its Implications Satoru Fujishige Research Institute for Mathematical Sciences Kyoto University, Japan 18th Aussois Workshop on Combinatorial Optimization Aussois, January 6–10, 2014 (Joint work with Shin-ichi Tanigawa, RIMS, Kyoto University) 1

The present talk will 1. introduce a new concept of transversal submodular func- tion, a generalization of ordinary submodular set function, 2. show a min-max relation between the minimum of a transver- sal submodular function and the maximum of the negative of a norm composed of ℓ 1 and ℓ ∞ norms, and 3. based on the min-max relation, give a unifying view over the recent results on generalizations of submodular set functions: [1] A. Huber and V. Kolmogorov: Towards minimizing k -submodular functions. Proceedings of ISCO 2012, LNCS 7422 (2012) 451–462. [2] F. Kuivinen: On the complexity of submodular func- tion minimisation on diamonds. Discrete Optimiza- tion 8 (2011) 459–477. 2

✟ ————————————————————————– V : a nonempty finite set U ≡ { U 1 , U 2 , · · · , U n } : a partition of V �✂✁ �☎✄ �✝✆ ✞✂✞☎✞ T ( ⊆ V ) : a subtransversal (or partial transversal ) of U ( | T ∩ U | ≤ 1 for all U ∈ U ) T : the set of all subtransversals of U U ( T ) = { U ∈ U | U ∩ T ̸ = ∅} ( ∀ T ∈ T ) U ( v ) : the unique U ∈ U that contains v ∈ V ————————————————————————– We consider a function f : T → R . → 3

✟ ————————————————————————– Consider two binary operations ▽ and △ on T satisfying the condition that for all T 1 , T 2 ∈ T T 1 ▽ T 2 ∈ T , U ( T 1 ▽ T 2 ) ⊆ U ( T 1 ) ∪ U ( T 2 ) , T 1 △ T 2 ∈ T , U ( T 1 △ T 2 ) ⊆ U ( T 1 ) ∩ U ( T 2 ) . Define a function f : T → R with f ( ∅ ) = 0 satisfying f ( T 1 ) + f ( T 2 ) ≥ f ( T 1 ▽ T 2 ) + f ( T 1 △ T 2 ) ( ∀ T 1 , T 2 ∈ T ) . We call f a transversal submodular function or a t-submodular function , for short. ————————————————————————– �✂✁✄�✆☎ �✞✝ ✠☛✡ ✠✌☞ → 4

Example 1 : k -submodular functions due to Huber and Kol- mogorov (ISCO 2012). ————————————————————————– For any T, T ′ ∈ T define binary operations ⊔ and ⊓ on T by T ⊔ T ′ = ( T ∪ T ′ ) \ ∪ { U ∈ U | | U ∩ ( T ∪ T ′ ) | = 2 } , T ⊓ T ′ = T ∩ T ′ . Let k = max {| U | | U ∈ U} . A function f : T → R is called k -submodular if ( ∀ T, T ′ ∈ T ) . f ( T ) + f ( T ′ ) ≥ f ( T ⊔ T ′ ) + f ( T ⊓ T ′ ) We assume f ( ∅ ) = 0 . ————————————————————————– → 5

Example 1 : k -submodular functions due to Huber and Kol- mogorov (ISCO 2012). ————————————————————————– For any T, T ′ ∈ T define binary operations ⊔ and ⊓ on T by T ⊔ T ′ = ( T ∪ T ′ ) \ ∪ { U ∈ U | | U ∩ ( T ∪ T ′ ) | = 2 } , T ⊓ T ′ = T ∩ T ′ . Let k = max {| U | | U ∈ U} . A function f : T → R is called k -submodular if ( ∀ T, T ′ ∈ T ) . f ( T ) + f ( T ′ ) ≥ f ( T ⊔ T ′ ) + f ( T ⊓ T ′ ) We assume f ( ∅ ) = 0 . ————————————————————————– Remark : Bouchet (1997) considered k -submodular functions (monotone nondecreasing and unit-increasing) to define a set system called a multimatroid as a generalization of delta- matroids. ————————————————————————– → 6

✠ ☎ ✌ � ✠ ✠ ✡ ✟ ✆ ☎ ✆ ✝ ☎ ✆ ✞ Example 2 : Submodular functions on product lattices and, in particular, diamonds due to Kuivinen ( Discrete Optimization , 2011). ————————————————————————– 0 U : a new element for each U ∈ U Put ˆ U = U ∪ { 0 U } for each U ∈ U . �✂✁ �☞☛ �✍✌ �✂✁ �☞☛ ✄✂✄✂✄ An arbitrary lattice L U = ( ˆ U, ∨ U , ∧ U ) with lattice operations, join ∨ U and meet ∧ U , for each U ∈ U 0 U : the minimum element of L U . 1 U : the maximum element of L U . ————————————————————————– → 7

� ✌ ✠ ✠ ✡ ✠ ✟ ✆ ☎ ✞ ✆ ☎ ✝ ✆ ☎ Let L = ⊗ U ∈U L U (= ( ⊗ U ∈U ˆ U, ∨ , ∧ )) be the product of lat- tices L U = ( ˆ U, ∨ U , ∧ U ) for U ∈ U . A function f : ⊗ U ∈U ˆ U → R is called a submodular function on product lattice L if f ( ˆ T ) + f ( ˆ T ′ ) ≥ f ( ˆ T ∨ ˆ T ′ ) + f ( ˆ T ∧ ˆ T ′ ) T ′ ∈ ⊗ U ∈U ˆ for all ˆ T, ˆ U . ————————————————————————– This function can be regarded as a special case of t-submodular functions by discarding minimum elements 0 U for all U ∈ U . �✂✁ �☞☛ �✍✌ �✂✁ �☞☛ ✄✂✄✂✄ → 8

A Min-Max Theorem for T-submodular Functions ————————————————————————– Let f : T → R be a t-submodular function . Define a function F : 2 U → R as follows. F ( X ) = min { f ( T ) | T ∈ T , U ( T ) ⊆ X} ( ∀X ⊆ U ) . ————————————————————————– Lemma 1 : F : 2 U → R is a submodular function on 2 U with F ( ∅ )=0 . ————————————————————————– (Proof) For any X , Y ⊆ U there exist T X , T Y ∈ T such that U ( T X ) ⊆ X , U ( T Y ) ⊆ Y , F ( X ) = f ( T X ) , F ( Y ) = f ( T Y ) . Hence we have F ( X ) + F ( Y ) = f ( T X ) + f ( T Y ) ≥ f ( T X ▽ T Y ) + f ( T X △ T Y ) ≥ F ( X ∪ Y ) + F ( X ∩ Y ) . We also have F ( ∅ ) = f ( ∅ ) = 0 . ✷ ————————————————————————– → 9

We can easily see that min { f ( T ) | T ∈ T } = min { F ( X ) | X ⊆ U} . Hence we have the following. ————————————————————————– Lemma 2 : min { f ( T ) | T ∈ T } = max { x ( U ) | x ≤ 0 , x ∈ P( F ) } , where P( F ) = { x ∈ R U | ∀X ⊆ U : x ( X ) ≤ F ( X ) } , the submodular polyhedron associated with submodular function F and x ( X ) = ∑ U ∈X x ( U ) . ————————————————————————– (Proof) This follows from Edmonds’ min-max theorem for submodular function minimization. ✷ ————————————————————————– It should be noted that since F is monotone non-increasing, every x ∈ P( F ) is nonpositive, so that we may suppress the condition x ≤ 0 appearing in Lemma 2. → 10

For any x ∈ R U define z x ∈ R V by ( ∀ v ∈ V ) . z x ( v ) = x ( U ( v )) Here it should be noted that x ( U ( v )) is the value of x ∈ R U for the coordinate U ( v ) ∈ U . ————————————————————————– Lemma 3 : Suppose we are given a nonpositive x ∈ R U , i.e., x ≤ 0 . Then, we have x ∈ P( F ) if and only if z x ∈ P( f ) , where P( f ) = { z ∈ R V | ∀ T ∈ T : z ( T )( ≡ ∑ z ( v )) ≤ f ( T ) } . v ∈ T ————————————————————————– (Proof) Suppose x ∈ P( F ) . Then, for any T ∈ T z x ( T ) = x ( U ( T )) ≤ F ( U ( T )) ≤ f ( T ) . Hence z x ∈ P( f ) . Conversely, suppose z x ∈ P( f ) for x ∈ R U with x ≤ 0 . Then, for any X ⊆ U and any T ∈ T such that U ( T ) ⊆ X we have x ( X ) ≤ x ( U ( T )) = z x ( T ) ≤ f ( T ) , where the first inequality holds since x ≤ 0 . This implies x ( X ) ≤ min { f ( T ) | T ∈ T , U ( T ) ⊆ X} = F ( X ) . Hence x ∈ P( F ) . ✷ ————————————————————————– → 11

✟ ————————————————————————– For any z ∈ R V define n ∑ || z || 1 , ∞ = max u ∈ U i | z ( u ) | . i =1 This defines a norm on R V , which is a composition of ℓ 1 and ℓ ∞ norms. �✂✁ �☎✄ �✝✆ ✞✂✞☎✞ ————————————————————————– Remark : || · || 1 , ∞ = || · || 1 if | U i | = 1 for all i = 1 , · · · , n , and || · || 1 , ∞ = || · || ∞ if n = 1 . → 12