Families of Mass Destruction Shagnik Das 1 aros 1 Tam´ as M´ esz´ 1 Freie Universit¨ at Berlin Symposium Diskrete Mathematik, Berlin 15th July 2016
Introducing the problem Some initial bounds An exact solution Concluding remarks Shattering sets Definition (Shattering) A family F shatters a set A if its members intersect A in every possible way: { F ∩ A : F ∈ F} = 2 A . Given a family F , Sh( F ) denotes the collection of sets it shatters. Definition (Families) A (small) example Ground set: [ n ] = { 1 , 2 , . . . , n } . Let F be the family {{ 1 } , { 1 , 2 } , { 2 , 3 } , { 2 , 4 } , Family F a collection of { 1 , 2 , 5 } , { 1 , 3 , 4 } , { 2 , 4 , 5 } , subsets: F ⊆ 2 [ n ] . { 3 , 4 , 5 } , { 1 , 3 , 4 , 5 }} . k-uniform if all members have � [ n ] � size k : F ⊆ . Then { 1 , 4 , 5 } is shattered. k S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Shattering sets Definition (Shattering) A family F shatters a set A if its members intersect A in every possible way: { F ∩ A : F ∈ F} = 2 A . Given a family F , Sh( F ) denotes the collection of sets it shatters. Definition (Families) A (small) example Ground set: [ n ] = { 1 , 2 , . . . , n } . Let F be the family {{ 1 } , { 1 , 2 } , { 2 , 3 } , { 2 , 4 } , Family F a collection of { 1 , 2 , 5 } , { 1 , 3 , 4 } , { 2 , 4 , 5 } , subsets: F ⊆ 2 [ n ] . { 3 , 4 , 5 } , { 1 , 3 , 4 , 5 }} . k-uniform if all members have � [ n ] � size k : F ⊆ . Then { 1 , 4 , 5 } is shattered. k S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Shattering sets Definition (Shattering) A family F shatters a set A if its members intersect A in every possible way: { F ∩ A : F ∈ F} = 2 A . Given a family F , Sh( F ) denotes the collection of sets it shatters. Definition (Families) A (small) example Ground set: [ n ] = { 1 , 2 , . . . , n } . Let F be the family {{ 1 } , { 1 , 2 } , { 2 , 3 } , { 2 , 4 } , Family F a collection of { 1 , 2 , 5 } , { 1 , 3 , 4 } , { 2 , 4 , 5 } , subsets: F ⊆ 2 [ n ] . { 3 , 4 , 5 } , { 1 , 3 , 4 , 5 }} . k-uniform if all members have � [ n ] � size k : F ⊆ . Then { 1 , 4 , 5 } is shattered. k S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Shattering sets Definition (Shattering) A family F shatters a set A if its members intersect A in every possible way: { F ∩ A : F ∈ F} = 2 A . Given a family F , Sh( F ) denotes the collection of sets it shatters. Definition (Families) A (small) example Ground set: [ n ] = { 1 , 2 , . . . , n } . Let F be the family {{ 1 } , { 1 , 2 } , { 2 , 3 } , { 2 , 4 } , Family F a collection of { 1 , 2 , 5 } , { 1 , 3 , 4 } , { 2 , 4 , 5 } , subsets: F ⊆ 2 [ n ] . { 3 , 4 , 5 } , { 1 , 3 , 4 , 5 }} . k-uniform if all members have � [ n ] � size k : F ⊆ . Then { 1 , 4 , 5 } is shattered. k S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Shattering sets Definition (Shattering) A family F shatters a set A if its members intersect A in every possible way: { F ∩ A : F ∈ F} = 2 A . Given a family F , Sh( F ) denotes the collection of sets it shatters. Definition (Families) A (small) example Ground set: [ n ] = { 1 , 2 , . . . , n } . Let F be the family {{ 1 } , { 1 , 2 } , { 2 , 3 } , { 2 , 4 } , Family F a collection of { 1 , 2 , 5 } , { 1 , 3 , 4 } , { 2 , 4 , 5 } , subsets: F ⊆ 2 [ n ] . { 3 , 4 , 5 } , { 1 , 3 , 4 , 5 }} . k-uniform if all members have � [ n ] � size k : F ⊆ . Then { 1 , 2 , 3 } is not shattered. k S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Shattering sets Definition (Shattering) A family F shatters a set A if its members intersect A in every possible way: { F ∩ A : F ∈ F} = 2 A . Given a family F , Sh( F ) denotes the collection of sets it shatters. Definition (Families) A (small) example Ground set: [ n ] = { 1 , 2 , . . . , n } . Let F be the family {{ 1 } , { 1 , 2 } , { 2 , 3 } , { 2 , 4 } , Family F a collection of { 1 , 2 , 5 } , { 1 , 3 , 4 } , { 2 , 4 , 5 } , subsets: F ⊆ 2 [ n ] . { 3 , 4 , 5 } , { 1 , 3 , 4 , 5 }} . k-uniform if all members have � [ n ] � size k : F ⊆ . Then { 1 , 2 , 3 } is not shattered. k S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks VC dimension and the Sauer–Shelah Lemma Definition (VC dimension) The VC dimension of a family F , denoted dim VC ( F ), is the size of the largest set it shatters: dim VC ( F ) = max {| A | : A ∈ Sh( F ) } . Corollary Lemma (Sauer, Shelah (1972)) For any set family F ⊆ 2 [ n ] , If dim VC ( F ) < k, then k − 1 | Sh ( F ) | ≥ |F| , � n � � |F| ≤ . i and the bound is best possible. i =0 Many applications: computational geometry, machine learning, ... S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks VC dimension and the Sauer–Shelah Lemma Definition (VC dimension) The VC dimension of a family F , denoted dim VC ( F ), is the size of the largest set it shatters: dim VC ( F ) = max {| A | : A ∈ Sh( F ) } . Corollary Lemma (Sauer, Shelah (1972)) For any set family F ⊆ 2 [ n ] , If dim VC ( F ) < k, then k − 1 | Sh ( F ) | ≥ |F| , � n � � |F| ≤ . i and the bound is best possible. i =0 Many applications: computational geometry, machine learning, ... S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks VC dimension and the Sauer–Shelah Lemma Definition (VC dimension) The VC dimension of a family F , denoted dim VC ( F ), is the size of the largest set it shatters: dim VC ( F ) = max {| A | : A ∈ Sh( F ) } . Corollary Lemma (Sauer, Shelah (1972)) For any set family F ⊆ 2 [ n ] , If dim VC ( F ) < k, then k − 1 | Sh ( F ) | ≥ |F| , � n � � |F| ≤ . i and the bound is best possible. i =0 Many applications: computational geometry, machine learning, ... S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks VC dimension and the Sauer–Shelah Lemma Definition (VC dimension) The VC dimension of a family F , denoted dim VC ( F ), is the size of the largest set it shatters: dim VC ( F ) = max {| A | : A ∈ Sh( F ) } . Corollary Lemma (Sauer, Shelah (1972)) For any set family F ⊆ 2 [ n ] , If dim VC ( F ) < k, then k − 1 | Sh ( F ) | ≥ |F| , � n � � |F| ≤ . i and the bound is best possible. i =0 Many applications: computational geometry, machine learning, ... S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximising the number of shattered sets Question Which families of m sets maximise the number of shattered sets? Why maximise? 1. Machines are dangerously smart 2. An old problem with a lot of history 3. Quite a bit of fun S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximising the number of shattered sets Question Which families of m sets maximise the number of shattered sets? Why maximise? 1. Machines are dangerously smart 2. An old problem with a lot of history 3. Quite a bit of fun S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximising the number of shattered sets Question Which families of m sets maximise the number of shattered sets? Why maximise? 1. Machines are dangerously smart 2. An old problem with a lot of history 3. Quite a bit of fun S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximising the number of shattered sets Question Which families of m sets maximise the number of shattered sets? Why maximise? A rose by any other name... 1. Machines are ◮ Universal sets, dangerously smart k -independent sets, covering arrays, k -faulty systems 2. An old problem with ◮ Used for software testing, a lot of history derandomisation 3. Quite a bit of fun S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximising the number of shattered sets Question Which families of m sets maximise the number of shattered sets? Why maximise? A rose by any other name... 1. Machines are ◮ Universal sets, dangerously smart k -independent sets, covering arrays, k -faulty systems 2. An old problem with ◮ Used for software testing, a lot of history derandomisation 3. Quite a bit of fun S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximally destructive animals A bull in a china shop S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximally destructive animals Ein Elefant im Porzellanladen S. Das FU Berlin
Introducing the problem Some initial bounds An exact solution Concluding remarks Maximally destructive animals Ein Elefant im Porzellanladen S. Das FU Berlin
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