Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Optimal assignments with supervisions Adi Niv Mathematics Department, Science Faculty Kibbutzim College of Education, Technology and the Arts (Joint work with M. Maccaig, S. Sergeev) January 24 th 2019 University of Birmingham Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity BASIC DEFINITIONS AND CONCEPTS Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Tropical linear algebra Consider real numbers R ∪ {−∞} equipped with a ⊙ b = a + b , a ⊕ b := max( a , b ) . Semifield with 0 = −∞ , 1 = 0. I.e. a − 1 = − a ∄ ⊖ a . and Applies to matrices and vectors entry-wise: ( A ⊕ B ) i , j := ( A i , j ⊕ B i , j ) � ( A ⊙ B ) i , j := A i , k ⊙ B kj k Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Jacobi identity Correspondence : I , J minor of A − 1 to J c , I c minor of A . Theorem (the classical identity) For A ∈ GL n ( F ) , I , J ⊆ [ n ] s.t. | I | = | J | = k ∧ k ∧ n − k ( DA − 1 D ) I , J = (det( A )) − 1 A Jc , Ic , where D i , i = ( − 1) i and D i , j = 0 for i � = j. (for instance) S. M. Fallat and C. R. Johnson, Totally Nonnegative Matrices. Princeton press, 2011. Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Jacobi identity Theorem (the tropical identity) n × n Let M ∈ R max and I , J ⊆ [ n ] s.t. | I | = | J | = k. Either: [ D (det( M ) − 1 adj( M )) D ] ∧ k I , J = det( M ) − 1 M ∧ n − k Jc , Ic Or: There exist distinct bijections π, σ ∈ S I , J such that [adj( M )] ∧ k � � I , J = adj( M ) i ,π ( i ) = adj( M ) i ,σ ( i ) . i ∈ I i ∈ I M. Akian, S. Gaubert and N, Tropical Compound Matrix Identities, LAA, 2018. Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity How did it form? The tropical determinant is actually the permanent with respect to ⊕ , ⊙ . That is � � � per( A ) = A i ,π ( i ) = max A i ,π ( i ) , π ∈ S n π ∈ S n i ∈ [ n ] i ∈ [ n ] Graphically: the permutation of optimal weight in the graph of A , Combinatorially: the ’optimal assignment problem’. Let π, τ be permutations of identical weight w . * In supertropical w ( π ) ⊕ w ( τ ) is sigular. * In symmetrized w ( π ) ⊕ w ( τ ) is singular if π and τ are permutations of opposite signs. Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity How did it form? 2013 - PhD (with L.Rowen) - Conjecture: Let A ∇ = per − 1 ( A )adj( A ) (sort of inverse). Then (supertropically) coefficient-wise per( A ) f A ∇ ( x ) = x n f A ( x − 1 ) ⊕ ‘ singular polynomial ′ . That is, ⊕ A ∇ I , I corresponds to ⊕ A I c , I c . [Y.Shitov ’On the Char. Polynomial of a Supertropical Adjoint Matrix’, LAA.] Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity How did it form? 2013 - PhD (with L.Rowen) - Conjecture: Let A ∇ = per − 1 ( A )adj( A ) (sort of inverse). Then (supertropically) coefficient-wise per( A ) f A ∇ ( x ) = x n f A ( x − 1 ) ⊕ ‘ singular polynomial ′ . That is, ⊕ A ∇ I , I corresponds to ⊕ A I c , I c . [Y.Shitov ’On the Char. Polynomial of a Supertropical Adjoint Matrix’, LAA.] 2015 - Postdoc (with M.Akian and S.Gaubert) - (symmetrized) Tropical Jacobi: [ D (det( M ) − 1 adj( M )) D ] ∧ k I , J = det( M ) − 1 M ∧ n − k ⊕ ‘ singular matrix ′ . Jc , Ic So, entry-wise , for every I , J , and including signs . Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity How did it form? 2013 - PhD (with L.Rowen) - Conjecture: Let A ∇ = per − 1 ( A )adj( A ) (sort of inverse). Then (supertropically) coefficient-wise per( A ) f A ∇ ( x ) = x n f A ( x − 1 ) ⊕ ‘ singular polynomial ′ . That is, ⊕ A ∇ I , I corresponds to ⊕ A I c , I c . [Y.Shitov ’On the Char. Polynomial of a Supertropical Adjoint Matrix’, LAA.] 2015 - Postdoc (with M.Akian and S.Gaubert) - (symmetrized) Tropical Jacobi: [ D (det( M ) − 1 adj( M )) D ] ∧ k I , J = det( M ) − 1 M ∧ n − k ⊕ ‘ singular matrix ′ . Jc , Ic So, entry-wise , for every I , J , and including signs . 2016-2018 (with McCaig and Sergeev) - Graph theory version: Every optimal (1 , k )-regular multigraph of M w.r.t. I , J either: corresponds to an optimal bijection w.r.t. I c , J c , or: there exists another optimal (1 , k )-regular w.r.t. I , J . [That is, combinatorially, without signs, which led to the application.] Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Definitions: digraphs A weighted digraph G is a pair ( V G , E G ) where V G is set of nodes and E G ⊆ V G × V G is set of directed edges on | V G | nodes (allowing loops and multiple edges). Weight: w ( i , j ) for each ( i , j ). A bipartite graph is a triple ( V H , 1 , V H , 2 , E H ) s.t. i ∈ V H , 1 ⇔ j ∈ V H , 2 for every ( i , j ) ∈ E H , weighted: w ( i , j ) for each ( i , j ). Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Associated digraphs Matrix M ∈ R n × n max − → weighted digraph G M = ( V , E ) , where V = [ n ] and E = { ( i , j ): M i , j � = 0 } , and weight w ( i , j ) = M i , j . Weighted digraph G = ([ n ] , E , w ) − → matrix M G , � w ( i , j ) ; if ( i , j ) ∈ E , where ( M G ) i , j = 0 ; otherwise . Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Digraphs and matrices M 1 , 1 M 1 , 2 M 1 , 1 M 1 , 2 M 1 , 3 1 2 M = M G = M 2 , 1 M 2 , 1 0 0 M 1 , 3 M 3 , 2 M 3 , 1 M 3 , 1 M 3 , 2 0 G = G M 3 Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Associated bipartite graphs Matrix M ∈ R m × n max − → bipartite graph G M = ( V H 1 , V H 2 , E H ) , | V H 1 | = m , | V H 2 | = n , and E H = { ( i , j ): M i , j � = −∞} , weight w ( i , j ) = M i , j . → matrix M G ∈ R m × n Bipartite graph G = ( V H 1 , V H 2 , E H ) − max | V H 1 | = m , | V H 2 | = n � w ( i , j ) ; if ( i , j ) ∈ E H , where ( M G ) i , j = 0 ; otherwise . Digraph DG = ([ n ] , E D ) ← → bipartite graph BG = ([2 n ] , E B ) , s.t. ( i , j + n ) ∈ E B for every ( i , j ) ∈ E D . Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Bipartite graphs and matrices 1 M 1 , 1 M 1 , 2 M 1 , 3 M = M 2 , 1 0 0 2 M 3 , 1 M 3 , 2 0 3 1 2 3 Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Definitions: assignment problems Let S n denote the set of permutations on [ n ], and S I , J denote the set of bijections from I ⊆ [ n ] to J ⊆ [ n ] (that is, | I | = | J | ). For M ∈ R n × n max tropical permanent is defined by � � � per( M ) = max M i ,π ( i ) = M i ,π ( i ) . π ∈ S n i ∈ [ n ] π ∈ S n i ∈ [ n ] A permutation π of maximal weight in per( M ) is an optimal permutation in M or G M . That is, � � per( M ) = M i ,π ( i ) = w ( i , π ( i )) . i ∈ [ n ] i ∈ [ n ] This is identical to the set of optimal assignments , i.e., optimal solutions to the assignment problem in the bipartite graph associated with M . Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Permutation subgraphs A non- 0 tropical ”summand” w ( π ) = � i ∈ [ n ] M i ,π ( i ) in per M , or in M ↔ permutation-subgraph of G M with V ( E π ) = [ n ], E π = { ( i , π ( i )) ∀ i ∈ [ n ] } . 6 1 2 5 3 4 (1 2 4)(5 3)(6) (and the same for path, cycle, bijection,...) Adi Niv Optimal assignments with supervisions
Basic definitions and concepts Optimal assignments with supervisions Tropical Jacobi identity Assignment subgraphs A non- 0 tropical ”summand” w ( π ) = � i ∈ [ n ] M i ,π ( i ) in per M ↔ assignment subgraph with V ( E π ) = [ n ] + [ n ], E π = { ( i , π ( i )) ∀ i ∈ [ n ] } . 1 2 3 (2 1 3) (and the same for path, cycle, bijection,...) Adi Niv Optimal assignments with supervisions
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