Topics in Combinatorial Optimization Orlando Lee Unicamp 18 de - PowerPoint PPT Presentation

Topics in Combinatorial Optimization Orlando Lee – Unicamp 18 de junho de 2014 Orlando Lee – Unicamp Topics in Combinatorial Optimization

Agradecimentos Este conjunto de slides foram preparados originalmente para o curso T´ opicos de Otimiza¸ c˜ ao Combinat´ oria no primeiro semestre de 2014 no Instituto de Computa¸ c˜ ao da Unicamp. Preparei os slides em inglˆ es simplesmente porque me deu vontade, mas as aulas ser˜ ao em portuguˆ es (do Brasil)! Agradecimentos especiais ao Prof. M´ ario Leston Rey. Sem sua ajuda, certamente estes slides nunca ficariam prontos a tempo. Qualquer erro encontrado nestes slide ´ e de minha inteira responsabilidade (Orlando Lee, 2014). Orlando Lee – Unicamp Topics in Combinatorial Optimization

Edmonds-Giles theorem We will describe a general framework involving digraphs and submodular functions that generalizes several combinatorial results. This framework includes MaxFlow-MinCut theorem, Lucchesi-Yonger theorem, minimum cost orientations of graphs and weighted matroid intersection theorem. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Crossing families Let C be a collection of subsets of a ground set V . A pair X , Y of subsets of V is crossing or cross if X − Y � = ∅ , Y − X � = ∅ , X ∩ Y � = ∅ and X ∪ Y � = ∅ . We say that C is crossing if X , Y ∈ C , X ∩ Y � = ∅ , X ∪ Y � = ∅ ⇒ X ∩ Y , X ∪ Y ∈ C . This is equivalent to require that X , Y ∈ C , X , Y cross ⇒ X ∩ Y , X ∪ Y ∈ C . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Crossing families Some simple examples of crossing families are: 2 V , 2 V − {∅ , V } , {{ v } : v ∈ V } or any collection of disjoint sets. A more interesting and important example is the following. Let D = ( V , A ) be a digraph. Then { X : X ⊆ V , r �∈ X } = 2 V − r for some r ∈ V and { X : X ⊆ 2 V − {∅ , V } , d in ( X ) = 0 } are a crossing families. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Crossing submodular functions Let C be a crossing familly on V . A function b : C �→ R is called submodular on crossing pairs or crossing submodular if b ( X ) + b ( Y ) � b ( X ∩ Y ) + b ( X ∪ Y ) for every X , Y ∈ C with X ∩ Y � = ∅ and X ∪ Y � = ∅ . Note that this is equivalent to require submodular on crossing pairs of C . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Submodular flows Let D = ( V , A ) be a digraph, let C be a crossing family on V and let b : C �→ R be a crossing submodular function. A submodular flow is a vector (function) x ∈ R A such that x ( δ in ( U )) − x ( δ out ( U )) � b ( U ) for each U ∈ C . ( ∗ ) The set of vectors in R A that satisfy ( ∗ ) is called submodular flow polyhedron. We will show that if b is integral then this polyhedron is integral. More precisely, we will show that the system above is TDI. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Incidence matrix of a family in a digraph Let D = ( V , A ) be a digraph and let C be a family of subsets of V . Let N be the C × A -matrix defined by:  1 if a enters X ,  N X , a = − 1 if a leaves X , 0 otherwise,  for each X ∈ C and each a ∈ A . We say that N is the incidence matrix of D , C . So x ∈ R A is a submodular flow if satisfies Nx � b . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Cross-free families A family C ⊆ 2 V is cross-free if for all X , Y ∈ C we have that X ⊆ Y or Y ⊆ X or X ∩ Y = ∅ or X ∪ Y = V , that is, no distinct members of C cross. The family C is laminar if for all X , Y ∈ C we have that X ⊆ Y or Y ⊆ X or X ∩ Y = ∅ , that is, no distinct members of C intersect. So a laminar family is also a cross-free family. Two members of a cross-free family may intersect as long as their union is V . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Cross-free families Theorem. Let D = ( V , A ) be a digraph and let C a cross-free family on V . Then N , the incidence matrix of D , C , is totally unimodular. Idea of the proof (see Schrijver’s book, Vol. A, p. 213-216) (a) Prove that certain matrices, known as network matrices, are totally unimodular. (b) Prove that if C is cross-free, then N is a network matrix, and hence N is totally unimodular. I have included the proofs in these slides but have not discussed in class. Orlando Lee – Unicamp Topics in Combinatorial Optimization

LP formulation Consider the following linear programming (LP). max � a ∈ A c ( a ) x ( a ) x ( δ in ( X )) − x ( δ out ( X )) � b ( X ) s.t. for every X ∈ C , l ( a ) � x ( a ) � u ( a ) for every a ∈ A . This is equivalent to: max cx s.t. Nx � b l � x � u Orlando Lee – Unicamp Topics in Combinatorial Optimization

Dual problem The dual problem (DP) is the following. � � � y X b ( X ) − min u ( a ) w ( a ) + l ( a ) z ( a ) X ∈C a ∈ A a ∈ A � � s.t. y X − y X − w ( a ) + z ( a ) = c ( a ) a ∈ A , X ∈C : a ∈ δ in ( X ) X ∈C : a ∈ δ out ( X ) X ∈ C , y X � 0 w ( a ) , z ( a ) � 0 a ∈ A . Or equivalently, min yb − 1 w + 1 z s.t yN − wI + zI = c y , w , z � 0 Orlando Lee – Unicamp Topics in Combinatorial Optimization

Some applications Let us describe how to model some well-known combinatorial optimization problems as a submodular flow problem. (a) Minimum Cost Circulation. Set C := {{ v } : v ∈ V } and b = 0 . (b) Minimum Cost Dijoin (Lucchesi-Younger). Set C := { X ⊆ V − {∅ , V } : d in ( X ) = 0 } and b = − 1 . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Some applications (c) Minimum Cost k -Arc-Connected Orientation. Suppose we are given a digraph D = ( V , A ) and a cost function c : A �→ R + . The value c ( a ) represents the cost of reversing the orientation of a . Suppose we have a target arc-connectivity k , that is, we want to reorient D so that the resulting digraph is k -arc-connected (that is, d out ( X ) � k for every X ⊆ 2 V − {∅ , V } ). Set C := 2 V − {∅ , V } b ( X ) := d in ( X ) − k . and You can check that a submodular flow for this pair C , b corresponds to a k -arc-connected reorientation. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Some applications (d) Weighted matroid intersection theorem. Let M 1 := ( E , r 1 ) and M 2 := ( E , r 2 ) be two matroids on the same ground set given by their respective independence oracles, and let c : E �→ R be a cost function. We want to find a common independent set I of both matroids which maximizes c ( I ). Note that if I is a common independent then χ I must be a feasible solution of the following LP: x ( x ( U )) � r 1 ( U ) for every U ⊆ E , for every U ⊆ E , x ( x ( U )) � r 2 ( U ) x ( e ) � 0 for every e ∈ E . Actually, this defines the polyhedron of the intersection of two matroids. We derive this from Edmonds-Giles submodular flow theeorem. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Some applications Construct two copies E ′ and E ′′ of the ground set E . For an element u ∈ E , let u ′ and u ′′ denote the corresponding element in the respective copy. Analogously, define U ′ and U ′′ for any U ⊆ E . Let D be a digraph with vertex set E ′ ∪ E ′′ and arc-set { ( u ′′ , u ′ ) : u ∈ E } , that is, an arc connects corresponding elements and goes from E ′′ to E ′ . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Some applications Let C := { U ′ : U ⊆ E } ∪ { E ′ ∪ U ′′ : U ⊆ E } . This is a crossing family. Now define b : C �→ Z + as follows: b ( U ′ ) = r 1 ( U ) for U ⊆ E , U � = E b ( E ′ ∪ U ′′ ) = r 2 ( U ) for U ⊆ E , U � = ∅ , b ( E ′ ) = min { r 1 ( E ) , r 2 ( E ) } . In our model we let x ( u ′′ , u ′ ) = 1 meaning that we choose u to be in I , our desired common independent set. You can check that a { 0 , 1 } -vector x is the incidence vector of a common independent set if and only if belongs to the submodular flow polyhedron defined above. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Submodular flow polyhedron Theorem. (Edmonds-Giles, 1977) Let D = ( V , A ) be a digraph, let C be a crossing family, let b a crossing submodular function defined on C , let c : A �→ Z be an arc cost function, and let l , u ∈ Z A arc capacities. Then the dual LP problem (DP) has an optimum solution (if it exists) which is integral. Proof. Suppose the dual has an optimum solution. Let y be an optimum solution of the dual which minimizes � y X | X || X | , X ∈C where X := V \ X . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Submodular flow polyhedron Let C > 0 := y + = { X ∈ C : y X > 0 } . We will show that C > 0 is cross-free. We need the following result. Theorem. If X , Y are subsets of V such that X �⊆ Y and Y ⊆ X , then | X || X | + | Y || Y | > | X ∩ Y || X ∩ Y | + | X ∪ Y || X ∪ Y | . Proof. See Theorem 2.1 in Schrijver’s book. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Submodular flow polyhedron Suppose for a contradiction that C > 0 is not cross-free. Let X , Y two members of C > 0 which cross. Let α := min { y X , y Y } > 0. Define y ′ on C by:  y S − α if S = X or S = Y ,  y ′ if S = X ∩ Y or S = X ∪ Y , S := y S + α othewise. y S  Then y ′ is a feasible dual solution (Exercise). Orlando Lee – Unicamp Topics in Combinatorial Optimization