MEP123: Master Equality Polyhedron with one, two or three rows Oktay G¨ unl¨ uk Mathematical Sciences Department IBM Research January, 2009 joint work with Sanjeeb Dash and Ricardo Fukasawa 1/17
Master Equality polyhedron Let n, r ∈ Z and n ≥ r > 0. MEP n � � � K 1 ( n, r ) = conv x ∈ Z 2 n +1 : ix i = r + i = − n ◮ K 1 ( n, r ) was first defined by Uchoa, Fukasawa, Lysgaard, Pessoa, Poggi de Arag˜ ao and Andrade (’06) in a slightly different form. ◮ Using simple cuts based on K 1 ( n, r ), they reduce the integrality gap for capacitated MST instances by more than 50% on average. 2/17
Master Equality polyhedron Let n, r ∈ Z and n ≥ r > 0. MEP n � � � K 1 ( n, r ) = conv x ∈ Z 2 n +1 : ix i = r + i = − n Gomory’s MCGP n − 1 � � � P 1 ( n, r ) = conv x ∈ Z n + : − nx − n + ix i = r i =1 Observation: MCGP is a lower dimensional face of MEP. 2/17
Gomory’s Master Cyclic Group Polyhedron � � � P 1 ( n, r ) = conv x ∈ Z n + : − nx − n + ix i = r i ∈ I G where I G = [1 , n − 1] ≡ { 1 , . . . , n − 1 } . Theorem (Gomory) � i ∈ I G π i x i ≥ 1 is a nontrivial facet defining inequality of P 1 ( n, r ) if and only if π is an extreme point of the following polytope : ∀ i, k ∈ I G , π i + π k ≥ π ( i + k ) mod n ∀ i, k ∈ I G , r = ( i + k ) mod n, π i + π k = π r Q = ∀ k ∈ I G , π k ≥ 0 π r = 1 . 3/17
A “Polar” description of MEP Theorem (DFG) � i ∈ I π i x i ≥ 1 is a nontrivial facet of K 1 ( n, r ) if and only if π is an extreme point of the following polyhedron : i + j ∈ I + π i + π j ≥ ∀ i, j ∈ I, π i + j , j, k, i + j + k ∈ I + π i + π j + π k ≥ π i + j + k , ∀ i ∈ I, π i + π j = ∀ i, j ∈ I, i + j = r π r , T = π r = 1 , π 0 = 0 , = 0 , π − n where I = [ − n, n ] and I + = [0 , n ]. 4/17
Some observations ◮ T and Q are not polars as they exclude trivial inequalities x ≥ 0. (they also impose ”complementarity” conditions π i + π j = π r for all i + j = r ) ◮ Their extreme points give all nontrivial facets. ◮ Q gives the convex hull of nontrivial facet coefficients (for MCGP) ◮ T gives the convex hull plus some directions (for MEP). ◮ They can be used for efficient separation via linear programming. ◮ Not all facets of MEP can be obtained by lifting facets of MCGP. 5/17
Some observations ◮ T and Q are not polars as they exclude trivial inequalities x ≥ 0. (they also impose ”complementarity” conditions π i + π j = π r for all i + j = r ) ◮ Their extreme points give all nontrivial facets. ◮ Q gives the convex hull of nontrivial facet coefficients (for MCGP) ◮ T gives the convex hull plus some directions (for MEP). ◮ They can be used for efficient separation via linear programming. ◮ Not all facets of MEP can be obtained by lifting facets of MCGP. 5/17
Some observations ◮ T and Q are not polars as they exclude trivial inequalities x ≥ 0. (they also impose ”complementarity” conditions π i + π j = π r for all i + j = r ) ◮ Their extreme points give all nontrivial facets. ◮ Q gives the convex hull of nontrivial facet coefficients (for MCGP) ◮ T gives the convex hull plus some directions (for MEP). ◮ They can be used for efficient separation via linear programming. ◮ Not all facets of MEP can be obtained by lifting facets of MCGP. 5/17
Pairwise subadditivity Regular subadditivity π i + π j ≥ π i + j ∀ i, j, i + j ∈ I = [ − n, n ] Relaxed subadditivity i + j ∈ I + = [0 , n ] π i + π j ≥ π i + j , ∀ i, j ∈ I, i + j + k ∈ I + π i + π j + π k ≥ π i + j + k , ∀ i, j, k ∈ I, ◮ Regular subadditivity ⇒ relaxed subadditivity ◮ All nontrivial facets satisfy regular subadditivity. ◮ If π satisfies either condition, then πx ≥ π r is valid for K 1 ( n, r ). ◮ Subadditivity constraints introduce additional extreme points. 6/17
Pairwise subadditivity Regular subadditivity ⇒ relaxed subadditivity: T subadditivity ⊆ T � relaxed pairwise subadditivity T = complementarity + normalization � regular pairwise subadditivity T subadditivity = complementarity + normalization T = conv.hull { π 1 , . . . , π k } + some unit directions � �� � all non-trivial facets T subadditivity = conv.hull { π 1 , . . . , π k , . . . , π t } + a smaller cone � �� � all non-trivial facets and more 7/17
Pairwise subadditivity Regular subadditivity ⇒ relaxed subadditivity: T subadditivity ⊆ T � relaxed pairwise subadditivity T = complementarity + normalization � regular pairwise subadditivity T subadditivity = complementarity + normalization T = conv.hull { π 1 , . . . , π k } + some unit directions � �� � all non-trivial facets T subadditivity = conv.hull { π 1 , . . . , π k , . . . , π t } + a smaller cone � �� � all non-trivial facets and more 7/17
Multiple rows Let n ∈ Z + , r ∈ Z m + , r � = 0 and r ≤ n 1 MEP � � � x ∈ Z | I | K m ( n, r ) = conv + : ix i = r i ∈ I where I = [ − n, n ] m . MCGP � � � x ∈ Z | I + | P m ( n, r ) = conv : ix i = r ( mod n ) + i ∈ I + where I G = [0 , n − 1] m \ { 0 } . 8/17
MCGP with multiple rows � � � x ∈ Z | I G | P m ( n, r ) = conv : ix i = r ( mod n ) + i ∈ I G where I G = [0 , n − 1] m \ { 0 } Theorem (Gomory) πx ≥ 1 is a nontrivial facet defining inequality of P m ( n, r ) if and only if π is an extreme point of the following polytope: ∀ i, k ∈ I G , π i + π k ≥ π ( i + k ) mod n ∀ i, k ∈ I G , r = ( i + k ) mod n , π i + π k = π r Q m = ∀ k ∈ I G π k ≥ 0 π r = 1 . 9/17
MEP with multiple rows � � � x ∈ Z | I | K m ( n, r ) = conv + : ix i = r i ∈ I where I = [ − n, n ] m and let I + = [0 , n ] m \ { 0 } Normalization As the dimension of K m ( n, r ) is | I | − m , any inequality πx ≥ β can be normalized so that π i = 0 for all i ∈ I N , where − n 0 0 0 − n 0 , , . . . , I N = . . . . . . . . . 0 0 − n Theorem After normalization all non-trivial facets can be written as πx ≥ 1 . 10/17
MEP with multiple rows � � � x ∈ Z | I | K m ( n, r ) = conv + : ix i = r i ∈ I where I = [ − n, n ] m and let I + = [0 , n ] m \ { 0 } Theorem Generalizing the ”non-trivial polar” T 1 for K 1 ( n, r ) � ≥ ∀ S ∈ S i ∈ S π i π S , T m = π i + π j = ∀ i, j ∈ I, i + j = r π r , π 0 = 0 , π r = 1 , π i = 0 , ∀ i ∈ I N requires large S (some | S | = O ( n ) ) if all S ∈ S satisfy � i ∈ S i ∈ I + . ◮ For MCGP, all | S | = 2; for MEP, all | S | ≤ 3. 10/17
Separation via nontrivial polars Definition A polaroid T of K m ( n, r ) is a polyhedral set such that: 1. All π ∈ T , satisfy the normalization conditions 2. If π ∈ T then πx ≥ 1 is valid for all x ∈ K m ( n, r ) 3. If πx ≥ 1 is facet-defining for K m ( n, r ), then π ∈ T . 11/17
Separation via nontrivial polars Definition A polaroid T of K m ( n, r ) is a polyhedral set such that: 1. All π ∈ T , satisfy the normalization conditions 2. If π ∈ T then πx ≥ 1 is valid for all x ∈ K m ( n, r ) 3. If πx ≥ 1 is facet-defining for K m ( n, r ), then π ∈ T . Nontrivial Polar ◮ Polaroid ⊆ Nontrivial Polar where Nontrivial Polar = { π ∈ R | I | : πx ≥ 1 for all x ∈ K m ( n, r ) } Nontrivial Polar = conv.hull { π 1 , . . . , π k } + a cone � �� � � �� � unit directions all non-trivial facets Polaroid = conv.hull { π 1 , . . . , π k , . . . , π t } + a smaller cone � �� � all non-trivial facets and more 11/17
Separation via nontrivial polars Definition A polaroid T of K m ( n, r ) is a polyhedral set such that: 1. All π ∈ T , satisfy the normalization conditions 2. If π ∈ T then πx ≥ 1 is valid for all x ∈ K m ( n, r ) 3. If πx ≥ 1 is facet-defining for K m ( n, r ), then π ∈ T . Let P denote the continuous relaxation of K m ( n, r ). Theorem Given a point x ∗ ∈ P , and a polaroid T of K m ( n, r ) . Then 1. x ∗ ∈ K m ( n, r ) can be checked by solving an LP over T , and, 2. if x ∗ �∈ K m ( n, r ) then a violated facet-defining inequality can be obtained by solving a second LP over T . 11/17
K m ( n, r ) with m = 1 , 2 T 1 is a polaroid for K 1 ( n, r ) π i + π j ≥ ∀ i, j, i + j ∈ I π i + j , T 1 = π i + π j = π r , ∀ i, j ∈ I, i + j = r π 0 = 0 , π r = 1 , π − n = 0 where I = [ − n, n ] T 2 is a polaroid for K 2 ( n, r ) π i + π j ≥ π i + j , ∀ i, j, i + j ∈ I T 2 = π i + π j = ∀ i, j ∈ I, i + j = r π r , � = π � � = 0 π 0 = 0 , π r = 1 , π � − n 0 0 − n where I = [ − n, n ] 2 12/17
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