MEP123: Master Equality Polyhedron with one, two or three rows

Transcription

1 1/17 MEP123: Master Equality Polyhedron with one, two or three rows Oktay Günlük Mathematical Sciences Department IBM Research January, 29 joint work with Sanjeeb Dash and Ricardo Fukasawa

2 2/17 Master Equality polyhedron Let n, r Z and n r >. MEP { K 1 (n, r) = conv x Z 2n+1 + : n i= } ix i = r K 1 (n, r) was first defined by Uchoa, Fukasawa, Lysgaard, Pessoa, Poggi de Aragão and Andrade ( 6) 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 5% on average.

3 2/17 Master Equality polyhedron Let n, r Z and n r >. MEP { K 1 (n, r) = conv x Z 2n+1 + : n i= } ix i = r Gomory s MCGP { n 1 } P 1 (n, r) = conv x Z n + : x + ix i = r i=1 Observation: MCGP is a lower dimensional face of MEP.

4 3/17 Gomory s Master Cyclic Group Polyhedron P 1 (n, r) = conv {x Z n + : x + } ix i = r i I G where I G = [1, n 1] {1,..., n 1}. Theorem (Gomory) i I G π ix 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: π i + π k π (i+k) mod i, k I n G, Q = π i + π k = π r i, k I G, r = (i + k) mod n, π k k I G, π r = 1.

5 4/17 A Polar description of MEP Theorem (DFG) i I π ix i 1 is a nontrivial facet of K 1 (n, r) if and only if π is an extreme point of the following polyhedron: T = π i + π j π i+j, i, j I, i + j I + π i + π j + π k π i+j+k, i I, j, k, i + j + k I + π i + π j = π r, i, j I, i + j = r π r = 1, π =, π =, where I = [, n] and I + = [, n].

6 5/17 Some observations T and Q are not polars as they exclude trivial inequalities x. (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.

7 5/17 Some observations T and Q are not polars as they exclude trivial inequalities x. (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.

8 5/17 Some observations T and Q are not polars as they exclude trivial inequalities x. (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.

9 6/17 Pairwise subadditivity Regular subadditivity π i + π j π i+j i, j, i + j I = [, n] Relaxed subadditivity π i + π j π i+j, i, j I, i + j I + = [, n] π i + π j + π k π i+j+k, i, j, k I, 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.

10 7/17 Regular subadditivity relaxed subadditivity: T subadditivity T Pairwise subadditivity T = { relaxed pairwise subadditivity complementarity + normalization T subadditivity = { regular pairwise subadditivity complementarity + normalization T subadditivity T = conv.hull{π 1,..., π }{{ k } } + some unit directions all non-trivial facets = conv.hull{π 1,..., π k,..., π }{{} t } + a smaller cone all non-trivial facets and more

11 7/17 Regular subadditivity relaxed subadditivity: T subadditivity T Pairwise subadditivity T = { relaxed pairwise subadditivity complementarity + normalization T subadditivity = { regular pairwise subadditivity complementarity + normalization T subadditivity T = conv.hull{π 1,..., π }{{ k } } + some unit directions all non-trivial facets = conv.hull{π 1,..., π k,..., π }{{} t } + a smaller cone all non-trivial facets and more

12 8/17 Multiple rows Let n Z +, r Z m +, r and r n1 MEP { K m (n, r) = conv x Z I + : } ix i = r i I where I = [, n] m. MCGP { P m (n, r) = conv x Z I+ + : } ix i = r (mod n) i I + where I G = [, n 1] m \ {}.

13 9/17 MCGP with multiple rows { P m (n, r) = conv x Z IG + : } ix i = r (mod n) i I G where I G = [, n 1] m \ {} 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: π i + π k π (i+k) mod i, k I G, n Q m π = i + π k = π r i, k I G, r = (i + k) mod n, π k k I G π r = 1.

14 1/17 MEP with multiple rows { K m (n, r) = conv x Z I + : } ix i = r i I where I = [, n] m and let I + = [, n] m \ {} Normalization As the dimension of K m (n, r) is I m, any inequality πx β can be normalized so that π i = for all i I N, where I N =.,.,...,. Theorem. After normalization all non-trivial facets can be written as πx 1...

15 1/17 MEP with multiple rows { K m (n, r) = conv x Z I + : } ix i = r i I where I = [, n] m and let I + = [, n] m \ {} Theorem Generalizing the non-trivial polar T 1 for K 1 (n, r) i S π i π S, S S T m = π i + π j = π r, i, j I, i + j = r π =, π r = 1, π i =, 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.

16 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.

17 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 {{} all non-trivial facets unit directions Polaroid = conv.hull{π 1,..., π k,..., π }{{} t } + a smaller cone all non-trivial facets and more

18 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.

19 12/17 K m (n, r) with m = 1, 2 T 1 is a polaroid for K 1 (n, r) π i + π j π i+j, i, j, i + j I T 1 = π i + π j = π r, i, j I, i + j = r π =, π r = 1, π = where I = [, 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 = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = where I = [, n] 2

20 12/17 K m (n, r) with m = 1, 2 T 1 is a polaroid for K 1 (n, r) π i + π j π i+j, i, j, i + j I T 1 = π i + π j = π r, i, j I, i + j = r π =, π r = 1, π = where I = [, 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 = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = where I = [, n] 2

21 13/17 K m (n, r) with m = 3 Ta 3 is NOT a polaroid for K 3 (n, r) π i + π j π i+j, i, j, i + j I Ta 3 π = i + π j = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = π [ ] = where I = [, n] 3 T 3 b is NOT a polaroid for K3 (n, r) π i + π j + π k π i+j+k, i, j, k, i + j + k I Tb 3 π = i + π j = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = π [ ] =

22 13/17 K m (n, r) with m = 3 Ta 3 is NOT a polaroid for K 3 (n, r) π i + π j π i+j, i, j, i + j I Ta 3 π = i + π j = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = π [ ] = where I = [, n] 3 T 3 b is NOT a polaroid for K3 (n, r) π i + π j + π k π i+j+k, i, j, k, i + j + k I Tb 3 π = i + π j = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = π [ ] =

23 14/17 K 3 (1, 2) Let a = b = c = and consider the point x and the inequality πx 1 where xa = x b = 1, x c = 2 and all other x i = πa = π b = π c = and all other π i = 1 (including π r) Example Note that, i I i x i = a + b + 2c = 2 and x K 3 (1, 2). Also, π satisfies all 2 and 3-term subadditivity conditions: πi + π j π i+j for all i, j, i + j I, πi + π j + π k π i+j+k for all i, j, k, i + j + k I, And yet, π x = 1!

24 15/17 In General Consider K m (n, r) and let I = [, n] m. k-term subadditivity We say that π R I satisfies k-term subadditivity if π i π S i S for all S I such that (i) S k and (ii) i I Validity via subadditivity It is possible to construct invalid cuts πx 1 for K m (n, r) where π satisfies the normalization conditions and k-subadditivity unless i S k max{2, 3 2 m 3 + 1} (for m 1, the lower bound is: 2, 2, 4, 7, 13, 25,...)

25 15/17 In General Consider K m (n, r) and let I = [, n] m. k-term subadditivity We say that π R I satisfies k-term subadditivity if π i π S i S for all S I such that (i) S k and (ii) i I Validity via subadditivity It is possible to construct invalid cuts πx 1 for K m (n, r) where π satisfies the normalization conditions and k-subadditivity unless i S k max{2, 3 2 m 3 + 1} (for m 1, the lower bound is: 2, 2, 4, 7, 13, 25,...)

26 16/17 K 3 (n, r) Theorem If π satisfies 4-term subadditivity, then πx 1 is valid for K 3 (n, r). T 3 is a polaroid for K 3 (n, r) π i + π j + π k + π l π i+j+k+l, i, j, k, l, i + j + k + l I T 3 π = i + π j = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = π [ ] = where I = [, n] 3

27 16/17 K 3 (n, r) Theorem If π satisfies 4-term subadditivity, then πx 1 is valid for K 3 (n, r). T 3 is a polaroid for K 3 (n, r) π i + π j + π k + π l π i+j+k+l, i, j, k, l, i + j + k + l I T 3 π = i + π j = π r, i, j I, i + j = r π =, π r = 1, π [ ] = π [ ] = π [ ] = where I = [, n] 3

28 Thank you... 17/17