Seven Things Everyone Should Know about Gomory

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Seven Things Everyone Should Know about Gomorys
Group Problem

• Ellis Johnson
• ISyE, Georgia Tech
• MIP 2008
• Columbia University
• August 4-7, 2008

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Topics

1. The asymptotic theorem
2. The p-nary group problem
3. The generality of the subadditive
   characterization of facets
4. The subadditive dual problem
5. Subadditive functions on the unit interval
6. Periodic subadditive functions on Rm with
   directional derivatives
7. LP problem with multiple rhs

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1. The Asymptotic Theorem

• Basically says that the group relaxation
• NtN b (mod B), tN gt 0 and integer
• minimize cNtN
• becomes optimum to the pure integer IP
  whenever b is far enough interior to the cone
  where b is optimum
• For non-degenerate basic solution becomes true if
  b is scaled up, i.e. multiplied by a large real
  scalar
• Right way to think of changing LP soln
• Not rounding basic variables

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1. The Asymptotic Theorem2

• Originated for knapsack problem during study of
  cutting stock problem
• a1t1 a2t2 antn s b
• Cyclic group relaxation
• a2t2 antn s b (mod a1)
• solves IP if b gt a1max1, a2, ,an
• This observation was first step to group problem
  development by Gomory
• Knapsack problem with fixed a1, , an and b
  growing is polynomially solvable

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1. The Asymptotic Theorem3

• Why Should You Know
• Knapsack case is the origin of the group problem
• Gives a sufficient condition for group problem to
  solve IP
• Focuses on changing non-basic variables in LP and
  not on rounding
• I gave knapsack case for optimization
  comprehensive exam and the students were clueless

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2. The p-nary Group Problem

• At b0 (mod p) for p a prime
• t gt 0 and integer
• ct z (min)
• Forget the objective for now we are interested
  in characterizing facets
• Can use p-nary arithmetic to bring to
• ItB NtN b (mod p), tj gt 0, integer
• Cannot update objective because it is real
  arithmetic, not mod p

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2. The p-nary Group Problem2

• The Blocking Group Problem
• Original ItB NtN b (mod p)
• Blocking -NTtB ItN 0 (mod p)
• -bTtB 1 (mod
  p)
• where tj gt 0, integer in both problems
• Two augmented matrices
• M I N b
• M -NT I 0
• -bT 0 1

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2. The p-nary Group Problem3

• Rows of M and M generate 2 dual row modules
• R Qp-1 p-1 R Qp-1 p-1
• 
  
• Q1 1 Q1
  1
• Q0 0 Q0
  0
• The rows of Qp-1 are solutions to Mtb and
  include all vertices of P(M,b)
• Gomory showed that for the binary and ternary
  cases, vertices are irreducible

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2. The p-nary Group Problem4

• The rows of R give valid inequalities
• Qkt k, for k 1,,p-1, to Mtb
• but there may be other facets to P(M,b)
• Fulkerson property
• P(M,b) t 0, Qp-1t p-1
• Regular implies Fulkerson property and implies
  the vertices are Qp-1
• Fulkerson property is symmetric in and is
  inherited by minors

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2. The p-nary Group Problem5

• Why Should You Know
• Applies to sub-problems not master
• Generalizes blocking clutters and blocking
  polyhedra
• Provides an algebraic framework for blocking
  problem
• Blocking problem is polytope whose vertices are
  all facets when Fulkerson property holds

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3. Generality of Subadditive

• The facets of the finite, Abelian group problem,
  Sgt(g) g0 ? 0, t(g) gt 0 and integer, are the
  vertices of the polytope
• p(g) gt 0, p(g0) 1, g ? 0,
• p(g) p(g0-g) p(g0) (complementary)
• p(g) p(h) p(gh) (subadditive)
• Valid inequality
• Subadditive cone p(g) 0, p(g)p(h)p(gh)
• Minimal cant lower any and stay valid
• Extreme in SAC n Minimal

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3. Generality of Subadditive2

• Works for semi-group problems (AJ) and naturally
  leads to multi-groups in that gh and hg may be
  different
• Generalized to additive systems
• We allow sum to be empty set infeasible element
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• Framework of valid, subadditive, minimal, and
  extreme (facet) is a general and powerful tool

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4. Subadditive Dual

• The group problem t(g) 0, S gt(g) g0
• min S c(g)t(g) where c(g) 0
• The subadditive dual problem max p(g0)
• p(g) c(g)
• p(g) gt 0, p(g) p(h) p(gh)
• p(g) p(g0-g) p(g0)
• Group problem can be solved by Dikstra but
  subadditive lifting uses a better dual
• In Dikstra, dual is distance to node
• Lifting gives a solution to the above dual

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4. Subadditive Dual2
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3
2

• The group problem

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4
0
0
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1
3
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1. 2
2. 2.5
3. 3
4. 5

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4. Subadditive Dual3

• The subadditive dual problem max p(4)
• p(1) 2, p(2) 5, p(3) 3, p(4) 7,
• p(g) gt 0, p(g) p(h) p(gh)
• p(1) p(3) p(4), 2p(2) p(4)
• Group problem is polynomially solvable
• Data is addition table, costs, and rhs
• Number of vertices, facets are exponential
• The dual problem is polynomially solvable
• Could solve as LP
• Lifting is generally faster than Dikstra

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4. Subadditive Dual4
1 0 1 0 1 0 0 0 0
1 -1 -2 -1 1 0
1 -2 1 -1 0
1 -1 1 -2 0
1 1 1 1
2 5 3 7 0 0 0 0 0 z
2
2.5
3
5
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4. Subadditive Dual5

• What is the primal LP solution?
• It is related to the validity proof for
  subadditive inequalities
• S p(g)t(g) Sg?h,j p(g)t(g) p(hj)
• p(h)(t(h)-1)
  p(j)(t(j)-1)
• Leads to notion of complementary linearity
• Complementary linearity If there is an optimum
  solution using h and j, then p(hj) p(h) p(j)
  

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4. Subadditive Dual6

• Why Should You Know
• Provides a correct and interesting dual to a
  combinatorial problem, e.g. the group problem
• Links to shortest path problem
• Leads to computational methods
• Applies to set packing, covering, etc.

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5. Subadditive functions on 0,1

• Straight line fill-in
• Functions with 2 slopes are extreme for any group
  problem that includes lower break-points
• The much-loved Gomory MIC

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5. Subadditive functions 0,12

• What is the correct infinite master problem?
• I suggest it should be (countable) sequences on
  0,1 such that Srt(r) is absolutely convergent
• What is problem with finite support?
• Set of vectors is not closed
• Closure of homogenous solutions is non-negative
  orthant (x(1) 1, x(n) -1/n)
• What is the dual space?

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5. Subadditive functions 0,13

• Dual space (of absolutely convergent sequences)
  is functions that satisfy a Lipchitz condition at
  the origin ( -)
• A function in this dual space will be continuous
  and have bounded derivates
• Dont need step functions
• Extreme functions in the subadditive
  complementary cone are facets
• Conjecture They are piece-wise linear (with
  finitely many pieces?)

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5. Subadditive functions 0,14

• Why Should You Know
• Allows generating valid inequalities for any IP
• Opens up study of the infinite group problem
• Used to generate exponential number of facets
  (2-slope)
• Leads to study of m-dimensional functions

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6. Subadditive Functions on Rm

• Originated as joint work with Gomory
• Can start with grid points and subadditive values
  from eg a facet
• Repeat in periodic fashion on Rm
• For any positively-homogenous function (gauge
  function) that lies above all of the points, we
  can fill-in by setting the gauge function on
  each of the points and taking the min, but may
  not be complementary

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7. LP with multiple rhs

• Wy e S, y 0, min cy
• Facets Sµ(w)y(w) h0 are from gauge µ
• Positively homogenous µ(?v) ?µ(v) for all ? gt
  0 and veRd
• Convex which here is ? subadditive
• h0 minseSµ(s)
• Gauge functions are support functions of convex
  sets g(vS) supaeSav
• S for facets are polyhedra
• S s s w bw

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7. LP with multiple rhs2

• Simplicial facets
• For one dimensional, have continuous variables s
  and s- and the unique minimal gauge function is
  MIC
• For 2-dimensional problems, there are many
  minimal gauge functions
• One class is simplicial ie triangular contours or
  equivalently triangular support

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7. LP with multiple rhs3

• MIC
• g(vS) supaeSav where S 1/(f-1),1/f
• For 2-d problems, one way to generalize MIC is to
  use triangles

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-3/2
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7. LP with multiple rhs4

• For 2-d problems, one way to generalize MIC is to
  use triangles