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  1. Seven Things Everyone Should Know about Gomory’s Group Problem Ellis Johnson ISyE, Georgia Tech MIP 2008 Columbia University August 4-7, 2008

  2. Topics • The asymptotic theorem • The p-nary group problem • The generality of the subadditive characterization of facets • The subadditive dual problem • Subadditive functions on the unit interval • Periodic subadditive functions on Rm with directional derivatives • LP problem with multiple rhs

  3. 1. The Asymptotic Theorem • Basically says that the group relaxation NtN≡ b (mod B), tN> 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 sol’n • Not rounding basic variables

  4. 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 > a1*max{1, 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

  5. 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

  6. 2. The p-nary Group Problem • At ≡ b0 (mod p) for p a prime t > 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> 0, integer • Cannot update objective because it is real arithmetic, not mod p

  7. 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> 0, integer in both problems • Two augmented matrices M = [ I N | b ] M* = [-NTI | 0 ] [-bT 0 | 1]

  8. 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 Mt*=b and include all vertices of P(M,b) • Gomory showed that for the binary and ternary cases, vertices are irreducible

  9. 2. The p-nary Group Problem4 • The rows of R give valid inequalities Qkt* ≥ k, for k = 1,…,p-1, to Mt*=b 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

  10. 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

  11. 3. Generality of Subadditive • The facets of the finite, Abelian group problem, Sgt(g) = g0 ≠ 0, t(g) > 0 and integer, are the vertices of the polytope p(g) > 0, p(g0) = 1, g ≠ 0, p(g) + p(g0-g) = p(g0) (complementary) p(g) + p(h) ≥p(g+h) (subadditive) • Valid inequality • Subadditive cone: p(g) ≥0, p(g)+p(h)≥p(g+h) • Minimal: can’t lower any and stay valid • Extreme in SAC ∩ Minimal

  12. 3. Generality of Subadditive2 • Works for semi-group problems (A&J) and naturally leads to multi-groups in that g+h and h+g may be different • Generalized to additive systems • We allow sum to be empty set = infeasible element ∞ • Framework of valid, subadditive, minimal, and extreme (facet) is a general and powerful tool

  13. 4. Subadditive Dual • The group problem: t(g) ≥ 0, Σ gt(g) = g0 min Σ c(g)t(g) where c(g) ≥ 0 • The subadditive dual problem: max p(g0) p(g) ≤ c(g) • p(g) > 0, p(g) + p(h) ≥p(g+h) 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

  14. 4. Subadditive Dual2 2 3 3 • The group problem 1 2 4 4 5 7 2 5 4 0 0 3 1 3 2 x x • 2 • 2.5 • 3 • 5 x x x x x

  15. 4. Subadditive Dual3 • The subadditive dual problem: max p(4) p(1) ≤ 2, p(2) ≤ 5, p(3) ≤ 3, p(4) ≤ 7, • p(g) > 0, p(g) + p(h) ≥p(g+h) 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

  16. 4. Subadditive Dual4

  17. 4. Subadditive Dual5 • What is the primal LP solution? • It is related to the validity proof for subadditive inequalities Σp(g)t(g) ≥ Σg≠h,jp(g)t(g) + p(h+j) + 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(h+j) = p(h) + p(j)

  18. 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.

  19. 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

  20. 5. Subadditive functions [0,1]2 • What is the correct infinite master problem? • I suggest it should be (countable) sequences on [0,1] such that Σrt(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?

  21. 5. Subadditive functions [0,1]3 • 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 • Don’t need step functions • Extreme functions in the subadditive complementary cone are facets • Conjecture: They are piece-wise linear (with finitely many pieces?)

  22. 5. Subadditive functions [0,1]4 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

  23. 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

  24. 7. LP with multiple rhs • Wy ε S, y ≥ 0, min cy • Facets Σμ(w)y(w) ≥ h0 are from gauge μ • Positively homogenous: μ(λv) = λμ(v) for all λ > 0 and vεRd • Convex which here is ↔ subadditive • h0 = minsεS{μ(s)} • Gauge functions are support functions of convex sets g(v|S*) = supa*εS*{a*∙v} • S* for facets are polyhedra S* = {s* | s* ∙w ≤ bw }

  25. 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

  26. 7. LP with multiple rhs3 • MIC g(v|S*) = supa*εS*{a*∙v} where S* = [1/(f-1),1/f] • For 2-d problems, one way to generalize MIC is to use triangles 1 -3/2 3

  27. 7. LP with multiple rhs4 • For 2-d problems, one way to generalize MIC is to use triangles