Revisiting Integer Decomposition, Integer Rounding and Total Dual Integrality

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RevisitingInteger Decomposition, Integer
Rounding and Total Dual Integrality

• András Sebo,
• CNRS, Grenoble

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Integer Decomposition (ID)

• V set of vectors. S ? 2V has the ID property,
• if v ? k conv(S) , (that is, v?S?S ?(S)?S,
  ?s?S ?(S) k)
• and v, k integer ?v S1 Sk , (Si?S, i1,k)
• Examples G(V,E), E ? 2V not ID
• matchings ? 2E
• matchings in bipartite graphs
• independent sets of matroids

ID
matchings ? 2E
E ? 2V
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Integer Rounding (IR)

• Def A system of inequalities S x 1
• 
  x ? 0
• has the IR property,if for any integer 1?n vector
  c ?min yT1 ? yS ? c, y ? 0 min yT1 y
  integer
• Example (boring) If b1, and S
• G(V,E), ?e ? 0,1V, (e ? E) , matchings not
  IR
• matchings in bipartite graphs IR
• independent sets of matroids IR

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ID and IR are always related

• Variant of Baum, Trotter (1981) S S down
• S x 1, x ? 0 has IR ? S has ID
• Proof ? v?s?S ?(S)?S, ?s?S ?(S) k
• Apply IR to cv to get better than y ?.
• ? add ?opt? - opt times 0 ? S and apply ID.

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TDI

• Ax b is TDI ,
• if for all c ?Zn, if whenever min yTb yAc
• exists it does have an integer optimum.
• Edmonds-Giles Then it has integer vertices
• S x 1, x ? 0 conv(S )x Ax
  b
• can be TDI or IR if TDI b1, if only
  IR,
• maybe noninteger b can be big

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INTEGER DECOMPOSITION SS down TDI
Suppose S S down?2V
is ID. conv(S ) x Ax b. Then k
conv(S ) x Ax kb, x ? 0 If in addition
Ax kb, 0 x 1 is TDI for all k
Edmonds type theorem

max union of k elements of S
min X k b(c) c ? rows of A covering V /
X
Greene-Kleitman type theorem
min?C?Cmin k b(c),V(C) ,c ?rows of A
covering V.
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Example 1 Bipartite Matchings

• S matchings of a graph G(V,E) ? 2E.
• Konigs theorem ID property
• Polyhedron conv(S )x?RE x(?(v)) 1, x ? 0
  
• S S down k TDI

Edmonds type theorem
max union of k elements of S
min X k c c ? stars covering E / X

Greene-Kleitman type theorem
min?C?C mink ,V(C) ,c ? stars covering V
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Example 2 Posets

• S family of antichains of a poset ? 2V.
• Dilworths theorem ID property
• Polyhedron conv(S )x?RV x(A) 1, x ? 0
• 
  A antichain

S S down k TDI Edmonds type theorem
max union of k elements of S
min X k c c ? chains covering V / X

Greene-Kleitman type theorem
min?C?C mink ,C ,c ? chains covering V
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Example 3 Matroids

• S family of independent sets of a matroid ?
  2V.
• Edmonds matroid partition ID property
• Polyhedron conv(S )x?RV x(U) r(U), x ? 0
  
• S S down k TDI

Edmonds type theorem
max union of k elements of S
min X k r(c) c ? ind. sets covering V /
X
Greene-Kleitman type theorem
min?C?C mink r(C),C ,c ? ind. sets covering
V.
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MIRUP

• Modified integer round up property (MIRUP)
• A system of inequalities Ax b (A mxn, b mx1)
• x
  ? 0
• has the MIRUP property , if for any c?Zn
• 1 ?min yb yA ? c, y ? 0 ? ?
• min yb yA ? c, y ? 0, y integer
• 1 BIGGER ERROR

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Reformulations to cones

• Hilbert basis (Hb) v1 , , vn is a Hilbert
  basis if
• any x?cone(v1 , , vn)?Zn is a nonneg. int.
  comb
• Schrijver TDI ? active rows form a Hb.
• Schrijver S IR ? is a Hb.
• Modified Hilbert basis in the def of Hb. ask
  that
• the coordinate sum of the int solution is 1 more

S 1 0 1
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Example 4 matchings in nonbip Goldberg(1973),
Andersen (1977), Seymour (1979)conjecture that
matchings have the MIRUP.
Example 5 matroid intersectionConjecture
of Aharoni and Berger (pers. comm)M1(S,F1),
M2(S,F2), S covered by k of Fi (i1,2).Then it
can be covered by k1 of F1? F2 .
Example 6 bin packingConjecture of Marcotte,
Sheithauer, Terno MIRUP
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Conclusion

• - ID (IR) combined with TDI , and IR 1 have
  combinatorial meanings.
• Stable sets of posets are an example.
  Generalizations ?
• To stable sets, paths, circuits Leads to
  proofs for graph theory thms and relating some
  conj (of Berge and Linial on path partitions).
• Do the solutions of the bin packing problem have
  the MIRUP property ?
• A method and some answer

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