Revisiting

Integer Decomposition, Integer Rounding and Total Dual Integrality

András Sebő, CNRS, Grenoble

Integer Decomposition (ID)

V set of vectors. S 2V has the ID property,if v k conv(S) , (that is, v=SS (S)S, sS (S) =k)

and v, k integer v= S1 + … + Sk , (SiS, i=1,…k)

Examples: G=(V,E), E 2V not ID : matchings 2E ’’ ’’

matchings in bipartite graphsindependent sets of matroids

E 2V

matchings 2E

ID:

Integer Rounding (IR)

Def: A system of inequalities S x ≤1 x 0 has the IR property,if for any integer 1n vector c :

min yT1 ; yS c, y 0 = min yT1; … y integer

Example (boring) : If b=1, and S := G=(V,E), e {0,1}V, (e E) , matchings : not IRmatchings in bipartite graphs : IRindependent sets of matroids : IR

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=sS (S)S, sS (S) =k

Apply IR to c:=v to get better than y := .

: add : opt - opt times 0 S and apply ID.

TDIAx ≤ b is TDI ,if for all c Zn, if whenever min {yTb : yA=c}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: b=1, if only IR,maybe noninteger b can be big

INTEGER DECOMPOSITION + ‘S=S down ’ +TDI

Suppose S = S down2V is ID.

conv(S ) =:{x : Ax ≤ b}. Thenk 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 =minCCmin{ k b(c),|V(C)| },c rows of A covering V.

Example 1: Bipartite MatchingsS := matchings of a graph G=(V,E) 2E.Kőnig’s theorem : ID property

Polyhedron : conv(S ):={xRE: x((v)) ≤ 1, x 0} S = S down & « k TDI »

=minCC min{{k ,|V(C)| },c stars covering V} Greene-Kleitman type theorem

=min{ |X| + k |c | : c stars covering E / X }max union of k elements of S = Edmonds type theorem :

Greene-Kleitman type theorem

Example 2: PosetsS := family of antichains of a poset 2V.Dilworth’s theorem : ID property

Polyhedron : conv(S ):={xRV: x(A) ≤ 1, x 0} A antichain

=minCC min{{k ,|C| },c chains covering V }

=min{ |X| + k |c | : c chains covering V / X }max union of k elements of S =

+ S = S down & « k TDI »Edmonds type theorem :

Greene-Kleitman type theorem

Example 3: MatroidsS := family of independent sets of a matroid 2V.Edmonds’ matroid partition : ID property

Polyhedron : conv(S ):={xRV: x(U) ≤ r(U), x 0} + S = S down & « k TDI »

=minCC min{k r(C),|C| },c ind. sets covering V.

=min{ |X| + k r(c) : c ind. sets covering V / X }

max union of k elements of S = Edmonds type theorem :

MIRUPModified integer round up property (MIRUP):

A system of inequalities Ax ≤ b (A: mxn, b: mx1) x 0 has the MIRUP property , if for any cZn :1+ {min yb : yA c, y 0 } min yb : yA c, y 0, y integer

1 BIGGER ERROR

Reformulations to cones

Hilbert basis (Hb): v1 , … , vn is a Hilbert basis if

any xcone(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 thatthe coordinate sum of the int solution is ≤ 1 more

S 10 1

Example 6 : bin packingConjecture of Marcotte, Sheithauer, Terno: MIRUP

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 (i=1,2).Then it can be covered by k+1 of F1 F2 .

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 …

SEE YOU ON WEDNESDAY

להתראות

A mercredi

до среды

Szerdán találkozunk