1. Revisiting Integer Decomposition, Integer Rounding and Total Dual Integrality András Sebő, CNRS, Grenoble

2. Integer Decomposition (ID) V set of vectors. S  2 V 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= S 1 + … + S k, (S i  S, i=1,…k) Examples: G=(V,E), E  2 V not ID : matchings  2 E ’’ ’’ matchings in bipartite graphs independent sets of matroids E  2 V matchings  2 E ID:

3. 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 y T 1  ; y S  c, y  0 = min y T 1; … y integer Example (boring) : If b=1, and S := G=(V,E),  e  {0,1} V, (e  E), matchings : not IR matchings in bipartite graphs : IR independent sets of matroids : IR

4. 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 c:=v to get better than y :=.  : add :  opt  - opt times 0  S and apply ID.

5. TDI Ax ≤ b is TDI, if for all c  Z n, if whenever min {y T b : 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

6. INTEGER DECOMPOSITION + ‘ S=S down ’ +TDI Suppose S = S down  2 V 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  C min{ k b( c ),|V(C)| }, c  rows of A covering V.

7. Example 1: Bipartite Matchings S := matchings of a graph G=(V,E)  2 E. Kőnig’s theorem : ID property Polyhedron : conv( S ):={x  R E : x(  (v)) ≤ 1, x  0} S = S down & « k TDI » =min  C  C 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 :

8. Greene-Kleitman type theorem Example 2: Posets S := family of antichains of a poset  2 V. Dilworth’s theorem : ID property Polyhedron : conv( S ):={x  R V : x(A) ≤ 1, x  0} A antichain =min  C  C 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 :

9. Greene-Kleitman type theorem Example 3: Matroids S := family of independent sets of a matroid  2 V. Edmonds’ matroid partition : ID property Polyhedron : conv( S ):={x  R V : x(U) ≤ r(U), x  0} + S = S down & « k TDI » =min  C  C 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 :

10. 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  Z n : 1+  {min yb : yA  c, y  0 }   min yb : yA  c, y  0, y integer 1 BIGGER ERROR

11. Reformulations to cones Hilbert basis (Hb): v 1, …, v n is a Hilbert basis if any x  cone(v 1, …, v n )  Z n 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

12. Example 6 : bin packing Conjecture 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 intersection Conjecture of Aharoni and Berger (pers. comm): M 1 =(S, F 1 ), M 2 =(S, F 2 ), S covered by k of F i (i=1,2). Then it can be covered by k+1 of F 1  F 2.

13. 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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