IPO Investigating Polyhedra by Oracles Matthias Walter - PowerPoint PPT Presentation

IPO – Investigating Polyhedra by Oracles Matthias Walter Otto-von-Guericke Universit¨ at Magdeburg Joint work with Volker Kaibel (OVGU) Aussois Combinatorial Optimization Workshop 2016

Motivation Finding Facets Adjacency Affine Hull PORTA & Polymake Approach max � c , x � s.t. x ∈ Z E + x ( δ ( v )) = 1 ∀ v ∈ V Recognized class of facets: x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 3 / 15

Motivation Finding Facets Adjacency Affine Hull PORTA & Polymake Approach max � c , x � s.t. x ∈ Z E Enumeration + x ( δ ( v )) = 1 ∀ v ∈ V  ∗   ∗   ∗   ∗   ∗   ∗   ∗  All extr. . . . . . . . . . . . . . . . . .               . . . . . . . points:               ∗ ∗ ∗ ∗ ∗ ∗ ∗ Recognized class of facets: x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 3 / 15

Motivation Finding Facets Adjacency Affine Hull PORTA & Polymake Approach max � c , x � All facets: s.t. x ∈ Z E ( ∗ . . . ∗ ) x ≤∗ Enumeration + x ( δ ( v )) = 1 ∀ v ∈ V ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗  ∗   ∗   ∗   ∗   ∗   ∗   ∗  ( ∗ . . . ∗ ) x ≤∗ All extr. . . . . . . . . . . . . . . . . . ( ∗ . . . ∗ ) x ≤∗               . . . . . . . points:               ∗ ∗ ∗ ∗ ∗ ∗ ∗ ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗ All equations: ( ∗ . . . ∗ ) x ≤∗ Convex hull tool ( ∗ . . . ∗ ) x = ∗ ( ∗ . . . ∗ ) x ≤∗ (e.g., double-description, lrs . beneath&beyond,. . . ) . ( ∗ . . . ∗ ) x ≤∗ . ( ∗ . . . ∗ ) x = ∗ ( ∗ . . . ∗ ) x ≤∗ . . . Recognized class of facets: ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗ x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd ( ∗ . . . ∗ ) x ≤∗ Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 3 / 15

Motivation Finding Facets Adjacency Affine Hull PORTA & Polymake Approach max � c , x � All facets: s.t. x ∈ Z E ( ∗ . . . ∗ ) x ≤∗ Enumeration + x ( δ ( v )) = 1 ∀ v ∈ V ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗  ∗   ∗   ∗   ∗   ∗   ∗   ∗  ( ∗ . . . ∗ ) x ≤∗ All extr. . . . . . . . . . . . . . . . . . ( ∗ . . . ∗ ) x ≤∗               . . . . . . . points:               ∗ ∗ ∗ ∗ ∗ ∗ ∗ ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗ All equations: ( ∗ . . . ∗ ) x ≤∗ Convex hull tool ( ∗ . . . ∗ ) x = ∗ ( ∗ . . . ∗ ) x ≤∗ (e.g., double-description, lrs . beneath&beyond,. . . ) . ( ∗ . . . ∗ ) x ≤∗ . ( ∗ . . . ∗ ) x = ∗ ( ∗ . . . ∗ ) x ≤∗ . . . Recognized class of facets: ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗ x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd ( ∗ . . . ∗ ) x ≤∗ Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 3 / 15

Motivation Finding Facets Adjacency Affine Hull IPO Approach max � c , x � s.t. x ∈ Z E + x ( δ ( v )) = 1 ∀ v ∈ V Recognized class of facets: x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 4 / 15

Motivation Finding Facets Adjacency Affine Hull IPO Approach max � c , x � s.t. x ∈ Z E + x ( δ ( v )) = 1 ∀ v ∈ V All equations: ( ∗ . . . ∗ ) x = ∗ . . . ( ∗ . . . ∗ ) x = ∗ Recognized class of facets: x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 4 / 15

Motivation Finding Facets Adjacency Affine Hull IPO Approach max � c , x � s.t. x ∈ Z E + x ( δ ( v )) = 1 ∀ v ∈ V All equations: Only some ( ∗ . . . ∗ ) x = ∗ useful facets: . . ( ∗ . . . ∗ ) x ≤∗ . ( ∗ . . . ∗ ) x = ∗ ( ∗ . . . ∗ ) x ≤∗ . . . Recognized class of facets: ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗ x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 4 / 15

Motivation Finding Facets Adjacency Affine Hull IPO Approach max � c , x � s.t. x ∈ Z E MIP + solver x ( δ ( v )) = 1 ∀ v ∈ V All equations: Only some ( ∗ . . . ∗ ) x = ∗ useful facets: . . ( ∗ . . . ∗ ) x ≤∗ . ( ∗ . . . ∗ ) x = ∗ ( ∗ . . . ∗ ) x ≤∗ . . . Recognized class of facets: ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗ x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 4 / 15

Motivation Finding Facets Adjacency Affine Hull IPO Approach max � c , x � s.t. x ∈ Z E MIP + solver x ( δ ( v )) = 1 ∀ v ∈ V All equations: Only some ( ∗ . . . ∗ ) x = ∗ useful facets: . . ( ∗ . . . ∗ ) x ≤∗ . ( ∗ . . . ∗ ) x = ∗ ( ∗ . . . ∗ ) x ≤∗ . . . Recognized class of facets: ( ∗ . . . ∗ ) x ≤∗ ( ∗ . . . ∗ ) x ≤∗ x ( δ ( S )) ≥ 1 ∀ S ⊆ V , | S | odd Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 4 / 15

Motivation Finding Facets Adjacency Affine Hull Facets Finding Facets Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 5 / 15

Motivation Finding Facets Adjacency Affine Hull Quadratic Matching Polytopes Consider the quadratic matching polytope of order n with one quadratic term: ( χ ( M ) , y ) ∈ { 0 , 1 } | E n | +1 : M matching in K n , y = x 1 , 2 x 3 , 4 � � P n := conv . hull Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 6 / 15

Motivation Finding Facets Adjacency Affine Hull Quadratic Matching Polytopes Consider the quadratic matching polytope of order n with one quadratic term: ( χ ( M ) , y ) ∈ { 0 , 1 } | E n | +1 : M matching in K n , y = x 1 , 2 x 3 , 4 � � P n := conv . hull Hupp, Klein & Liers, ’15 obtained a bunch of facets: x ( δ ( v )) ≤ 1 for all v ∈ V n . x e ≥ 0 for all e ∈ E n . y ≤ x 1 , 2 and y ≤ x 3 , 4 . (Note that y ≥ x 1 , 2 + x 3 , 4 − 1 is no facet.) Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 6 / 15

Motivation Finding Facets Adjacency Affine Hull Quadratic Matching Polytopes Consider the quadratic matching polytope of order n with one quadratic term: ( χ ( M ) , y ) ∈ { 0 , 1 } | E n | +1 : M matching in K n , y = x 1 , 2 x 3 , 4 � � P n := conv . hull Hupp, Klein & Liers, ’15 obtained a bunch of facets: x ( δ ( v )) ≤ 1 for all v ∈ V n . x e ≥ 0 for all e ∈ E n . y ≤ x 1 , 2 and y ≤ x 3 , 4 . (Note that y ≥ x 1 , 2 + x 3 , 4 − 1 is no facet.) x ( E [ S ]) + y ≤ | S |− 1 for certain odd S . 2 x ( E [ S ]) ≤ | S |− 1 for certain odd S . 2 x ( E [ S ]) + x ( E [ S \ { 1 , 2 } ]) + x 3 , 4 − y ≤ | S | − 2 for certain odd S . x ( E [ S ]) + x 2 , a + x 3 , a + x 4 , a + y ≤ | S | 2 for certain even S and nodes a . x 1 , 2 + x 1 , a + x 2 , a + x ( E [ S ]) + x 3 , 4 + x 3 , b + x 4 , b − y ≤ | S | 2 + 1 for certain even S and certain nodes a , b . Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 6 / 15

Motivation Finding Facets Adjacency Affine Hull Some are Missing! Excerpt from their paper: Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 7 / 15

Motivation Finding Facets Adjacency Affine Hull Some are Missing! Excerpt from their paper: Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 7 / 15

Motivation Finding Facets Adjacency Affine Hull QMP: An IP Model param n := 6; set V := { 1 to n }; set E := { <u,v> in V*V with u < v }; set F := { <1,2>,<3,4>,<1,5>,<2,5>,<3,6>,<4,6>,<1,3>,<2,4> }; var x[E] binary; var y binary; maximize weights: 10*x[1,2] + 10*x[3,4] + 2*x[1,5] + 2*x[2,5] + 2*x[3,6] + 2*x[4,6] + 4*x[1,3] + 4*x[2,4] -10*y + sum <u,v> in E-F: -1000*x[u,v]; subto degree: forall <w> in V: (sum <u,v> in E with u == w or v == w: x[u,v]) <= 1; subto product1: y <= x[1,2]; subto product2: y <= x[3,4]; subto product3: y >= x[1,2] + x[3,4] - 1; Matthias Walter IPO – Investigating Polyhedra by Oracles Aussois 2016 8 / 15