Orbitopes. Marc Pfetsch. joint work with Volker Kaibel. Zuse Institute Berlin

Transcription

1 Orbitopes Marc Pfetsch joint work with Volker Kaibel Zuse Institute Berlin

2 What this talk is about We introduce orbitopes. A polyhedral way to break symmetries in integer programs.

3 Introduction 2 Orbitopes 3 Cyclic Groups 4 Symmetric Groups 5 Coloring Polyhedra Outline

4 Motivation: Graph Coloring Given: graph G = (V, E) without isolated nodes, positive integer C C-coloring: function V {,..., C}, such that no two adjacent nodes receive the same color (number). Graph coloring: Find minimal C such that C-coloring exits. Well known NP-hard problem.

5 Variables: x ij = y j = Integer Programming Formulation iff node i receives color j iff color j is used min C j= y j x ij + x kj y j {i, k} E, j {,..., C} C x ij = i V j= x ij {, } i V, j {,..., C} y j {, } j {,..., C}

6 Variables: x ij = y j = Integer Programming Formulation iff node i receives color j iff color j is used min C j= y j x ij + x kj y j {i, k} E, j {,..., C} C x ij = i V j= x ij {, } i V, j {,..., C} y j {, } j {,..., C} Highly symmetric: Can arbitrarily permute numbers for colors without changing the structure and objective.

7 Branch-and-Cut. Solve LP-relaxation: min CX y j j= x ij + x kj y j {i, k} E, j {,..., C} CX x ij = i V j= x ij i V, j {,..., C} y j j {,..., C} 2. Add valid inequalities (cuts). Example: clique inequalities: S V a clique, x ij for all j {,..., C}. i S 3. Perform branching branching tree Example: Two branches: x ij = and x ij = for some i, j. 4. Perform bounding: cut off parts of the tree.

8 Symmetry strikes twice... weak LP-relaxations: x ij = C, y j = 2 C gives value: 2 unnecessarily large branch-andbound tree

9 Potential Applications Line planning in public transport Periodic timetabling Planning the UMTS radio interface Chip-verification Frequency assignment in mobile phone networks Location / routing areas Wavelength assignment in optical networks Block decomposition of matrices

10 Example: Line planning Line planning model that keeps track of transfers:

11 Other Approaches Margot [22]: modified branch-and-bound algorithm. Avoids redundant parts in the tree (using group theory). Work for constraint propagation, e.g., automatic symmetry detection. Exist many symmetry breaking inequalities for particular problems.

12 Introduction 2 Orbitopes 3 Cyclic Groups 4 Symmetric Groups 5 Coloring Polyhedra Outline

13 Lexicographic Ordering of /-Matrices M p,q := {, } [p] [q] S n = symmetric group G subgroup of S q, acting by permuting columns : lexicographic ordering M max p,q (G) = -maximal matrices within their orbits under the group action. M p,q := {(x ij ) M p,q : q j= x ij for all i} M = p,q := {(x ij ) M p,q : q j= x ij = for all i} M p,q := {(x ij ) M p,q : q j= x ij for all i}.

14 Orbitopes full orbitope O p,q (G) := conv M max p,q (G) packing orbitope O p,q(g) := conv(m max p,q (G) M p,q) partitioning orbitope O = p,q(g) := conv(m max p,q (G) M = p,q) covering orbitope O p,q(g) := conv(m max p,q (G) M p,q)

15 Main Goal More Detailed Plan of Talk Add inequality descriptions of orbitopes to particular integer programs in order to remove symmetry. Will discuss only packing and partitioning orbitopes.. Orbitopes for cyclic permutations C q Linear optimization over orbitopes Complete linear description 2. Orbitopes for the symmetric group S q Linear optimization over orbitopes Complete linear description Facets 3. Coloring Polyhedra

16 Introduction 2 Orbitopes 3 Cyclic Groups 4 Symmetric Groups 5 Coloring Polyhedra Outline

17 For cyclic group G = C q : Packing Lexicographically Maximal Matrices X M p,q M max p,q (C q ) iff first column lexicographically not smaller than the others. Partitioning X M = p,q M max p,q (C q ) iff it has a -entry at position (, ).

18 Optimization over Orbitopes Proposition Linear optimization over M max p,q (C q ) M p,q and over M max p,q (C q ) M = p,q can be solved in time O(pq). Proof. For partitioning orbitopes: Maximize linear objective given by c Q [p] [q]. Take matrix with at position (, ). For each row, put at c-maximal position. For packing orbitopes: just a bit more involved

19 Linear Description Partitioning Theorem O = p,q(c q ) is completely described by x = and x j = for all 2 j q, x ij for all 2 i p and j [q], x(row i ) = for all 2 i p. This system is non-redundant. row i := {(i, ), (i, 2),..., (i, q)} = ith row x(s) := (i,j) S x ij

20 Proof Proof. Constraints arise from LP-relaxation. Constraint matrix for x(row i ) = is an interval matrix: C A The whole constraint matrix is totally unimodular.

21 Theorem O p,q(c q ) is completely described by x and x j = for all 2 j q, x ij for all 2 i p and j [q], x(row i ) for all 2 i p, the inequalities Linear Description Packing q i x ij x k j=2 k= for all 2 i p. This system is non-redundant.

22 Proof Sketch Constraints arise from LP-relaxation. Constraint matrix with suitable ordering of variables: Show: the whole constraint matrix is totally unimodular. Use Ghouila-Houri criterion: For each subset of columns, assign +/ weights to columns such that weighted sum is a vector with components in {,, +}. C A

23 Introduction 2 Orbitopes 3 Cyclic Groups 4 Symmetric Groups 5 Coloring Polyhedra Outline

24 Lexicographically Maximal Matrices For symmetric group G = S q : X M max p,q (S q ) iff columns are non-increasing lexicographically ordered.

25 Proposition Optimization over Orbitopes Linear optimization over M max p,q (S q ) M p,q and over M max p,q (S q ) M = p,q can be solved in time O(p 2 q). Proof: By dynamic programming: Maximize linear objective given by c Q [p] [q]. M = optimal matrix not lexicographically sorted T = optimal matrix lexicographically sorted in last part max {c(m) + c kj + c(t ) : k = i,..., p}

26 IP Formulation for Partitioning Can fix variables: For graph coloring, Méndez-Díaz & Zabala [2] introduced: This forces lexicographic ordering. Can be strengthened...

27 IP Formulation for Partitioning IP formulation: column inequalities:

28 IP Formulation for Partitioning IP formulation: column inequalities: + = + Variable bounds

29 IP Formulation for Partitioning IP formulation: column inequalities: + = + Variable bounds Does not provide a complete linear description!

30 Exponential class of inequalities: Shifted Column Inequalities shifted column (SC) bar Shifted Column Inequality (SCI) Shiftings:

31 Separation of SCIs Proposition The separation problem for SCIs can be solved in time O(pq). Proof. Dynamic programming: Case Case 2 min{ +, }.

32 Linear Description Partitioning Theorem O = p,q(s q ) is completely described by: x ij for all i [p], j [q], x(row i ) = for all i [p], x(b) x(s) for all SCIs with SC S and bar B.

33 Facets Theorem dim O = p,q(s q ) = (p q )(q ). 2 x ij defines a facet unless i = j < q. x(b) x(s) defines a facet unless:

34 Packing Orbitopes Proposition O = p,q(s q ) and O p,q (S q ) are affinely isomorphic via orthogonal projection removing the first row and column. Theorem O p,q(s q ) is completely described by x ij for all i [p], j [q], x(row i ) for all i [p], x(b) x(s) for all SCIs with SC S and bar B.

35 Packing Orbitopes Facets Theorem dim O p,q(s q ) = (p q )q, i.e., full-dimensional 2 x ij defines a facet unless i = j < q. x(row i ) always defines a facet. x(b) x(s) defines a facet unless:

36 Introduction 2 Orbitopes 3 Cyclic Groups 4 Symmetric Groups 5 Coloring Polyhedra Outline

37 Back to Coloring Exist lots of valid inequalities. Can combine SCIs with clique inequalities: Polyhedral study is involved... (Work of Yuri Faenza.)

38 Coloring Polyhedra I CY n (G) = conv{(x, y) {, } V [n] : q x ij = i V j= x ij + x kj y j n y j i= x ij {i, k} E, j [q] j [q] y j+ y j j [q ]}. =

39 Coloring Polyhedra I CY n (G) = conv{(x, y) {, } V [n] : q x ij = i V j= x ij + x kj y j n y j i= x ij {i, k} E, j [q] j [q] y j+ y j j [q ]}. Theorem (Méndez-Díaz and Zabala (2)) dim CY n (G) = V 2 χ(g).

40 Coloring Polyhedra II CO n (G) = conv{(x, y) {, } V [n] : q x ij = i V j= x ij + x kj y j n y j i= x ij q i x il l=j k= x k,j {i, k} E, j [q] j [q] i V, j [q]}. =

41 Dimension The dimension of CO n (G) projected to x-variables depends on the order of the nodes. Example: Proposition (Faenza (26)) The dimension of the star on n nodes is ( ) n 2 if the center is labeled or 2 ( n 2) 2 otherwise.

42 Similar Problems k-colorable subgraph problem What is the largest (induced) subgraph that can be colored with k colors? k-partitioning Complete graph G = (V, E) with weights w e, find partition of V into k parts V,..., V k such that k is minimal. i,j= i j e E(V i,v j ) w e Currently investigated by Matthias Peinhardt.

43 Outlook Continue work on coloring branch-and-cut algorithm. Continue work on full orbitopes. Continue work on k-partitioning. Study covering orbitopes. Investigate interplay with special polyhedra. Applications.