Polytopes Associated with Symmetry Handling Christopher Hojny joint - - PowerPoint PPT Presentation

Polytopes Associated with Symmetry Handling

Christopher Hojny

joint work with Marc Pfetsch

Technische Universität Darmstadt Department of Mathematics 20th Combinatorial Optimization Workshop, Aussois 2016

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 1

Motivation

◮ Consider a necklace with n black (0) and white (1) beads. ◮ Associate with a necklace a vector in {0, 1}n. ◮ necklaces x, x′ ∈ {0, 1}n are “equal” ⇔ ∃ cyclic shift γ such that γ(x) = x′.

?

=

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 2

Motivation

◮ Consider a necklace with n black (0) and white (1) beads. ◮ Associate with a necklace a vector in {0, 1}n. ◮ necklaces x, x′ ∈ {0, 1}n are “equal” ⇔ ∃ cyclic shift γ such that γ(x) = x′.

?

=

1 shift

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 2

Motivation

◮ Consider a necklace with n black (0) and white (1) beads. ◮ Associate with a necklace a vector in {0, 1}n. ◮ necklaces x, x′ ∈ {0, 1}n are “equal” ⇔ ∃ cyclic shift γ such that γ(x) = x′.

?

=

2 shifts

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 2

Motivation

◮ Consider a necklace with n black (0) and white (1) beads. ◮ Associate with a necklace a vector in {0, 1}n. ◮ necklaces x, x′ ∈ {0, 1}n are “equal” ⇔ ∃ cyclic shift γ such that γ(x) = x′.

?

=

3 shifts

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 2

Motivation

◮ Consider a necklace with n black (0) and white (1) beads. ◮ Associate with a necklace a vector in {0, 1}n. ◮ necklaces x, x′ ∈ {0, 1}n are “equal” ⇔ ∃ cyclic shift γ such that γ(x) = x′.

?

=

3 shifts Question: How can we characterize (the convex hull of) a representative system of “distinct” necklaces?

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 2

Representative Systems

Generalized Problem What can be said about conv

• {x ∈ {0, 1}n : x is orbit representative}
• if Γ ≤ Sn acts on {0, 1}n?

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 3

Lexicographic Representative Systems

Friedman’s approach [Friedman, 2007]:

◮ define ¯

c = (2n−1, 2n−2, ... , 2, 1)⊤ ∈ Rn

◮ Friedman inequality for γ ∈ Γ:

¯ c⊤x ≥ ¯ c⊤γ(x)

⇔

• i∈[n]

(2n−γ(i) − 2n−i)xi ≤ 0

◮ x ∈ {0, 1}n with ¯

c⊤x ≥ ¯ c⊤γ(x) ∀ γ ∈ Γ are lexicographically maximal in their Γ-orbit.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 4

Application in Binary Programs

◮ Let Γ ≤ Sn be a symmetry group of a binary program, i.e.,

◮ γ ∈ Γ transforms feasible solutions x to feasible solutions γ(x), ◮ γ(x) and x have the same objective value for each feasible x.

◮ Add Friedman’s inequalities for each γ ∈ Γ. ◮ Restricted binary program contains only solutions which are lexicographically

maximal w.r.t. Γ.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 5

Application in Binary Programs

◮ Let Γ ≤ Sn be a symmetry group of a binary program, i.e.,

◮ γ ∈ Γ transforms feasible solutions x to feasible solutions γ(x), ◮ γ(x) and x have the same objective value for each feasible x.

◮ Add Friedman’s inequalities for each γ ∈ Γ. ◮ Restricted binary program contains only solutions which are lexicographically

maximal w.r.t. Γ. But:

◮ Coefficients of Friedman inequalities are exponentially large. ◮ If Γ is large, many inequalities have to be added.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 5

Further Symmetry Handling Techniques

Examples:

◮ isomorphism pruning [Margot, 2002] ◮ orbital branching [Ostrowski et al., 2011] ◮ symmetry breaking inequalities [Liberti, 2012]

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 6

Outline

Symmetry Breaking Polytopes Optimization Complexity over Symresacks Separation of Minimal Cover Inequalities for Symresacks Relaxed Symretopes

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 7

Symmetry Breaking Polytopes

Definition The symmetry breaking polytope (symretope) for Γ ≤ Sn is S(Γ) := conv

• {x ∈ {0, 1}n : ¯

c⊤x ≥ ¯ c⊤γ(x) ∀ γ ∈ Γ}

• .

A relaxed symretope for Γ is any polytope S with conv

• S ∩ {0, 1}n

= S(Γ).

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 8

Symmetry Breaking Polytopes

Definition The symmetry breaking polytope (symretope) for Γ ≤ Sn is S(Γ) := conv

• {x ∈ {0, 1}n : ¯

c⊤x ≥ ¯ c⊤γ(x) ∀ γ ∈ Γ}

• .

A relaxed symretope for Γ is any polytope S with conv

• S ∩ {0, 1}n

= S(Γ). Proposition (cf. [Babai and Luks, 1983]) The linear optimization problem over S(Γ) is NP-hard.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 8

Symmetry Breaking Polytopes

Definition The symmetry breaking polytope (symretope) for Γ ≤ Sn is S(Γ) := conv

• {x ∈ {0, 1}n : ¯

c⊤x ≥ ¯ c⊤γ(x) ∀ γ ∈ Γ}

• .

A relaxed symretope for Γ is any polytope S with conv

• S ∩ {0, 1}n

= S(Γ). Questions:

◮ Can we find a complete linear description of S(Γ)? ◮ Can we find tractable IP-formulations for S(Γ) via relaxed symretopes? ◮ Are all Friedman inequalities necessary for a relaxed symretope?

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 8

Symresacks

Special case: only one permutation, knapsack polytope Definition The symresack for a permutation γ ∈ Sn is Pγ := conv

• {x ∈ {0, 1}n : ¯

c⊤x ≥ ¯ c⊤γ(x)}

• =

conv

• {x ∈ {0, 1}n : x γ(x)}
• .

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 9

Symresacks

Special case: only one permutation, knapsack polytope Definition The symresack for a permutation γ ∈ Sn is Pγ := conv

• {x ∈ {0, 1}n : ¯

c⊤x ≥ ¯ c⊤γ(x)}

• =

conv

• {x ∈ {0, 1}n : x γ(x)}
• .

Examples:

◮ γ = (1, 2) ∈ Sn:

Pγ = conv

• {x ∈ {0, 1}n : x1 ≥ x2}
• Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 9

Symresacks

Special case: only one permutation, knapsack polytope Definition The symresack for a permutation γ ∈ Sn is Pγ := conv

• {x ∈ {0, 1}n : ¯

c⊤x ≥ ¯ c⊤γ(x)}

• =

conv

• {x ∈ {0, 1}n : x γ(x)}
• .

Examples:

◮ γ = (1, 2) ∈ Sn:

Pγ = conv

• {x ∈ {0, 1}n : x1 ≥ x2}
• ◮ γ = (1, 2)(3, 4) ... (2n − 1, 2n) ∈ S2n:

Pγ = conv

• {x ∈ {0, 1}2n : (x1, x3, ... , x2n−1) (x2, x4, ... , x2n)}
• This is the orbisack On, see [Kaibel and Loos, 2011].

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 9

Orbisacks

Orbisack On := conv

• {X = (X 1, X 2) ∈ {0, 1}n×2 : X 1 X 2}
• Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 10

Orbisacks

Orbisack On := conv

• {X = (X 1, X 2) ∈ {0, 1}n×2 : X 1 X 2}
• Properties of vertices of On:

◮ either all rows are constant

     

1 1 1 1 1 1

     

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 10

Orbisacks

Orbisack On := conv

• {X = (X 1, X 2) ∈ {0, 1}n×2 : X 1 X 2}
• Properties of vertices of On:

◮ either all rows are constant ◮ or the first non-constant row c

is (1, 0), c is called critical row

     

1 1 1 1

     

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 10

Orbisacks and Symresacks

Orbisackoidal Symresack Qγ := conv

• {X = (X 1, X 2) ∈ {0, 1}n×2 : X 1 X 2, X 2 = γ(X 1)}
• Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 11

Orbisacks and Symresacks

Orbisackoidal Symresack Qγ := conv

• {X = (X 1, X 2) ∈ {0, 1}n×2 : X 1 X 2, X 2 = γ(X 1)}
• Lemma

Pγ and Qγ are linearly equivalent.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 11

Optimization over Qγ and Pγ

Theorem The linear optimization problem max{W ⊤X : X ∈ Qγ}, W ∈ Rn×2, can be solved in time O(n2) for any γ ∈ Sn. Proof idea:

◮ each vertex of Qγ is a vertex of On ◮ fix a critical row c ∈ [n + 1] ◮ find a maximizer in Qγ with critical row c in time O(n) ◮ iterate over all possible critical rows

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 12

Optimization over Qγ and Pγ

Theorem The linear optimization problem max{W ⊤X : X ∈ Qγ}, W ∈ Rn×2, can be solved in time O(n2) for any γ ∈ Sn. Proof idea:

◮ each vertex of Qγ is a vertex of On ◮ fix a critical row c ∈ [n + 1] ◮ find a maximizer in Qγ with critical row c in time O(n) ◮ iterate over all possible critical rows

Corollary The linear optimization problem max{w⊤x : x ∈ Pγ}, w ∈ Rn, can be solved in time O(n2) for any γ ∈ Sn.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 12

Consequences of the Optimization Algorithm

◮ There is an extended formulation of Pγ of size O(n2). ◮ The algorithm can be adapted to optimize over S(Γ) in time O(|Γ|n|Γ|).

◮ If |Γ| ∈ O(1), optimization over S(Γ) can be done in polynomial time. ◮ Examples for such symretopes: geometric symmetries in fixed dimension, e.g.,

cyclic group, or full orbitopes with a fixed number of columns [Kaibel and Pfetsch, 2008]

◮ There is an extended formulation of S(Γ) of size O(n|Γ|).

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 13

Outline

Symmetry Breaking Polytopes Optimization Complexity over Symresacks Separation of Minimal Cover Inequalities for Symresacks Relaxed Symretopes

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 14

Minimal Cover Inequalities for Symresacks

◮ How do covers for Friedman inequalities ¯

c⊤x ≥ ¯ c⊤γ(x) look like?

◮ C ⊆ [n] is cover

⇔

¯ c⊤χC ≤ ¯ c⊤γ(χC) + 1

◮ cover inequality for Friedman inequality:

• i∈C+

xi −

• i∈C−

xi ≤ |C+| − 1, where C+ = {i ∈ [n] : i ≥ γ(i)} and C− = C \ C+

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 15

Separation of Minimal Cover Inequalities

Classical approach: Formulate separation problem of minimal cover inequalities for ¯ x ∈ Rn as IP , see [Crowder et al., 1983]: max

• i:i≥γ(i)

¯ xiyi −

• i:i<γ(i)

¯ xi(1 − yi) − (

• i:i≥γ(i)

yi − 1) ¯ c⊤γ(y) − ¯ c⊤y ≥ 1, y ∈ {0, 1}n.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 16

Separation of Minimal Cover Inequalities

Classical approach: Formulate separation problem of minimal cover inequalities for ¯ x ∈ Rn as IP , see [Crowder et al., 1983]: max

• i:i≥γ(i)

¯ xiyi −

• i:i<γ(i)

¯ xi(1 − yi) − (

• i:i≥γ(i)

yi − 1) ¯ c⊤γ(y) − ¯ c⊤y ≥ 1

⇔

y ≺ γ(y), y ∈ {0, 1}n.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 16

Separation of Minimal Cover Inequalities

Classical approach: Formulate separation problem of minimal cover inequalities for ¯ x ∈ Rn as IP , see [Crowder et al., 1983]: max

• i:i≥γ(i)

¯ xiyi −

• i:i<γ(i)

¯ xi(1 − yi) − (

• i:i≥γ(i)

yi − 1) ¯ c⊤γ(y) − ¯ c⊤y ≥ 1

⇔

y ≺ γ(y), y ∈ {0, 1}n. Theorem The separation problem of minimal cover inequalities and ¯ x ∈ Rn for Pγ can be solved in time O(n2) for all γ ∈ Sn.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 16

IP-formulations for Symretopes

An IP-formulation for S(Γ) is given by SC(Γ) := conv

γ∈Γ

{x ∈ {0, 1}n : x fulfills all cover inequalities for Pγ}

• since Pγ = conv
• {x ∈ {0, 1}n : x fulfills all cover inequalities for Pγ}
• .

Properties:

◮ SC(Γ) is defined by inequalities with coefficients in {0, ±1}. ◮ The LP-relaxation of SC(Γ) can be separated in time O(|Γ|n2).

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 17

Outline

Symmetry Breaking Polytopes Optimization Complexity over Symresacks Separation of Minimal Cover Inequalities for Symresacks Relaxed Symretopes

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 18

Friedman Inequalities and Facets

Proposition

◮ (i, i + 1) ∈ Γ, i ∈ [n − 1]: Friedman inequality for (i, i + 1) defines a facet of S(Γ). ◮ γ ∈ Γ not a transposition: Friedman inequality for γ does not define a facet

• f S(Γ).

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 19

Friedman Inequalities and Facets

Proposition

◮ (i, i + 1) ∈ Γ, i ∈ [n − 1]: Friedman inequality for (i, i + 1) defines a facet of S(Γ). ◮ γ ∈ Γ not a transposition: Friedman inequality for γ does not define a facet

• f S(Γ).

Example 1

◮ Γ = Sn ◮ only next neighbor transpositions define facets ◮ Friedman inequality for next neighbor transposition (i, i + 1):

(2n−(i+1) − 2n−i)xi + (2n−i − 2n−(i+1))xi+1 ≤ 0

⇔ −xi + xi+1 ≤ 0

◮ S(Sn) = {x ∈ [0, 1]n : −xi + xi+1 ≤ 0 ∀i ∈ [n − 1]}

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 19

Friedman Inequalities and Facets

Proposition

◮ (i, i + 1) ∈ Γ, i ∈ [n − 1]: Friedman inequality for (i, i + 1) defines a facet of S(Γ). ◮ γ ∈ Γ not a transposition: Friedman inequality for γ does not define a facet

• f S(Γ).

Example 2

◮ Γ = Cn, n ≥ 3 ◮ no Friedman inequality is facet defining ◮ But not all Friedman inequalities from

conv

• {x ∈ {0, 1}n : ¯

c⊤x ≥ ¯ c⊤γ(x) ∀ γ ∈ Cn}

• can be dropped to define a relaxed symretope.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 19

Sufficient Criterion

Proposition Let Γ = γ1, ... , γm ≤ Sn. Then S⋆ := m

i=1 S(γi) is a relaxed symretope for Γ if for

each x ∈ S⋆ ∩ Zn and γ ∈ Γ there is a sequence of powers of generators such that

γ = γp1

i1 ... γpk ik ,

i1, ... , ik ∈ [m], p1, ... , pk ∈ Z≥0, and such that

• γ

pj ij ... γpk ik

• (x) ∈ S(γij−1)

∀j ≤ k.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 20

Sufficient Criterion

Proposition Let Γ = γ1, ... , γm ≤ Sn. Then S⋆ := m

i=1 S(γi) is a relaxed symretope for Γ if for

each x ∈ S⋆ ∩ Zn and γ ∈ Γ there is a sequence of powers of generators such that

γ = γp1

i1 ... γpk ik ,

i1, ... , ik ∈ [m], p1, ... , pk ∈ Z≥0, and such that

• γ

pj ij ... γpk ik

• (x) ∈ S(γij−1)

∀j ≤ k.

Examples:

◮ Γ = Sn: Friedman inequalities for (i, i + 1), i ∈ [n − 1], sufficient ◮ Γ = An: Friedman inequalities for (i, i + 1, i + 2), i ∈ [n − 2], sufficient

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 20

Conclusion and Outline

◮ preliminary computational experiments indicate effectiveness of symretopes in

symmetry handling

◮ Questions:

◮ Can we find a complete linear description of S(Γ)? () ◮ Can we find tractable IP-formulations for S(Γ) via relaxed symretopes? () ◮ Are all Friedman inequalities necessary for a relaxed symretope? () Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 21

Conclusion and Outline

◮ preliminary computational experiments indicate effectiveness of symretopes in

symmetry handling

◮ Questions:

◮ Can we find a complete linear description of S(Γ)? () ◮ Can we find tractable IP-formulations for S(Γ) via relaxed symretopes? () ◮ Are all Friedman inequalities necessary for a relaxed symretope? ()

Thank you for your attention!

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 21

Literature

Babai, L. and Luks, E. M. (1983). Canonical labeling of graphs. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC ’83, pages 171–183, New York, NY, USA. ACM. Crowder, H., Johnson, E. L., and Padberg, M. (1983). Solving large-scale zero-one linear programming problems. Operations Research, 31(5):803–834. Friedman, E. J. (2007). Fundamental domains for integer programs with symmetries. In Dress, A., Xu, Y., and Zhu, B., editors, Combinatorial Optimization and Applications, volume 4616 of Lecture Notes in Computer Science, pages 146–153. Springer Berlin Heidelberg. Kaibel, V. and Loos, A. (2011). Finding descriptions of polytopes via extended formulations and liftings. In Mahjoub, A. R., editor, Progress in Combinatorial Optimization. Wiley. Kaibel, V. and Pfetsch, M. E. (2008). Packing and partitioning orbitopes. Mathematical Programming, 114(1):1–36. Liberti, L. (2012). Reformulations in mathematical programming: automatic symmetry detection and exploitation. Mathematical Programming, 131(1-2):273–304. Margot, F . (2002). Pruning by isomorphism in branch-and-cut. Mathematical Programming, 94(1):71–90. Ostrowski, J., Linderoth, J., Rossi, F ., and Smriglio, S. (2011). Orbital branching. Mathematical Programming, 126(1):147–178.

Aussois 2016 | Christopher Hojny: Polytopes Associated with Symmetry Handling | 22