Petersen Graph - Department of Mathematicss-dstolee1/Presentations/Sto10-Symmetric... · Let...

Petersen Graph(10, 3, 0, 1)

Paley(13) Graph(13, 6, 2, 3)

Hoffman-Singleton Graph(50, 7, 0, 1)

Higman-Sims Graph(100, 22, 0, 6)

?

(99, 14, 1, 2)

1 Need a computer!2 Fit into standard optimization problems.3 Formulations are symmetric.4 Standard methods fail!

Symmetric Optimization

Derrick Stolee

March 12, 2010

Discrete Structures Symmetry Techniques Implementation Deliverables and Results

Outline

1 Discrete Structures

2 Symmetry Techniques

Orbital Branching

Symmetric Solutions

Symmetry-Breaking Cuts

3 Implementation

4 Deliverables and Results


Strongly Regular Graphs

Let n, k , λ, µ be non-negative integers.

A (n, k , λ, µ) strongly regular graph is an undirected graph G onn vertices, each of degree k , with the conditions

If uv ∈ E(G), then there are λ vertices x so thatux ∈ E(G) and xv ∈ E(G).If uv 6∈ E(G), then there are µ vertices x so thatux ∈ E(G) and xv ∈ E(G).


Strongly Regular Graphs: Example

Petersen Graph(10, 3, 0, 1)


Directed Strongly Regular Graphs

Let n, k , t , λ, µ be non-negative integers.

A (n, k , t , λ, µ) directed strongly regular graph is a directedgraph G on n vertices, each of in- and out-degree k , with theconditions

If uv ∈ E(G), then there are λ vertices x so thatux ∈ E(G) and xv ∈ E(G).

If uv 6∈ E(G), then there are µ vertices x so thatux ∈ E(G) and xv ∈ E(G).

For each vertex v there are t vertices u so thatuv ∈ E(G) and vu ∈ E(G).


Directed Strongly Regular Graphs: Example

(6, 2, 1, 0, 1)


Parameters with Unknown Existence

(a) Undirected Parameters

n k λ µ

85 14 3 299 14 1 2

115 18 1 3162 21 0 3

(b) Directed Parameters

n k t λ µ

22 9 6 3 424 10 5 3 527 7 4 1 227 10 6 3 428 6 3 2 128 7 2 1 230 11 9 2 5

Table: Parameters for strongly regular and directed strongly regulargraphs of unknown existence.


Adjacency Matrix of Graphs

A graph is described by a 0-1 adjacency matrix.

A =

0 1 1 0 0 01 0 0 1 0 00 0 0 0 1 10 0 0 0 1 11 0 0 1 0 00 1 1 0 0 0


Matrix Equations for SRGs

A = A> (G undirected)AJ = JA = kJ (k -regular)

A2 + (µ− λ)A = kI + µ(J − I) (common neighbors)


Matrix Equations for DSRGs

AJ = JA = kJ (k -regular)

A2 + (λ− µ)A = tI + µ(J − I) (two-paths)


Matrix Equations for DSRGs

AJ = JA = kJ (k -regular)

A2 + (λ− µ)A = tI + µ(J − I) (two-paths)

Note: If t = k , we have the definition of a strongly-regular graph!(A becomes symmetric)


Equation Systems for DSRGs

Let xi,j = Ai,j = 1[vivj ∈ E(G)] =

{1 vivj ∈ E(G)0 otherwise

.


Equation Systems for DSRGs

Let xi,j = Ai,j = 1[vivj ∈ E(G)] =

{1 vivj ∈ E(G)0 otherwise

.

∑j 6=i

xi,j = ∑j 6=i

xj,i = k (1) (∀i ∈ {1, . . . , n})

∑a 6=i

xi,axa,i = t − µ (2) (∀i , j ∈ {1, . . . , n}, i 6= j)

∑a 6=i,j

xi,axa,j + (λ− µ)xi,j = µ (3) (∀i , j ∈ {1, . . . , n}, i 6= j)


Optimization Systems for DSRGs

These equations are quadratic.Forms a Quadratically-Constrained Program.Convert to Integer Linear Program.

Requires O(n3) variables and constraints.


What is a symmetric system?

Symmetry exists in an optimization system when there arenon-trivial automorphisms of the system.

(i.e., a permutation σ of the variables stabilizes the constraints).


What symmetries in a formulation of discrete object?

In the case of discrete objects (graphs, block designs,coverings, packings), the automorphisms of the system aregiven by:

A base set S (i.e. Vertices)A variable set X (i.e. Edge indicator variables)A map from σ ∈ Sym(S) to σ ∈ Sym(X )All such σ are automorphisms of the system.


Symmetry in the Formulation of SRGs

Take any σ ∈ Sym(V (G)).

xi,jσ7−→ xσ(i),σ(j).





∑j 6=i

xi,jσ7−→ ∑

j 6=σ(i)xσ(i),j = k

∑a 6=i

xi,axa,iσ7−→ ∑

a 6=σ(i)xσ(i),axa,σ(i) = t − µ

∑a 6=i,j

xi,axa,j + (λ− µ)xi,jσ7−→

∑a 6=σ(i),σ(j)

xσ(i),axa,σ(j) + (λ− µ)xσ(i),σ(j) = µ





(n2) = n(n−1)

2 variables (or n(n− 1) directed variables).+ n! = n · (n− 1) · (n− 2) · · · 3 · 2 · 1 relabelings.= Lots of symmetry!


Outline



Orbital Branching

Symmetric Solutions


3 Implementation



Why do standard methods fail?

What is branch-and-bound?


Why do standard methods fail?

Methods such as branch-and-bound or cutting planes reducethe search space. Any feasible solution x∗ can be permuted toσ(x∗), giving many new solutions.


Why do standard heuristics fail?

For example, heuristics use information from continuousrelaxation to “guess” in the integer search.

Symmetric systems give the averaging effect for continuoussolutions: variables xi in orbit concentrate around the meanexpected value.

For instance, in a strongly-regular graph, we need 12nk edges,

and “most” continuous solutions have each

xi,j ≈k

n− 1=

nk2(n

2).


Outline



Orbital Branching

Symmetric Solutions


3 Implementation



Variable Assignments

There are three types of variable assignments.

1 Fixed variables: set by a branching procedure orpresolving.

2 Set or implied variables: set by constraint propagation onfixed values.

3 Unassigned variables: no value set.

Put them in a tuple (F , I, U).


Standard Branching

At each node, select a variable xi to branch.

Each branch transitions an unassigned variable to a fixedvariable.


Orbits with Variable Assignments


Orbital Branching

Partition unassigned variables into orbits O1, . . . ,Ok .If xa, xb ∈ Oi , there exists a permutation σ ∈ Aut(S | V ) so thatσ(xa) = xb.


Orbital Branching

Introduced in [OLRS07].Implementation no longer available.Will re-implement.Applying it to a new problem.We propose an extension.


Outline



Orbital Branching

Symmetric Solutions


3 Implementation



Symmetry as a Constraint

We’d like to add a constraint to the system that forces thesolution to also be symmetric.

The solution is symmetric if there are non-trivial automorphismsthat preserve the values of the solution.

If we knew the entire automorphism group, we could setequality between variables in orbit.

However, this is likely too strong!


An Example

Consider a (99, 14, 1, 2) strongly regular graph.


An Example


1

3 2


An Example


?

1

3 2


Partial Permutations

DefinitionA partial permutation on S is a map π : S → P(S).

A partial permutation π extends to a permutation π where π isa bijection of S and π(v) ∈ π(v) for all v ∈ S.


Partial Permutations

During search, π(v) are updated when xi,j 6= xa,b.

If |π(i)| = 1, we have a unique extension at this coordinate!


Outline



Orbital Branching

Symmetric Solutions


3 Implementation



Symmetry-Breaking Cuts: Previous Attempts

A cut is an added constraint to reduce the search space.


Symmetry-Breaking Cuts: Previous Attempts

Previous strategies take two forms:

Static cuts.Dynamic cuts.

Both attempt to remove symmetry, but retain a solution fromeach isomorphism class.


Part of a New Strategy

A new strategy:

Dynamic cuts on continuous relaxation.

Attempt to break symmetry in the continuous optimum; combataveraging effect.


Interaction with Heuristics

These cuts would not affect constraint propagation in integralsolution.

Symmetry-broken cts. optimal x∗.Used for branching rules.Used to differentiate orbits.


Outline



Orbital Branching

Symmetric Solutions


3 Implementation



Structure for Implementation

Implementation strategy:

1 An object-oriented framework:

1 Optimization system.

2 Symmetry model.

3 Constraint algorithm.




2 Generalized branching framework:

1 Standard branching.

2 Orbital branching.

3 Partial permutations.




3 Integration with continuous solvers:

1 COIN-OR

2 IBM ILOG CPLEX




4 Integration with TREESEARCH [Sto10]:

1 Abstract tree-based search.

2 Automated statistic tracking.

3 Execution on supercomputing resources.


Outline



Orbital Branching

Symmetric Solutions


3 Implementation



Deliverables

An open framework for solving symmetric optimizationproblems.Performance analysis of old and new techniques.Search for new graphs.

One exists, and its properties are...Many exist, and the enumeration is...None exist with these automorphisms...None exist, due to the full search...

Lower bounds on performance improvement.


Current Progress

Built object framework for search.Centered on TREESEARCH.Standard Branching and Orbital Branching.Constraint algorithms: GAC, SGAC.No continuous solver.


Current Progress: Performance


Searching for a (22, 9, 6, 3, 4) DSRG

The search is underway.Running on the Open Science Grid.Up to 1500 simultaneous parallel processes!


Standard Branching to Depth 10: 1024 Nodes.


Orbital Branching to Depth 28: 682 Nodes.

(6, 2, 1, 0, 1)

(14, 6, 3, 2, 3)

(15, 5, 2, 1, 2)

(18, 6, 3, 0, 3)

(20, 4, 1, 0, 1)


Acknowledgements

Thanks to those who helped with this work:

Advisors Stephen Hartke and Vinod VariyamHolland Computing Center faculty and staff

David SwansonBrian BocklemanDerek Weitzel

Kathryn Stolee


Thanks

Thanks for listening!

Special thanks to

Advisors Stephen Hartke and Vinod VariyamCommittee members Christina Falci, Jamie Radcliffe,Stephen Scott.


Stephen G. Hartke, Hannah Kolb, Jared Nishikawa, andDerrick Stolee.Automorphism groups of a graph and a vertex-deletedsubgraph.under submission, 2009.

Derrick Stolee.A distributed search for strongly regular graphs, Classproject, Cluster and grid computing, 2009.

Derrick Stolee.TREESEARCH users guide, 2010.

Written proposal available by request.


Self-Citations

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