Case study 5: all interval series

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Case study 5 all interval series

• Results due to Simonis, Puget Regin

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All interval series

• Prob007 at www.csplib.org
• Comes from musical composition
• Traced back to Alban Berg
• Extensively used by Ernst Krenek
• Op.170 Quaestio temporis

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All interval series

• Take the 12 standard pitch classes
• c, c, d, ..
• Represent them by numbers 0, .., 11
• Find a sequence so each occurs once
• Each difference occurs once

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All interval series

• Can generalize to any n (not just 12)
• Find Sn, a permutation of 0,n)
• such that Sn1-Sn are all distinct
• Finding one solution is easy

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All interval series

• Can generalize to any n (not just 12)
• Find Sn, a permutation of 0,n) such that
  Sn1-Sn are all distinct
• Finding one solution is easy
• n,1,n-1,2,n-2,.., floor(n/2)2,floor(n/2)-1,flo
  or(n/2)1,floor(n/2)
• Giving the differences n-1,n-2,..,2,1
• Challenge is to find all solutions!

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Basic CSP model

• What are the variables?

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Basic CSP model

• What are the variables?
• Si j if the ith note is j
• What are the constraints?

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Basic CSP model

• What are the variables?
• Si j if the ith note is j
• What are the constraints?
• Si in 0,n)
• All-different(S1,S2, Sn)
• Forall ilti Si1 - Si / Si1 - Si
• Will this model be any good? If so, why?
• If not, why not?

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Improving basic CSP model

• Is it worth introducing any auxiliary variables?
• Are there any loose or messy constraints we could
  better (more compactly?) express via some
  auxiliary variables?

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Improving basic CSP model

• Is it worth introducing any auxiliary variables?
• Yes, variables for the pairwise differences
• Di Si1 - Si
• Now post single large all-different constraint
• Di in 1,n-1
• All-different(D1,D2,Dn-1)

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Dual or combined model?

• Would a dual model be useful?

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Dual or combined model?

• Would a dual model be useful?
• We already have GAC on permutations so not there!
• Would an 0/1 model be useful?

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Dual or combined model?

• Would a dual model be useful?
• We already have GAC on permutations so not there!
• Would an 0/1 model be useful?
• No obvious benefits here?

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Break symmetry

• Does the problem have any symmetry?

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Break symmetry

• Does the problem have any symmetry?
• Yes, we can reverse any sequence
• S1, S2, Sn is an all-inverse series
• Sn, , S2, S1 is also
• How do we eliminate this symmetry?

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Break symmetry

• Does the problem have any symmetry?
• Yes, we can reverse any sequence
• S1, S2, , Sn is an all-inverse series
• Sn, , S2, S1 is also
• How do we eliminate this symmetry?
• As with Golomb ruler!
• D1 lt Dn-1

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Break symmetry

• Does the problem have any other symmetry?

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Break symmetry

• Does the problem have any other symmetry?
• Yes, we can invert the numbers in any sequence
• 0, n-1, 1, n-2, map x onto n-1-x
• n-1, 0, n-2, 1,
• How do we eliminate this symmetry?

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Break symmetry

• Does the problem have any other symmetry?
• Yes, we can invert the numbers in any sequence
• 0, n-1, 1, n-2, map x onto n-1-x
• n-1, 0, n-2, 1,
• How do we eliminate this symmetry?
• S1 lt S2

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Implied constraints

• Are there useful implied constraints to add?

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Implied constraints

• Are there useful implied constraints to add?
• Hmm, unlike Golomb ruler, we only have
  neighbouring differences
• So, no need to consider transitive closure

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Implied constraints

• Are there useful implied constraints to add?
• Hmm, unlike Golomb ruler, we are not optimizing
• So, no need to improve propagation for
  optimization variable

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Performance

• Basic model is poor
• Refined model able to compute all solutions up to
  n14 or so
• GAC on all-different constraints very beneficial
• As is enforcing GAC on Di Si1-Si
• This becomes too expensive for large n
• So use just bounds consistency (BC) for larger n

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Case study 6 progressive party problem

• Model due to Barabra Smith

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Progressive party problem

• Progressive does not mean hippy!
• But people progress round the party
• Yacht club party
• Small number of host boats
• Crews of other boats circulate round the hosts

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Progressive party problem

• There are h hosts, and g guests
• Each host boat has a maximum capacity of people
• Each guest boat has a crew size
• In each of t time periods
• Each guest boat visits one of the h hosts
• Subject to capacity constraints on hosts
• In addition, no two guests should meet twice and
  no guest visit the same host twice

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Basic CSP model

• What are the variables?

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Basic CSP model

• What are the variables?
• Hit j if guest boat i visits host j at time t
• Why not the dual model of which guest visits
  host j at time t?

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Basic CSP model

• What are the variables?
• Hit j if guest boat i visits host j at time t
• Why not the dual model of which guest visits
  host j at time t?
• But several guests visit each host so would need
  to deal with set of guests for each host?

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Basic CSP model

• What are the constraints?
• No guest boat visits a host boat twice
• For all i . All-different(Hi1,Hi2,..,Hit)

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Basic CSP model

• What are the constraints?
• No guest boat visits a host boat twice
• For all i . All-different(Hi1,Hi2,..,Hit)
• Host boat capacity is not exceeded

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Basic CSP model

• What are the constraints?
• No guest boat visits a host boat twice
• For all i . all-different(Hi1,Hi2,..,Hit)
• Host boat capacity is not exceeded
• Hmm, hosts are values to Hit variables so would
  need to look at all Hit variables simultaneously!

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Improving basic CSP model

• Is it worth introducing any auxiliary variables?
• Could we express capacity constraints via some
  auxiliary variables?

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Improving basic CSP model

• Is it worth introducing any auxiliary variables?
• Could we express capacity constraints via some
  auxiliary variables?
• Of course, I wouldnt be asking the question
  otherwise

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Improving basic CSP model

• Introduce auxiliary 0/1 variables that can be
  summed
• Vijt 1 iff Hitj

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Improving basic CSP model

• Introduce auxiliary 0/1 variables that can be
  summed
• Vijt 1 iff Hitj
• Capacity constraint now easy to specify
• Sum_i Vijt crew_i lt capacity_j
• where crew_i is crew size of boat j and
  capacity_j is capacity of host boat j

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What about other constraints?

• Still have to specify that no two guest boats
  meet twice

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What about other constraints?

• Still have to specify that no two guest boats
  meet twice
• Quantify over all pairs of guest, all host boats
  and all time periods
• O(g2 h t) constraints

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Improving basic CSP model

• Is it worth introducing any auxiliary variables?
• Could we express dont meet twice constraints
  more compactly/efficiently via some auxiliary
  variables?

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Improving basic CSP model

• Is it worth introducing any auxiliary variables?
• Could we express dont meet twice constraints
  more compactly/efficiently via some auxiliary
  variables?
• Yes, of course. I wouldnt be asking otherwise!

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Improving CSP model

• Introduce auxiliary 0/1 variables to represent
  meetings
• Mklt 1 iff HktHlt
• I.e. if guest k meets guest l at time t

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Improving CSP model

• Introduce auxiliary 0/1 variables to represent
  meetings
• Mklt 1 iff HktHlt
• I.e. if guest k meets guest l at time t
• Meeting constraints easily specified
• Sum_t Mklt lt 1

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Basic methodology

• Devise basic CSP model
• What are the variables? What are the constraints?
• Introduce auxiliary variables if needed
• Consider dual or combined models
• Break symmetry
• Introduce implied constraints

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Break symmetry

• Does the problem have any symmetry?

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Break symmetry

• Does the problem have any symmetry?
• Yes, of course. The world is full of symmetry!

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Break symmetry

• Host boats of the same capacity are symmetric
• Guest boats of the same crew size are symmetric
• Guest boat symmetry is more important than host
  boat. Why?

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Break symmetry

• Does the problem have any other symmetry?

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Break symmetry

• Does the problem have any other symmetry?
• Yes, of course. Time periods are symmetric

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Break symmetry

• How do we deal with symmetry of time periods?

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Break symmetry

• How do we deal with symmetry of time periods?
• Largest guest boat visits host boats in order
• Similar trick to assigning first row of
  orthogonal Latin square
• Why largest?

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Break symmetry

• Guest boats with the same crew size are symmetric
• How do we break this symmetry?

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Break symmetry

• Guest boats with the same crew size are symmetric
• If jltk have same crew size then
• Hj1 lt Hk1

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Break symmetry

• Guest boats with the same crew size are symmetric
• If jltk have same crew size then
• Hj1 lt Hk1
• They might still meet at t1!
• We cannot rule this out
• If they do, we still have this symmetry

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Break symmetry

• Guest boats with the same crew size are symmetric
• If jltk have same crew size then
• Hj1 lt Hk1
• They might still meet at t1!
• Hj1ltHk1 or Hj2ltHk2
• They cannot meet at t1 and t2

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Break symmetry

• Host boats with the same capacity are symmetric
• How do we break this symmetry?

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Break symmetry

• Host boats with the same capacity are symmetric
• Partial solution
• If jltk have identical capacity, then guest boat 1
  visits k implies it also visits j

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Break symmetry

• Host boats with the same capacity are symmetric
• Partial solution
• If jltk have identical capacity, then guest boat 1
  visits k implies it also visits j
• Can you improve on this?

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Solving choices

• Which variables to branch on?
• Hit, Vijt or Mijt ?

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Solving choices

• Which variables to branch on?
• Hit and let others follow
• What variable ordering heuristic

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Solving choices

• Which variables to branch on?
• Hit and let others follow
• What variable ordering heuristic
• Fail first
• What value ordering?

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Solving choices

• Which variables to branch on?
• Hit and let others follow
• What variable ordering heuristic
• Fail first
• What value ordering?
• Host boats by their spare capacity
• Alias succeed first

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Performance

• Party held before problem solved!
• 13 hosts, 29 guests, 6 time periods
• Heuristic tweaked to assign each period in turn
• Solution found for 7 period problem
• They could have partied longer
• 12 host problem easy to show insoluble
• Simple capacity argument

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Conclusions

• Constraint programming can improve your social
  life!
• Modelling is an art but there are patterns
• Develop basic model
• Use auxiliary variables to represent constraints
  compactly/efficiently
• Consider dual, combined and 0/1 models
• Break symmetry
• Add implied constraints