Golomb rulers Mark ticks on a ruler Distance between any two ticks (not just neighbouring ticks) is distinct Applications in radio-astronomy, cystallography, …
Golomb rulers Simple solution Exponentially long ruler Ticks at 0,1,3,7,15,31,63,… Goal is to find minimal length rulers turn optimization problem into sequence of satisfaction problems Is there a ruler of length m? Is there a ruler of length m-1? ….
Optimal Golomb rulers Known for up to 23 ticks Distributed internet project to find large rulers 0,1 0,1,3 0,1,4,6 0,1,4,9,11 0,1,4,10,12,17 0,1,4,10,18,23,25 Solutions grow as approximately O(n^2)
Modelling the Golomb ruler Variable, Xi for each tick Value is position on ruler Naïve model with quaternary constraints For all i>j,k>l>j |Xi-Xj| \= |Xk-Xl|
Problems with naïve model Large number of quaternary constraints O(n^4) constraints Looseness of quaternary constraints Many values satisfy |Xi-Xj| \= |Xk-Xl| Limited pruning
A better non-binary model Introduce auxiliary variables for inter-tick distances Dij = |Xi-Xj| O(n^2) ternary constraints Post single large non-binary constraint alldifferent([D11,D12,…]). Tighter constraints and denser constraint graph
Other modeling issues Symmetry A ruler can always be reversed! Break this symmetry by adding constraint: D12 < Dn-1,n Also break symmetry on Xi X1 < X2 < … Xn Such tricks important in many problems
Other modelling issues Additional (implied) constraints Don’t change set of solutions But may reduce search significantly E.g. D12 < D13, D23 < D24, … E.g. D1k at least sum of first k integers Pure declarative specifications are not enough!
Solving issues Labeling strategies often very important Smallest domain often good idea Focuses on “hardest” part of problem Best strategy for Golomb ruler is instantiate variables in strict order Heuristics like fail-first (smallest domain) not effective on this problem!
Model (re)formulation Can we automate the process of refining and improving a model? Identify and breaking symmetries Inferring implied constraints … CGRASS proof planner hopes to do this!
The `Introduce` Method Preconditions 1.Exp occurs more than once in the constraint set. 2.someVariable = Exp not already present. Post-conditions 1.Generate new var, x, domain defined by Exp. 2.Add constraint: x = Exp. Motivation: solver propagates through newly introduced variable
Other automatic methods Symmetry identification and removal Order ticks Variable elimination Generalization of Gaussian elimination Automatically turns naïve Golomb model into a reasonable one!
Something to try at home? Circular (or modular) Golomb rulers Inter-tick distance variables more central, removing rotational symmetry? 2-d Golomb rulers All examples of “graceful” graphs
Summary Benefits of non-binary constraints Compact, declarative models Efficient and effective constraint propagation Supported by many constraint toolkits alldifferent, atmost, cardinality, … Large space of models Non-binary constraints only make this worse? Automatic tools can help with model selection and reformulation
Summary (II) Modelling decisions: Auxiliary variables Implied constraints Symmetry breaking constraints More to constraints than just declarative problem specifications!
Case study II: permutation problems Wide variety of scheduling, assignment and routing problems involve finding a permutation Plus satisfying some additional constraints Illustrates some of the fundamental decisions that must be made when modelling!
Modelling choices Need to decide variables, domains and constraints Often difficult choice for something even as basic as the decision variables? E.g. consider scheduling the World Cup. Are vars=games, vals=times or vars=times, vals=games ?
Permutation problems |vars|=|vals| each var has unique val many examples scheduling timetabling routing assignment problems permute vars for vals which do we choose? TSP problem = find permutation of cities which makes a tour of minimum length
Meta-motivation Methodology for comparing models based on definition of constraint tightness Other applications comparing implied constraints impact of reformulation
Comparing models Choice of model affects amount of constraint propagation tighter model more pruning and propagation Need dynamic tightness measure domains shrink as we descend down search tree other constraints may prune domains
Constraint tightness Pruning depends on local consistency enforced higher consistencies will infer implied constraints missing from looser models Introduce tightness measure: parameterized by level of consistency enforced considers domains changing in size
Definition of tightness model 1 is as tight as model 2 wrt A-consistency iff given any domains model 1 is A-consistent -> model 2 is A-consistent written A 1 A 2
How does this compare to the previous definition? Tightness measure introduced by Debruyne & Bessiere [IJCAI- 97] A-consistency is tighter than B-consistency iff given any model the model is A-consistent -> the model is B-consistent We fix models but vary domains
Properties of new ordering Partial ordering reflexive A1 A1 transitive A1 A2 & A2 A3 implies A1 A3 Defined relations tighter A 1 A 2 iff A 1 A 2 & not A 2 A 1 equivalence A 1 = A 2 iff A 1 A 2 & A 2 A 1 incomparable A A 2 iff neither A 1 A 2 nor A 2 A 1
Further properties Monotonicity AC 1u2 AC 1 AC 1n2 adding constraints can only tighten a model Fixed point AC 1 AC 2 implies AC 1u2 = AC 1 combining a looser model with a tighter model doesn’t help!
Extensions Ordering extends to search algorithms E.g. MAC 1 MAC 2 iff, given any domains, MAC on model 1 visits no more nodes than MAC on model 2 assume equivalent var and val ordering Similar monotonicity and fixed point properties Ordering extends to different consistencies applied to the different models E.g. GAC 1 AC 2
Permutation models primal model x i x j for all i,j primal alldiff model alldifferent(x1,x2,…) primal/dual models (dual) variables associated with each (primal) value “Modelling a Permutation Problem”, Barbara M Smith, ECAI'2000 Workshop on Modelling and Solving Problems with Constraints
Primal model n primal variables each with n values O(n^2) binary constraints x i x j x5x5 x6x6 x1x1 x2x2 x3x3 x4x4
Primal all-different model n primal variables each with n values one non-binary constraint all-different( x 1, x 2,..) x6x6 x5x5 x4x4 x3x3 x2x2 x1x1
Primal/dual model n primal variables each with n values n dual variables one for each primal value each dual value associated with a primal variable n^2 channelling constraints x i= j iff d j =i no other constraints needed! x5x5 x1x1 x2x2 x3x3 x4x4 d5d5 d1d1 d2d2 d3d3 d4d4
Other reason to channel Channelling between models frequent modelling technique Some constraints easier to specify in one model Others easier to specify in a (dual) model Channelling maintains consistency between the 2 models
Multiple permutation problems Some problems consist of several permutations order n quasigroup (or Latin square) has 2n intersecting permuations each of size n Following results extend to such cases
Some notation primal not equals primal alldiff c channelling constraints c channelling constraints & primal not equals c channelling constraints & primal alldiff c channelling constraints & primal/dual not equals...
SAT models n Boolean vars, X ij true iff x i =j primal SAT model O(n) clauses, each var takes at least one val O(n^3) clauses, no primal var takes two vals O(n^3) clauses, no two primal vars take same val channelling SAT model no need for (dual) Boolean vars as Xij can be used again [Gomes et al 2001] and [Bejar & Manya 2000] report promising experimental results using SAT models of permutation problems with the Davis Putnam (DP) tree search algorithm
Theoretical results MAC tighter than DP, DP as tight as FC
Asymptotic results Tightness ordering reflects asymptotic cost E.g. GAC AC c AC O(n^4) O(n^3) O(n^2) In each case, using best known algorithm Hence we need to run experiments to know if extra pruning is worth the cost!
Experimental results [Smith 2000] reports promising results for MAC c on Langford’s problem I studied 3 other permutation problems all interval series circular Golomb rulers quasigroups (Latin squares) All 4 problems in
Experimental results using Sicstus FD constraint library channelling is a standard non-binary constraint MACc best on “looser” problems Langford’s problem, all interval series Maintaining GAC better on “tighter” problems circular Golomb rulers, mean (but not median) quasigroup performance
Extensions injective mappings more vals than vars, each var takes unique val can introduce dummy vars to make permutation channelling constraints useful in many other problems often (but not always) bijective propagate pruning rapidly between models
Conclusions Many ways to model and solve even something as simple as a permutation problem Hard even to decide what are the variables! Theory and experiment needed to compare models properly Dominance/asymptotic results only take us so far Best model may not be a single model! Channel between two (or more?) models