Constrainedness Including slides from Toby Walsh

Constraint satisfaction Constraint satisfaction problem (CSP) is a triple where: –V is set of variables –Each X in V has set of values, D_X Usually assume finite domain {true,false}, {red,blue,green}, [0,10], … –C is set of constraints Goal: find assignment of values to variables to satisfy all the constraints

Constraint solver Tree search –Assign value to variable –Deduce values that must be removed from future/unassigned variables Constraint propagation –If any future variable has no values, backtrack else repeat Number of choices –Variable to assign next, value to assign Some important refinements like nogood learning, non-chronological backtracking, …

Constraint propagation Arc-consistency (AC) –A binary constraint r(X1,X2) is AC iff for every value for X1, there is a consistent value (often called support) for X2 and vice versa E.g. With 0/1 domains and the constraint X1 =/= X2 Value 0 for X1 is supported by value 1 for X2 Value 1 for X1 is supported by value 0 for X2 … –A problem is AC iff every constraint is AC

Tree search Backtracking (BT) Forward checking (FC) Backjumping (BJ, CBJ, DB) Maintaining arc-consistency (MAC) Limited discrepancy search (LDS) Non-chronological backtracking & learning Probing …

Modelling Choose a basic model Consider auxiliary variables –To reduce number of constraints, improve propagation Consider combined models –Channel between views Break symmetries Add implied constraints –To improve propagation

Propositional Satisfiability SAT –does a truth assignment exist that satisfies a propositional formula? –special type of constraint satisfaction problem Variables are Boolean Constraints are formulae –NP-complete 3-SAT –formulae in clausal form with 3 literals per clause –remains NP-complete (x1 v x2) & (-x2 v x3 v -x4) x1/ True, x2/ False,...

Random 3-SAT –sample uniformly from space of all possible 3- clauses –n variables, l clauses Which are the hard instances? –around l/n = 4.3 What happens with larger problems? Why are some dots red and others blue?

Random 3-SAT Varying problem size, n Complexity peak appears to be largely invariant of algorithm –backtracking algorithms like Davis-Putnam –local search procedures like GSAT What’s so special about 4.3?

Random 3-SAT Complexity peak coincides with solubility transition –l/n < 4.3 problems under- constrained and SAT –l/n > 4.3 problems over- constrained and UNSAT –l/n=4.3, problems on “knife-edge” between SAT and UNSAT

Shape –“Sharp” (in a technical sense) [Friedgut 99] Location –2-SAT occurs at l/n=1 [Chavatal & Reed 92, Goerdt 92] –3-SAT occurs at 3.26 < l/n < Theoretical results

“But it doesn’t occur in X?” X = some NP-complete problem X = real problems X = some other complexity class

“But it doesn’t occur in X?” X = some NP-complete problem Phase transition behaviour seen in: –TSP problem (decision not optimization) –Hamiltonian circuits (but NOT a complexity peak) –number partitioning –graph colouring –independent set –...

“But it doesn’t occur in X?” X = real problems Phase transition behaviour seen in: –job shop scheduling problems –TSP instances from TSPLib –exam Edinburgh –Boolean circuit synthesis –Latin squares (alias sports scheduling) –...

“But it doesn’t occur in X?” X = some other complexity class Phase transition behaviour seen in: –polynomial problems like arc-consistency –PSPACE problems like QSAT and modal K –...

Algorithms at the phase boundary What do we understand about problem hardness at the phase boundary? How can this help build better algorithms?

Kappa Defined for an ensemble of problems

Kappa (motivation)

Kappa When every state is a solution When every state is not a solution When there is a unique solution

Kappa When every state is a solution When every state is not a solution When there is a unique solution Easy Soluble Easy Insoluble On the knife edge Hard

An Example: random CSP’s Each of n variables has a uniform domain size m There is a probability p1 of a constraint between a pair of variables There is a probability p2 that a pair of values conflict (when a constraint exists)

An Example: random CSP’s

Can rearrange this formula so that we find the value of p2 such that we are at the phase transition, i.e. when kappa = 1 Then perform experiments to see if it is accurate predictor i.e. we put theory to the test

Kappa as a heuristic Minimise that

An Example: not random CSP’s

Kappa is general (3 examples) SAT with n variables, l clauses, a literals/clause Graph colouring, n vertices, e edges, m colours Number partitioning, n numbers, range (0,l], m bags with same sum

Kappa as a heuristic Minimise that

assume we only measure domain size (denominator) assume we only measure top line (numerator) assume we measure top and bottom line (numerator and denominator But that’s costly to do. Is there a low cost surrogate?

Looking inside search Three key insights –constrainedness “knife- edge” –backbone structure –2+p-SAT Suggests branching heuristics –also insight into branching mistakes

Constrainedness knife-edge kappa against depth/n

Constrainedness knife-edge Seen in other problem domains –number partitioning, … Seen on “real” problems –exam timetabling (alias graph colouring) Suggests branching heuristic –“get off the knife-edge as quickly as possible” –minimize or maximize-kappa heuristics must take into account branching rate, max-kappa often therefore not a good move!

Minimize constrainedness Many existing heuristics minimize-kappa –or proxies for it For instance –Karmarkar-Karp heuristic for number partitioning –Brelaz heuristic for graph colouring –Fail-first heuristic for constraint satisfaction –… Can be used to design new heuristics –removing some of the “black art”

Backbone Variables which take fixed values in all solutions –alias unit prime implicates Let f k be fraction of variables in backbone –l/n < 4.3, f k vanishing (otherwise adding clause could make problem unsat) –l/n > 4.3, f k > 0 discontinuity at phase boundary!

Backbone Search cost correlated with backbone size –if f k non-zero, then can easily assign variable “wrong” value –such mistakes costly if at top of search tree Backbones seen in other problems –graph colouring –TSP –… Can we make algorithms that identify and exploit the backbone structure of a problem?

2+p-SAT 2-SAT is polynomial (linear) but 3-SAT is NP-complete –2-SAT, unlike 3-SAT, has no backbone discontinuity Morph between 2-SAT and 3- SAT –fraction p of 3-clauses –fraction (1-p) of 2-clauses 2+p-SAT maps from P to NP –p>0, 2+p-SAT is NP-complete