CSCE 580 Artificial Intelligence Ch.5: Constraint Satisfaction Problems

CSCE 580Artificial IntelligenceCh.5: Constraint Satisfaction Problems Fall 2008 Marco Valtorta mgv@cse.sc.edu

Acknowledgment • The slides are based on the textbook [AIMA] and other sources, including other fine textbooks and the accompanying slide sets • The other textbooks I considered are: • David Poole, Alan Mackworth, and Randy Goebel. Computational Intelligence: A Logical Approach. Oxford, 1998 • A second edition (by Poole and Mackworth) is under development. Dr. Poole allowed us to use a draft of it in this course • Ivan Bratko. Prolog Programming for Artificial Intelligence, Third Edition. Addison-Wesley, 2001 • The fourth edition is under development • George F. Luger. Artificial Intelligence: Structures and Strategies for Complex Problem Solving, Sixth Edition. Addison-Welsey, 2009

Constraint satisfaction problems (CSPs) • Standard search problem: • state is a "black box“ – any data structure that supports successor function, heuristic function, and goal test • CSP: • state is defined by variablesXi with values from domainDi • goal test is a set of constraints specifying allowable combinations of values for subsets of variables • Simple example of a formal representation language • Allows useful general-purpose algorithms with more power than standard search algorithms

Example: Map-Coloring • VariablesWA, NT, Q, NSW, V, SA, T • DomainsDi = {red,green,blue} • Constraints: adjacent regions must have different colors • e.g., WA ≠ NT, or (WA,NT) in {(red,green),(red,blue),(green,red), (green,blue),(blue,red),(blue,green)}

Example: Map-Coloring • Solutions are complete and consistent assignments, e.g., WA = red, NT = green,Q = red,NSW = green,V = red,SA = blue,T = green

Constraint graph • Binary CSP: each constraint relates two variables • Constraint graph: nodes are variables, arcs are constraints

Varieties of CSPs • Discrete variables • finite domains: • n variables, domain size d  O(dn) complete assignments • e.g., Boolean CSPs, incl.~Boolean satisfiability (NP-complete) • infinite domains: • integers, strings, etc. • e.g., job scheduling, variables are start/end days for each job • need a constraint language, e.g., StartJob1 + 5 ≤ StartJob3 • Continuous variables • e.g., start/end times for Hubble Space Telescope observations • linear constraints solvable in polynomial time by linear programming

Varieties of constraints • Unary constraints involve a single variable, • e.g., SA ≠ green • Binary constraints involve pairs of variables, • e.g., SA ≠ WA • Higher-order constraints involve 3 or more variables, • e.g., cryptarithmetic column constraints

Example: Cryptarithmetic • Variables: F T U W R O X1 X2 X3 • Domains: {0,1,2,3,4,5,6,7,8,9} • Constraints: Alldiff (F,T,U,W,R,O) • O + O = R + 10 ·X1 • X1 + W + W = U + 10 · X2 • X2 + T + T = O + 10 · X3 • X3 = F, T ≠ 0, F≠ 0

Real-world CSPs • Assignment problems • e.g., who teaches what class • Timetabling problems • e.g., which class is offered when and where? • Transportation scheduling • Factory scheduling • Notice that many real-world problems involve real-valued variables

Standard search formulation (incremental) Let's start with the straightforward approach, then fix it States are defined by the values assigned so far • Initial state: the empty assignment { } • Successor function: assign a value to an unassigned variable that does not conflict with current assignment  fail if no legal assignments • Goal test: the current assignment is complete • This is the same for all CSPs • Every solution appears at depth n with n variables use depth-first search • Path is irrelevant, so can also use complete-state formulation • b = (n – l)d at depth l, hence n! ·dn leaves The result in (4) is grossly pessimistic, because the order in which values are assigned to variables does not matter. There are only dn assignments.

Backtracking search • Variable assignments are commutative}, i.e., [ WA = red then NT = green ] same as [ NT = green then WA = red ] • Only need to consider assignments to a single variable at each node  b = d and there are dn leaves • Depth-first search for CSPs with single-variable assignments is called backtracking search • Backtracking search is the basic uninformed algorithm for CSPs • Can solve n-queens for n≈ 25

Backtracking search

Backtracking example

Improving backtracking efficiency • General-purpose methods can give huge gains in speed: • Which variable should be assigned next? • In what order should its values be tried? • Can we detect inevitable failure early?

Most constrained variable • Most constrained variable: choose the variable with the fewest legal values • a.k.a. minimum remaining values (MRV) heuristic

Most constraining variable • Tie-breaker among most constrained variables • Most constraining variable: • choose the variable with the most constraints on remaining variables

Least constraining value • Given a variable, choose the least constraining value: • the one that rules out the fewest values in the remaining variables • Combining these heuristics makes 1000 queens feasible

Forward checking • Idea: • Keep track of remaining legal values for unassigned variables • Terminate search when any variable has no legal values

Constraint propagation • Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures: • NT and SA cannot both be blue! • Constraint propagation repeatedly enforces constraints locally

Arc consistency • Simplest form of propagation makes each arc consistent • X Y is consistent iff for every value x of X there is some allowed y

Arc consistency • Simplest form of propagation makes each arc consistent • X Y is consistent iff for every value x of X there is some allowed y • If X loses a value, neighbors of X need to be rechecked

Arc consistency • Simplest form of propagation makes each arc consistent • X Y is consistent iff for every value x of X there is some allowed y • If X loses a value, neighbors of X need to be rechecked • Arc consistency detects failure earlier than forward checking • Can be run as a preprocessor or after each assignment

Arc consistency algorithm AC-3 • Time complexity: O(n2d3), where n is the number of variables and d is the maximum variable domain size, because: • At most O(n2) arcs • Each arc can be inserted into the agenda (TDA set) at most d times • Checking consistency of each arc can be done in O(d2) time

Generalized Arc Consistency Algorithm • Three possible outcomes: • One domain is empty => no solution • Each domain has a single value => unique solution • Some domains have more than one value => there may or may not be a solution • If the problem has a unique solution, GAC may end in state (2) or (3); otherwise, we would have a polynomial-time algorithm to solve UNIQUE-SAT • UNIQUE-SAT or USAT is the problem of determining whether a formula known to have either zero or one satisfying assignments has zero or has one. Although this problem seems easier than general SAT, if there is a practical algorithm to solve this problem, then all problems in NP can be solved just as easily [Wikipedia; L.G. Valiant and V.V. Vazirani, NP is as Easy as Detecting Unique Solutions. Theoretical Computer Science, 47(1986), 85-94.] • Thanks to Amber McKenzie for asking a question about this!

Local search for CSPs • Hill-climbing, simulated annealing typically work with "complete" states, i.e., all variables assigned • To apply to CSPs: • allow states with unsatisfied constraints • operators reassign variable values • Variable selection: randomly select any conflicted variable • Value selection by min-conflicts heuristic: • choose value that violates the fewest constraints • i.e., hill-climb with h(n) = total number of violated constraints

Local search for CSP function MIN-CONFLICTS(csp, max_steps) return solution or failure inputs: csp, a constraint satisfaction problem max_steps, the number of steps allowed before giving up current an initial complete assignment for csp for i= 1 to max_stepsdo ifcurrent is a solution for csp then return current var a randomly chosen, conflicted variable from VARIABLES[csp] value the value v for var that minimize CONFLICTS(var,v,current,csp) set var = value in current return failure

Example: 4-Queens • States: 4 queens in 4 columns (44 = 256 states) • Actions: move queen in column • Goal test: no attacks • Evaluation: h(n) = number of attacks • Given random initial state, can solve n-queens in almost constant time for arbitrary n with high probability (e.g., n = 10,000,000)

Min-conflicts example 2 h=5 h=3 h=1 • Use of min-conflicts heuristic in hill-climbing

Min-conflicts example 3 • A two-step solution for an 8-queens problem using min-conflicts heuristic • At each stage a queen is chosen for reassignment in its column • The algorithm moves the queen to the min-conflict square breaking ties randomly

Advantages of local search • The runtime of min-conflicts is roughly independent of problem size. • Solving the millions-queen problem in roughly 50 steps. • Local search can be used in an online setting. • Backtrack search requires more time

Summary • CSPs are a special kind of problem: • states defined by values of a fixed set of variables • goal test defined by constraints on variable values • Backtracking = depth-first search with one variable assigned per node • Variable ordering and value selection heuristics help significantly • Forward checking prevents assignments that guarantee later failure • Constraint propagation (e.g., arc consistency) does additional work to constrain values and detect inconsistencies • Iterative min-conflicts is usually effective in practice

Problem structure • How can the problem structure help to find a solution quickly? • Subproblem identification is important: • Coloring Tasmania and mainland are independent subproblems • Identifiable as connected components of constrained graph. • Improves performance

Problem structure • Suppose each problem has c variables out of a total of n. • Worst case solution cost is O(n/c dc), i.e. linear in n • Instead of O(d n), exponential in n • E.g. n= 80, c= 20, d=2 • 280 = 4 billion years at 1 million nodes/sec. • 4 * 220= .4 second at 1 million nodes/sec

Tree-structured CSPs • Theorem: if the constraint graph has no loops then CSP can be solved in O(nd 2) time • Compare difference with general CSP, where worst case is O(d n)

Tree-structured CSPs • In most cases subproblems of a CSP are connected as a tree • Any tree-structured CSP can be solved in time linear in the number of variables. • Choose a variable as root, order variables from root to leaves such that every node’s parent precedes it in the ordering. (label var from X1 to Xn) • For j from n down to 2, apply REMOVE-INCONSISTENT-VALUES(Parent(Xj),Xj) • For j from 1 to n assign Xj consistently with Parent(Xj )

Nearly tree-structured CSPs • Can more general constraint graphs be reduced to trees? • Two approaches: • Remove certain nodes • Collapse certain nodes

Nearly tree-structured CSPs • Idea: assign values to some variables so that the remaining variables form a tree. • Assume that we assign {SA=x}  cycle cutset • And remove any values from the other variables that are inconsistent. • The selected value for SA could be the wrong one so we have to try all of them

Nearly tree-structured CSPs • This approach is worthwhile if cycle cutset is small. • Finding the smallest cycle cutset is NP-hard • Approximation algorithms exist • This approach is called cutset conditioning.

Nearly tree-structured CSPs • Tree decomposition of the constraint graph in a set of connected subproblems. • Each subproblem is solved independently • Resulting solutions are combined. • Necessary requirements: • Every variable appears in at least one of the subproblems • If two variables are connected in the original problem, they must appear together in at least one subproblem • If a variable appears in two subproblems, it must appear in each node on the path

Summary • CSPs are a special kind of problem: states defined by values of a fixed set of variables, goal test defined by constraints on variable values • Backtracking=depth-first search with one variable assigned per node • Variable ordering and value selection heuristics help significantly • Forward checking prevents assignments that lead to failure. • Constraint propagation does additional work to constrain values and detect inconsistencies. • The CSP representation allows analysis of problem structure. • Tree structured CSPs can be solved in linear time. • Iterative min-conflicts is usually effective in practice.

Dynamic Programming Dynamic programming is a problem solving method which is especially useful to solve the problems to which Bellman’s Principle of Optimality applies: “An optimal policy has the property that whatever the initial state and the initial decision are, the remaining decisions constitute an optimal policy with respect to the state resulting from the initial decision.” The shortest path problem in a directed staged network is an example of such a problem

Shortest-Path in a Staged Network The principle of optimality can be stated as follows: If the shortest path from 0 to 3 goes through X, then: 1. that part from 0 to X is the shortest path from 0 to X, and 2. that part from X to 3 is the shortest path from X to 3 The previous statement leads to a forward algorithm and a backward algorithm for finding the shortest path in a directed staged network

CSCE 580 Artificial Intelligence Ch.5: Constraint Satisfaction Problems