1	Constraint Satisfaction Problems Chapter 5 Sections 1 – 3
2	Outline Constraint Satisfaction Problems (CSP) Backtracking search for CSPs Local search for CSPs
3	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 variables X_i with values from domain D_i 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
4	Example: Map-Coloring Variables WA, NT, Q, NSW, V, SA, T Domains D_i = {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)}
5	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
6	Constraint graph Binary CSP: each constraint relates two variables Constraint graph: nodes are variables, arcs are constraints
7	Varieties of CSPs Discrete variables finite domains: n variables, domain size d à 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
8	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
9	Example: Task Scheduling
10	Example: Cryptarithmetic Variables: F T U W R O X₁ X₂X₃ 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
11	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
12	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 à 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 à use depth-first search Path is irrelevant, so can also use complete-state formulation
13	CSP Search tree size b = (n - l )d at depth l, hence n! · dⁿ leaves
14	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 Fix an order in which we’ll examine the variables à b = d and there are dⁿ leaves Depth-first search for CSPs with single-variable assignments is called backtracking search Is the basic uninformed algorithm for CSPs Can solve n-queens for n ≈ 25
15	Backtracking search
16	Backtracking example
17	Backtracking example
18	Backtracking example
19	Backtracking example
20	Exercise - paint the town! Districts across corners can be colored using the same color.
21	Constraint Graph
22	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?
23	Most constrained variable Most constrained variable: choose the variable with the fewest legal values a.k.a. minimum remaining values (MRV) heuristic
24	Most constraining variable Tie-breaker among most constrained variables Most constraining variable: choose the variable with the most constraints on remaining variables
25	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
26	Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values
27	Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values
28	Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values
29	Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values
30	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
31	Arc consistency Simplest form of propagation makes each arc consistent X àY is consistent iff for every value x of X there is some allowed y
32	More on arc consistency Arc consistency is based on a very simple concept if we can look at just one constraint and see that x=v is impossible … obviously we can remove the value x=v from consideration How do we know a value is impossible? If the constraint provides no support for the value e.g. if D_x = {1,4,5} and D_y = {1, 2, 3} then the constraint x > y provides no support for x=1 we can remove x=1 from D_x
33	Arc consistency Simplest form of propagation makes each arc consistent X àY is consistent iff for every value x of X there is some allowed y Arcs are directed, a binary constraint becomes two arcs NSW Þ SA arc originally not consistent, is consistent after deleting blue
34	Arc consistency Simplest form of propagation makes each arc consistent X àY 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 (re)checked
35	Arc Consistency Propagation When we remove a value from D_x, we may get new removals because of it E.g. D_x = {1,4,5}, D_y = {1, 2, 3}, D_z= {2, 3, 4, 5} x > y, z > x As before we can remove 1 from D_x, so D_x= {4,5} But now there is no support for D_z = 2,3,4 So we can remove those values, D_z= {5}, so z=5 Before AC applied to y-x, we could not change D_z This can cause a chain reaction
36	Arc consistency Simplest form of propagation makes each arc consistent X àY 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 (re)checked Arc consistency detects failure earlier than forward checking Can be run as a preprocessor or after each assignment
37	Arc consistency algorithm AC-3 Time complexity: O(n²d³)
38	Time complexity of AC-3 CSP has n²directed arcs Each arc X_i,X_jhas d possible values. For each value we can reinsert the neighboring arc X_k,X_iat most d times because X_i has d values Checking an arc requires at most d² time O(n² * d * d²) = O(n²d³)
39	Maintaining AC (MAC) Like any other propagation, we can use AC in search i.e. search proceeds as follows: establish AC at the root when AC3 terminates, choose a new variable/value re-establish AC given the new variable choice (i.e. maintain AC) repeat; backtrack if AC gives domain wipe out The hard part of implementation is undoing effects of AC
40	Special kinds of Consistency Some kinds of constraint lend themselves to special kinds of arc-consistency Consider the all-different constraint the named variables must all take different values not a binary constraint can be expressed as n(n-1)/2 not-equals constraints We can apply (e.g.) AC3 as usual But there is a much better option
41	All Different Suppose D_x= {2,3} = D_y, D_z = {1,2,3} All the constraints x¹y, y¹z, z¹x are all arc consistent e.g. x=2 supports the value z = 3 The single ternary constraint AllDifferent(x,y,z) is not! We must set z = 1 A special purpose algorithm exists for All-Different to establish GAC in efficient time Special purpose propagation algorithms are vital
42	K-consistency Arc Consistency (2-consistency) can be extended to k-consistency 3-consistency (path consistency): any pair of adjacent variables can always be extended to a third neighbor. Catches problem with D_x, D_y and D_z, as assignment of Dz = 2 and Dx = 3 will lead to domain wipe out. But is expensive, exponential time n-consistency means the problem is solvable in linear time As any selection of variables would lead to a solution In general, need to strike a balance between consistency and search. This is usually done by experimentation.
43	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
44	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)
45	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
46	Midterm test Five questions, first hour of class (be on time!) Topics to be covered (CSP is not on the midterm): Chapter 2 – Agents Chapter 3 – Uninformed Search Chapter 4 – Informed Search Not including the parts of 4.1 (memory-bounded heuristic search) and 4.5 Chapter 6 – Adversarial Search Not including 6.5 (games with chance)
47	Homework #1 Due today by 23:59:59 in the IVLE workbin. Late policy given on website. Only one submission will be graded, whichever one is latest. Your tagline is used to generate the ID to identify your agent on the scoreboard. If you don’t have an existing account fill out: https://mysoc.nus.edu.sg/~eform/new and send me e-mail ASAP.
48	Checklist for HW #1 Does it compile? Is my code in a single file? Did I comment my code so that it’s understandable to the reader? Is the main class appropriately named? Did I place a unique tagline so I can identify my player on the scoreboard?