CS 4700: Foundations of Artificial Intelligence

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CS 4700Foundations of Artificial Intelligence

• Carla P. Gomes
• gomes_at_cs.cornell.edu
• Module
• CSP2
• (Reading RN Chapter 5)

2

• Encodings
• Global Constraints
• Redundant Constraints
• Streamlining
• Complexity

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• Case Study

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Latin Squares An Abstraction for Real World
Applications
Given an N X N matrix, and given N colors, a
Latin Square of order N is a colored matrix,
such that -all cells are colored. - each
color occurs exactly once in each row. - each
color occurs exactly once in each column.
Latin Square (Order 4)
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Fiber Optic Networks
Nodes connect point to point fiber optic links

• Wavelength
• Division
• Multiplexing (WDM)
• the most promising
• technology for the
• next generation of
• wide-area
• backbone networks.

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Routing in Fiber Optic Networks
Input Ports
Output Ports
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1
2
2
3
3
4
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Routing Node
How can we achieve conflict-free routing in each
node of the network?
Dynamic wavelength routing is an NP-hard problem.
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Application Example Routers in Fiber Optic
Networks
Dynamic wavelength routing in Fiber Optic
Networks can be directly mapped into the Latin
Square Completion Problem.
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Design of Statistical Experiments

• We have 5 treatments for growing beans. We want
  to know what treatments are effective in
  increasing yield, and by how much.
• The objective is to eliminate bias and distribute
  the treatments somewhat evenly over the test
  plot.
• Latin Square Analysis of Variance

A D E B C C
B A E D D
C B A E E A
C D B B E
D C A
() Already in use in this sub-plot
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Underlying Latin Square structure Characterizes
many real world applications
Scheduling and timetabling
Sudoku
Many more applications
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Latin Squares An Abstraction for Real World
Applications
Given an N X N matrix, and given N colors, a
Latin Square of order N is a colored matrix,
such that -all cells are colored. - each
color occurs exactly once in each row. - each
color occurs exactly once in each column.
How do you generate a Latin square, say a 5x5? Is
it computationally hard?
Latin Square (Order 4)
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Completing Latin Squares A Hard Computational
Problem
Given a partial assignment of colors (10 colors
in this case), can the partial quasigroup (latin
square) be completed so we obtain a full
quasigroup? Example
Deciding if a Latin Square is completable is
NP-Complete
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How to Encode the Latin Square Completionas a
CSP?
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Latin Square Completion as a Binary CSP
How do we represent the Latin Square problem as
a binary CSP?
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Domain Reduction in LSCP
Forward Checking
Arc Consistency
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Other Examples of Applications of Graph Coloring
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Scheduling of Final Exams
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2
7
3
6
5
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What are the constraints between courses? Find a
valid coloring
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Scheduling of Final Exams

• How can the final exams at Cornell be scheduled
  so that no student has two exams at the same
  time?
• A vertex correspond to a course
• An edge between two vertices denotes that there
  is at least one common
• Student in the courses they represent
• Each time slot for a final exam is represented by
  a different color.
• A coloring of the graph corresponds to a valid
  schedule of the exams.

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Frequency Assignments

• T.V. channels 2 through 13 are assigned to
  stations in North
• America so that no no two stations within 150
  miles can
• operate on the same channel. How can the
  assignment of
• channels be modeled as a graph coloring?

• A vertex corresponds to one station
• There is a edge between two vertices if they are
  located within 150 miles
• of each other
• Coloring of graph --- corresponds to a valid
  assignment of channels
• each color represents a different channel.

Many other problems can be represented as graph
coloring
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Index Registers

• In efficient compilers the execution of loops can
  be sped
• up by storing frequently used variables
  temporarily in
• index registers in the central processing unit,
  instead of
• the regular memory. For a given loop, how many
  index registers are needed?

• Each vertex corresponds to a variable in the
  loop.
• An edge between two vertices denotes the fact
  that the corresponding variables must be stored
  in index registers at the same time during the
  execution of the loop.
• Chromatic number of the graph gives the number of
  index registers needed.

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Other Representations
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Latin Square Completion as an n-ary CSP

• Variables -
• Constraints -

Why would we use this formulation?
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Exploiting Structure for Domain Reduction

• A very successful strategy for domain reduction
  in CSP is to exploit the structure of groups of
  constraints and treat them as a sub-problem.
• Example using Network Flow Algorithms
• All-different constraints (Alldiff)

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Exploiting Structure in QCP ALLDIFF as Global
Constraint
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Exploiting Structure Arc Consistency vs. All Diff
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Global Constraints alldiff

• Example
• X1 1,2,3 X6 1,2,3,4,5,6,7,8,9
• X2 1,2,3,4,5,6 X7 1,2,3,4,5,6,7,8,9
• X3 1,2,3,4,5,6,7,8,9 X8 1,2,3
• X4 1,2,3,4,5,6 X9 1,2,3,4,5,6
• X5 1,2,3
• It is clear that constraint propagation based on
  maintenance of node-, arc- or even
  path-consistency would not eliminate any
  redundant label.
• Yet, it is very easy to infer such elimination
  with a global view of the constraint!

Example from Pedro Barahona
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Global Constraints alldiff

• Variables X1, X5 and X8 may only take values 1, 2
  and 3. Since there are 3 values for 3 variables,
  these must be assigned these values which must
  then be removed from the domain of the other
  variables.
• Now, variables X2, X4 and X9 may only take values
  4, 5 e 6, that must be removed from the other
  variables domains.

X1 1,2,3 X2 1,2,3,4,5,6 X3
1,2,3,4,5,6,7,8,9 X4 1,2,3,4,5,6 X5 1,2,3
X6 1,2,3,4,5,6,7,8,9 X7 1,2,3,4,5,6,7,8,9 X8
1,2,3 X9 1,2,3,4,5,6
X1 1,2,3 X2 1,2,3,4,5,6 X3
1,2,3,4,5,6,7,8,9 X4 1,2,3, 4,5,6 X5 1,2,3
X6 1,2,3,4,5,6,7,8,9 X7 1,2,3,4,5,6,7,8,9 X8
1,2,3 X9 1,2,3,4,5,6
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Global Constraints alldiff

• In this case, these prunings could be obtained,
  by maintaining (strong) 4-consistency.
• For example, analysing variables X1, X2, X5 and
  X8, it would be easy to verify that from the d4
  potential assignments of values to them, no
  assignment would include X21, X22, nor X23,
  thus leading to the prunning of X2 domain.
• However, such maintenance is usually very
  expensive, computationally. For each combination
  of 4 variables, d4 tuples whould be checked, with
  complexity O(d4).
• In fact, in some cases, n-strong consistency
  would be required, so its naïf maintenance would
  be exponential on the number of variables,
  exactly what one would like to avoid in search!

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Global Constraints all_diff

• Key Ideas
• Formulate the problem as Matching on a
  bipartite graph
• For each variable-value pair, there is an arc in
  the bipartite graph.
• A subset of arcs of a graph is called a
  matching, if no two arcs have a vertex in
• common. A matching of maximum cardinality is
  called a maximum matching.
• Note that when we have a bipartite graph a
  matchibng that covers all the nodes is a
• maximum matching.
• For any solution of the all_diff constraint there
  is one and only one maximum
• matching.

See Regin AAAI 1994
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Global Constraints all_diff

• Example A,B 1..2, C 1..3, D 2..5, E
  3..6,
• all_diff(A,B,C,D,E).

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Global Constraints all_diff
Given a CSP P, it is diff-arc-consistent iff
every arc belongs to a matching which covers
all the nodes of the graph associated with P.
Given a matching M, an arc in M is a matching
arc. Every arc not in M is free. A node or vertex
is matched if it is incident to a matching arc
and free otherwise. An alternating path or
cycle is a simple path or cycle whose arcs or
edges are alternately matching and free. The
length of an alternating path or cycle is the
number of arcs or edges it contains. An arc or
edge which belongs to every matching is vital.
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Global Constraints all_diff
Given a CSP P, it is diff-arc-consistent iff
every arc belongs to a matching which covers
all the nodes of the graph associated with P.

• The propagation (domain filtering) is done
  according to the following principles
• If an arc does not belong to any maximal
  matching, then it does not belong to any all_diff
  solution.
• Given a maximal matching, an arc belongs to any
  maximal matching iff it belongs (Berge 1970)
• To an even alternating cycle or
• To an even alternating path, starting at a free
  node.

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Global Constraints all_diff

• Example For the maximal matching (MM) shown

• 6 is a free node
• 6-E-5-D-4 is an even alternating path,
  alternating arcs from the MM (E-5, D-4) with arcs
  not in the MM (D-5, E-6)
• A-1-B-2-A is an even alternating cycle
• E-3 does not belong to any even alternating cycle
• E-3 does not belong to any even alternating path
  starting in a free node (6)
• E-3 may be filtered out!

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Global Constraints all_diff

• Compaction
• Before this analysis, the graph may be
  compacted, aggregating, into a single node,
  equivalent nodes, i.e. those belonging to even
  alternating cycles.
• Intuitively, for any solution involving these
  variables and values, a different solution may be
  obtained by permutation of the corresponding
  assignments.

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Global Constraints all_diff

• A-1-B-2-A is an even alternating cycle
• By permutation of variables A and B, the
  solution ltA,B,C,D,Egt lt1,2,3,4,5gt becomes
  ltA,B,C,D,Egt lt2,1,3,4,5gt
• Hence, nodes A e B, as well as nodes 1 and 2 may
  be grouped together (as may the nodes D/E and
  4/5).

With these grouping the graph becomes much more
compact
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Global Constraints all_diff

• Analysis of the compacted graph shows that

• Arc D/E - 3 may be filtered out (notice that
  despite belonging to cycle D/E - 3 - C - 1/2 -
  D/E, this cycle is not alternating.
• Arcs D/E - 1/2 and C - 1/2 may also be filtered.

The compact graph may thus be further simplified
to
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Global Constraints all_diff

• By expanding back the simplified compact graph,
  one gets the graph on the right that

Which immediately sets C3 and, more generaly,
filters the initial domains to A,B 1..2,
C 1,2,3, D 2,3,4,5, E 3,4,5,6
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Global Constraints all_diff

• Upon elimination of some labels (arcs), possibly
  due to other constraints, the all_diff constraint
  propagates such prunings, incrementally. There
  are 3 situations to consider
• Elimination of a vital arc (the only arc
  connecting a variable node with a value node)
  The constraint cannot be satisfied.

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Global Constraints all_diff

• Elimination of a non-vital arc which is a not
  member to the maximal matching
• Eliminate the arcs that do not belong any more to
  an alternating cycle or path.

Arc A-4 does not belong to the even alternating
path started in node 6. D-5 also leaves this
path, but it still belongs to an even alternating
cycle.
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Global Constraints all_diff

• Elimination of a non-vital arc which is not a
  member to the maximal matching
• Determine a new maximal matching and restart from
  there.

A new maximal matching includes arcs D-5 and E-6.
In this matching, arcs E-4 and E-5 do not belong
to even alternating paths or alternating cycles.
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Global Constraints

• Alldiff ? example of a global constraint.
• Global since they typically involve several
  variables and constraints
• (as in the alldiff). Poor choice of namethey
  are still local
• Global constraints encapsulate a sub-problem,
  (typically, why?) solved efficiently (polytime).
• Network flow algorithms are used for several
  global constraints.
• Lots of global constraints, e.g., element
  constraint, cardinality constraint, sequence
  constraint, etc, etc

Active Research Area
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Catalog of global constraints (300)
http//www.emn.fr/xinfo/sdemasse/gccat/