Caching in Backtracking Search

1
Caching in Backtracking Search

• Fahiem BacchusUniversity of Toronto

2
Introduction

• Backtracking search needs only space linear in
  the number of variable (modulo the size of the
  problem representation).
• However, its efficiency can greatly benefit from
  using more space to cache information computed
  during search.
• Caching can provable yield exponential
  improvements in the efficiency of backtracking
  search.
• Caching is an any-space feature. We can use as
  much or as little space for caching as we want
  without affecting soundness or completeness.
• Unfortunately, caching can also be time consuming.

How do we exploit the theoretical potential of
caching in practice?
3
Introduction

• We will examine this question for
• The problem of finding a single solution
• And for problems that require considering all
  solutions
• Counting the number of solutions/computing
  probabilities
• Finding optimal solutions.
• We will look at
• The theoretical advantages offered by caching.
• Some of the practical issues involved with
  realizing these theoretical advantages.
• Some of the practical benefits obtained so far.

4
Outline

• Caching when searching for a single solution.
• Clause learning in SAT.
• Theoretical results.
• Its practical application and impact.
• Clause learning in CSPs.
• Caching when considering all solutions.
• Formula caching for sum of products problems
• Theoretical results.
• Practical application

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1. Caching when searching for a single solution
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1.1 Clause Learning in SAT
Fahiem Bacchus, University of Toronto
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Clause Learning in SAT (DPLL)

• Clause learning is the most successful form of
  caching when searching for a single solution
  Marques-Silva and Sakallah, 1996 Zhang et
  al., 2001.
• Has revolutionized DPLL SAT solvers (i.e.,
  Backtracking SAT solvers).

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Clause Learning in SAT
1. Branch on a variable
2. Perform Propagation
Unit Propagation
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Clause Learning in SAT

• Every inferred literal is labeled with a clausal
  reason.
• The clausal reason for a literal is a subset of
  the previous literals on the path whose setting
  implies the literal

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Clause Learning in SAT
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Clause Learning in SAT

• Clause learning occurs when a contradiction is
  reached.
• This involves a sequence of resolution steps.
• Any implied literal in a clausal reason can be
  resolved away by resolving the clause with the
  clausal reason for the implied literal.

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12
Clause Learning in SAT

• SAT solvers utilize a particular sequence of
  resolutions against the conflict clause.
• 1-UIP learning Zhang et al., 2001iteratively
  resolve away the deepest implied literal in the
  clause until the clause contains only one literal
  from the level the contradiction was generated.

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Far Backtracking in SAT
1-UIP Clause

• Once the 1-UIP clause is learnt the SAT Solver
  backtracks to the level this clause became unit.

• It then uses the clause to force a new literal.

• Performs UP

• Continues its search.

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Theoretical Power of Clause Learning

• The power of clause learning has been examined
  from the point of view of the theory of proof
  complexity Cook Reckhow 1977.
• This area looks at the question of how large
  proofs can become and their relative sizes in in
  different propositional proof systems.
• DPLL with Clause learning performing resolution
  (a particular type or resolution).
• Various restricted versions of resolution have
  been well studied.
• Buresh-Oppenhiem, Pitassi 2003 contains a nice
  review of previous results and a number of new
  results in this area.

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Theoretical Power of Clause Learning

• Every DPLL search tree refuting an UNSAT instance
  contains a TREE-Resolution.
• TREE-Resolution proofs can be exponentially
  larger than REGULAR-Resolutions proofs.
• REGULAR-Resolutions proofs can be exponentially
  larger than general (unrestricted) resolution
  proofs.

? UNSAT formulas min_size(DPLL Search Tree)
min_size(TREE-Resolution) gtgt
min_size(REGULAR-Resolution) gtgt
min_size(general resolution)
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Theoretical Power of Clause Learning

• Furthermore every TREE-Resolution proof is a
  REGULAR-Resolution proof and every
  REGULAR-Resolution proof is a general resolution
  proof.

? UNSAT formulas min_size(DPLL Search Tree)
min_size(TREE-Resolution)
min_size(REGULAR-Resolution)
min_size(general resolution)
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Theoretical Power of Clause Learning

• Beame, Kautz, and Sabharwal 2003 showed that
  clause learning can SOMETIMES yield exponentially
  smaller proofs than REGULAR.
• Unknown if general resoution proofs are some
  times smaller.

? UNSAT formulas min_size(DPLL Search Tree)
min_size(TREE-Resolution) gtgt
min_size(REGULAR-Resolution) gtgt
min_size(Clause Learning DPLL Search
Tree) min_size(general resolution)
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Theoretical Power of Clause Learning

• It is still unknown if REGULAR or even TREE
  resolutions can sometimes be smaller than the
  smallest Clause Learning DPLL Search tree.

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Theoretical Power of Clause Learning

• It is also easily observed Beame, Kautz, and
  Sabharwal 2003 that with restarts clause
  learning can make the DPLL Search Tree as small
  as the smallest general resolution proof on any
  formula.

• UNSAT formulas min_size(Clause Learning
  Restarts DPLL Search Tree)
  min_size(general resolution)

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Theoretical Power of Clause Learning

• In sum. Clause Learning, especially with
  restarts, has the potential to yield exponential
  reductions in the size of the DPLL search tree.
• With clause learning DPLL can potentially solve
  problems exponentially faster.
• That this can happen in practice has been
  irrefutably demonstrated by modern SAT solvers.
• Modern SAT solvers have been able to exploit the
  theoretical potential of clause learning.

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Theoretical Power of Clause Learning

• The theoretical advantages of clause learning
  also hold for CSP backtracking search
• So the question that arises is can the
  theoretical potential of clause learning also be
  exploited in CSP solvers.

Fahiem Bacchus, University of Toronto
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1.1 Clause Learning in CSPs
Fahiem Bacchus, University of Toronto
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Clause Learning in CSPs

• Joint work with George Katsirelos who just
  completed his PhD with me NoGood Processing in
  CSPs
• Learning has been used in CSPs, but have not had
  the kind of impact Clause Learning has had in
  SAT.Decther 1990 T. Schiex G. Verfaillie
  1993 Frost Dechter 1994 Jussien Barichard
  2000
• This work has investigated NoGood learning.

A NoGood is a set of variable assignments that
cannot be extended to a solution.
Fahiem Bacchus, University of Toronto
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NoGood Learning

• NoGood Learning is NOT Clause Learning.
• It is strictly less powerful.
• To illustrate this let us consider encoding a CSP
  as a SAT problem, and compare what Clause
  Learning will do on the SAT encoding to what
  NoGood Learning would do.

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Propositional Encoding of a CSPthe propositions.

• A CSP consists of a
• Set of variables Vi and constraints Cj
• Each variable has a domain of values DomVi
  d1, , dm.
• Consider the set of propositions Vidj one for
  each value of each variable.
• Vidj means that Vi has been assigned the value
  dj.
• True when the assignment has been made.
• (Vidj) means that Vi has not been assigned the
  value dj
• True when dj has been pruned from Vis domain.
• if Vi has been assigned a different value, all
  other values (including dj) are pruned from its
  domain.
• Usually write Vi?dj instead of (Vidj).
• We encode the CSP using clauses over these
  assignment propositions.

Fahiem Bacchus, University of Toronto
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Propositional Encoding of a CSPthe clauses.

• For each variable V with DomVd1,,dk we have
  the following clauses
• (Vd1,Vd2,,Vdk) (must have a value)
• For every pair of values (di, dk) the clause (V ?
  di, V ? dk) (has a unique value)
• For each constraint C(X1,,Xk) over some set of
  variables we have the following clauses
• For each assignment to its variables that
  falsifies the constraint we have a clause
  blocking that assignment.
• If C(a,b,,k) FALSE then we have the clause
  (X1 ? a, X2 ? b, , Xk ? k)
• This is the direct encoding of Walsh 2000.

Fahiem Bacchus, University of Toronto
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DPLL on this Encoded CSP.

• Unit Propagation on this encoding is essentially
  equivalent to Forward Checking on the original
  CSP.

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DPLL on the encoded CSP

• Variables Q, X, Y, Z, ...
• DomQ 0,1
• DomX,Y,Z 1,2,3
• Constraints
• Q X Y 3
• Q X Z 3
• Q Y Z 3

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DPLL on the encoded CSP

• Clause learning

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DPLL on the encoded CSP

• Clause learning

A 1-UIP Clause
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DPLL on the encoded CSP

• This clause is not a NoGood!
• It asserts that we cannot have Q 0, Y 2, and
  Z ? 1 simultaneously.
• This is a set of assignments and domain prunings
  that cannot lead to a solution.
• A NoGood is only a set of assignments.
• To obtain a NoGood we have to further resolve
  awayZ 1 from the clause.

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DPLL on the encoded CSP

• NoGood learning

• This clause is a NoGood. It says that we cannot
  have the set of assignments Q 0, X 1, Y 2
• NoGood learning requires resolving the conflicts
  back to the decision literals.

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NoGoods vs. Clauses (Generalized NoGoods)

• Unit propagation over a collection of learnt
  NoGoods is ineffective.
• Nogoods are clauses containing negated literals
  only, e.g.,(Z ? 1, Y ? 0, X ? 3). If one of
  these clauses becomes unit, e.g., (X ? 3), the
  forced literal can only satisfy other NoGood
  clauses, it can never reduce the length of those
  clauses.
• A single clause can represent an exponential
  number of NoGoods
• (Q ? 1, Z 1, Y 1) is equivalent to (Domain
  1, 2, 3)(Q ? 1, Z ? 2, Y ? 2) (Q ? 1, Z ? 3, Y
  ? 2) (Q ? 1, Z ? 2, Y ? 3) (Q ? 1, Z ? 3, Y ? 3)

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NoGoods vs. Clauses (Generalized NoGoods)

• The 1-UIP clause can prune more branches during
  the future search than the NoGood clause
  Katsirelos 2007.
• Clause Learning can yield super-polynomially
  smaller search trees than NoGood Learning
  Katsirelos 2007

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Encoding to SAT

• With all of these benefits of clause learning
  over NoGood learning the natural question isWhy
  not encode CSPs to SAT and immediately obtain the
  benefits of Clause Learning already implemented
  in modern SAT solvers?

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Encoding to SAT

• The SAT theory produced by the direct encoding is
  not very effective.
• Unit Prop. on this encoding only achieves Forward
  Checking (a weak form of propagation).
• Under the direct encoding constraints of arity k
  yield 2O(k) clauses. Hence the resultant SAT
  theory is too large.
• No direct way of exploiting propagators.
• Specialized polynomial time algorithms for doing
  propagation on constraints of large arity.

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Encoding to SAT

• Some of these issues can be address by better
  encodings, e.g., Bacchus 2007, Katsirelos
  Walsh 2007, Quimper Walsh 2007. But overall
  complete conversion to SAT is currently
  impractical.

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Clause Learning in CSPs without encoding

• We can perform Clause Learning in a CSP solver by
  the following steps
• The CSP solver must keep track of the
  chronological sequence of variable assignments
  and value prunings made as we descend each path
  in the search tree.

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Clause Learning in CSPs without encoding

• Each item must be labeled with a clausal reason
  consisting of items previously falsified along
  the path.

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Clause Learning in CSPs without encoding

• Contradictions are labeled by falsified clauses,
  e.g., Domain Wipe Outs can be labeled by the must
  have a value clause.
• From this information clause learning can be
  performed whenever a contradiction is reached.
• These clauses can be stored in a clausal database
• Unit Propagation can be run on this database as
  new value assignments or value prunings are
  preformed.
• The inferences of Unit Propagation agument the
  other constraint propagation done by the CSP
  solver.

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Higher Levels of Local Consistency

• Note that this technique works irrespective of
  kinds of inference performed during search.
• That is, we can use any kind of inference we want
  to infer a new value pruning or new variable
  assignmentas long as we can label the inference
  with a clausal reason.
• This raises the question of how do we generate
  clausal reasons for other forms inference.
• Katsirelos 2007 answers this question for the
  most commonly used form of inference Generalized
  Arc Consistency.
• Including ways of obtain clausal reasons from
  various types of GAC propagators, ALL-DIFF, GCC.
  

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Some Empirical Data Katsirelos 2007

• GAC with NoGood learning helps a bit.
• GAC with clause learning but where GAC labels it
  inferences with NoGoods offers only minor
  improvements.
• To get significant improvements must do clause
  learning as well have proper clausal reasons from
  GAC.

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Observations

• Caching techniques have great potential, but to
  make them effective in practice it can sometimes
  require resolving a number of different issues.
• This work goes a long ways towards achieving the
  goal of exploiting the theoretical potential of
  Clause Learning.

Prediction Clause learning will play a
fundamental role in the next generation of CSP
solvers, and these solvers will often be orders
of magnitude more effective than current solvers.
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Open Issues

• Many issues remain open. Here we mention only
  one Restarts.
• As previously pointed out, clause learning gains
  a great deal more power with restarts. With
  restarts it can be as powerful as unrestricted
  resolution.
• Restarts play an essential role in the
  performance of SAT solvers. Both full restarts
  and partial restarts.

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Search vs. Inference

• With restarts and clause learning, the
  distinction of search vs. inference is turned on
  its head.
• Now search is performing inference.
• Instead the distinction becomes systematic vs.
  opportunistic inference.
• Enforcing a high level of consistency during
  search is performing systematic inference.
• Searching until we learn a good clause is
  opportunistic.
• Sat solvers perform very little systematic
  inference, only Unit Propagation, but they
  perform lots of opportunistic inference.
• CSP solvers essentially do the opposite.

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One Open Question

• In SAT solvers opportunistic inference is
  feasible if a learnt clause turns out not to be
  useful it doesnt matter much as the search to
  learn that clause did not take much time. Search
  (nodes/second rate) is very fast.
• In CSP solvers the enforcement of higher levels
  of local consistency makes restarts and
  opportunistic inference very expensive. Search
  (nodes/second rate) is very slow.

Is high levels of consistency really the most
effective approach for solving CSP once Clause
learning is available?
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2. Formula Caching when considering all
solutions.
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Considering All Solutions?

• One such class of problems are those that can be
  expressed as Sum-Of-Product problems Decther
  1999.
• Finite Set of Variables, V1, V2, , Vn
• A finite domain of values for each variable
  DomVi.
• A finite set of real valued local functions f1,
  f2, , fm.
• Each function is local in the sense that it only
  depends on a subset of the variables. f1(V1,
  V2), f2(V2, V4, V6),
• The locality of the functions can be exploited
  algorithmically.

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Sum of Products

• The sum of products problem is to compute from
  this representation

• The local functions assign a value to every
  complete instantiation of the variables (the
  product) and we want to compute some amalgamation
  of these values
• A number of different problems can be cast as
  instances of sum-of-product Decther 1999.

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Sum of ProductsExamples

• CSPs count the number of solutions.
• Inference in Bayes Nets.
• Optimization the functions are sub-objective
  functions returning real values and the global
  objective is to maximize the sum of the
  sub-objects (cf. soft constraints, generalized
  additive utility).

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Algorithms Brief History Arnborg et al. 1988

• It had long been noted that various NP-Complete
  problems on graphs were easy on Trees.
• With the characterization of NP completeness
  systematic study of how to extend these
  techniques beyond trees started in the 1970s
• A number of Dynamic Programming algorithms were
  developed for partial K-trees which could solve
  many hard problem in time linear in the size of
  the graph (but exponential in K)

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Algorithms Brief History Arnborg et al. 1988

• These ideas were made systematic by Robertson
  Seymour who wrote a series of 20 articles to
  prove Wagners conjecture 1983.
• Along the way they defined the concept of Tree
  and Branch decompositions and the graph
  parameters Tree-Width and Branch-Width.
• It was subsequently noted that partial k-Trees
  are equivalent to the class of graphs with tree
  width k. So all of the dynamic programming
  algorithms developed for partial K-Trees worked
  for tree width k graphs.
• The notion of tree width has been exploited many
  areas of computer science and combinatorics
  optimization. .

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Three types of Algorithms

• These algorithms all take one of three basic
  forms all of which achieve the same kinds of
  tree-width complexity guarantees.
• To understand these forms we first introduce the
  notion of a branch decomposition (which is
  somewhat easier to utilize than
  tree-decompositions when dealing the local
  functions with arity greater than 2)

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Branch Decomposition

• Start with m leaf nodes one for each of the local
  functions.
• Map each local function to some leaf node.

f3
f6
f1
f4
f2
f7
f5
55
Branch Decomposition

• Label each leaf node with the variables in the
  scope of the associated local function.

V5,V6,V7
V4, V5
V3,V7
V1,V3
V4,V6
V8,V6
V5, V2
f3
f6
f1
f4
f2
f7
f5
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Branch Decomposition

• Build a binary tree on top of these nodes.

V5,V6,V7
V4, V5
V3,V7
V1,V3
V4,V6
V8,V6
V5, V2
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Branch Decomposition

• Then label the rest of the nodes of the tree.

V4,V6,V3
V4,V6,V3
V4, V5
V5,V6,V3
V3,V4,V6
V5,V6,V7
V4, V5
V3,V7
V1,V3
V4,V6
V8,V6
V5, V2
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Internal Labels
B variables in the rest of the tree. (Not in
subtree under the node)
A
B
B
A variables in the subtree below
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Internal Labels
A ? B
v
v
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Branch Width

• Width of a particular decomposition is the size
  of the minimal label.
• Branch width is the minimal width over all
  possible branch decompositions.
• Branch width is no more than the tree-width.

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Algorithms Dynamic Programming

• Bottom up Dynamic Programming, e.g., Join Tree
  algorithms in Bayesian Inference

V4,V6,V3
V4,V6,V3
V4, V5
V5,V6,V3
V3,V4,V6
V5,V6,V7
V4, V5
V3,V7
V1,V3
V4,V6
V8,V6
V5, V2
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Algorithms Variable Elimination

• Linearize the bottom up process Variable
  Elimination.

V4,V6,V3
V4,V6,V3
V4, V5
V5,V6,V3
V3,V4,V6
V5,V6,V7
V4, V5
V3,V7
V1,V3
V4,V6
V8,V6
V5, V2
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Algorithms Instantiation and Decomposition

• Instantiate variables starting at the top (V4, V6
  and V3) and decompose the problem.

V4,V6,V3
V4,V6,V3
V5,V2,V7
V8,V1
V4, V5
V5,V6,V3
V3,V4,V6
V5,V6,V7
V4, V5
V3,V7
V1,V3
V4,V6
V8,V6
V5, V2
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Instantiation and Decomposition

• A number of works have used this approach
• Pseudo Tree Search Freuder Quinn 1985
• Counting Solutions Bayardo Pehoushek 2000
• Recursive Conditioning Darwiche 2001
• Tour Merging Cook Seymour 2003
• AND-OR Search Dechter Mateescu 2004

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Instantiation and Decomposition

• Solved by AND/OR search as we instantiate
  variables we examine the residual sub-problem.
• If the sub-problem consists of disjoint parts
  that share no variables (components) we solve
  each component in a separate recursion.

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Theoretical Results

• With the right ordering this approach can solve
  the problem in time 2O(w log n)and linear
  space, were w is the branch (tree) width of the
  instance.
• If the solved components are cached so that they
  do not have to be solved again the approach can
  solve the problem in time nO(1)2O(w).
• But now we need nO(1)2O(w) space.

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Solving Sum-Of-Products with Backtracking

• In joint work with Toniann Pitassi Shannon
  Dalmao we showed that caching is in fact
  sufficient to achieve these bounds with standard
  backtracking search. Bacchus, et al. 2003
• AND/OR decomposition of the search tree is not
  necessary (and may be harmful). Instead an
  ordinary decision tree can be searched.
• Once again Caching again provides a significant
  increase in the theoretical power of
  backtracking.

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Simple Formula Caching

• As assumptions are made during search the problem
  is reduced.
• In Simple Formula Caching we cache every solved
  residual formula, and if we encounter the same
  residual formula again we utilize its cached
  value instead of solving the same sub-problem
  again.
• Two residual formulas are the same if
• They contain the same (unassigned) variables.
• All instantiated variables in the remaining
  constraints (constraints with at least one
  unassigned variable) are instantiated to the same
  value.

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Simple Formula Caching

• C1(X,Y), C2(Y,Z) C3(Y,Q) Xa,Yb ? C2(Yb,Z)
  C3(Yb,Q)
• C1(X,Y), C2(Y,Z) C3(Y,Q) Xb,Yb ? C2(Yb,Z)
  C3(Yb,Q)
• These residual formulas are the same even though
  we obtained them from different instantiations.

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Simple Formula Caching
BTSimpleCache(F) if InCache(F), return
CachedValue(F) else Pick a variable V in F,
value 0 for d in DomainV value value
BTSimpleCache(FVd) AddToCache(F, value) return

• Runs in time and space 2O(w log n)

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Component Caching

• We can achieve the same performance as AND/OR
  decomposition, i.e., 2O(w log n) time with linear
  space or nO(1)2O(w) time with nO(1)2O(w) space by
  examining the residual formula for disjoint
  Components.
• We cache these disjoint components as they are
  solved.
• We remove any solved component from the residual
  formula.

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Component Caching

• Since components are no longer solved in a
  separate recursion we have to be a bit cleverer
  about identifying the value of these components
  from the search computation.
• This can be accomplished by using the cache in a
  clever way, or by dependency tracking techniques.

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Component Caching

• There are some potential advantages of searching
  a single tree rather than and AND/OR tree.
• With an AND/OR tree one has to make a commitment
  to which component to solve first.
• The wrong decision when doing Bayesian Inference
  or optimization with Branch and Bound can be
  expensive
• In the single tree the components are solved in
  an interleaved manner.
• This also provides more flexibility with respect
  to variable ordering.

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Bayesian Inference via Backtracking Search

• These ideas were used to build a fairly
  successful Bayes Net Reasoner. Bacchus et al.
  2003.
• Better performance however would require
  exploiting more of the structure internal to the
  local functions.

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Exploiting Micro Structure

• C1(A,Y,Z) TRUE ? A0 ? Y 1 A 1 ?
  Y0 Z 1Then C(A0,Y,Z) is in fact not a
  function of Z. That is C(A0,Y,Z) ? C(A0,Y)
• C2(X,Y,Z) TRUE ? X Y Z 3Then C(X3, Y,
  Z) is already satisfied.

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Exploiting Micro Structure

• In both cases if we could detect this during
  search we could potentially
• Generate more components, e.g., if we could
  reduce C1(A0,Y,Z) to a C1(A0,Y) perhaps Y and Z
  would be in different components.
• Generate more cache hits, e.g., if the residual
  formula differs from a cached formula only
  because it contains C2(X3,Y,Z), recognizing that
  constraint is already satisfied would allow us to
  ignore it and generate the cache hit.

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Exploiting Micro Structure

• It is interesting to note that if we encode to
  CNF we do get to exploit more of the micro
  structure (structure internal to the constraint).
• Clauses with a true literal are satisfied and can
  be removed from the residual formula.
• Bayes Net Reasoners using CNF encodings have
  displayed very good performance Chavira
  Darwiche 2005.

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Exploiting Micro Structure

• Unfortunately, as pointed out before, encoding in
  CNF can result in an impractical blowup in the
  size of the problem representation.
• Practical techniques for exploiting the micro
  structure remain a promising area for further
  research.
• Some promising results by Kitching to detect when
  a symmetric version of a current component has
  already been solved Kitching Bacchus 2007,
  but more work to be done.

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Observations

• Component caching solvers are the most effective
  ways of exactly computing the number of solutions
  of a SAT formula.
• Allow solution of certain types of Bayesian
  Inference problems not solvable by other methods.
• Have shown promise in solving decomposable
  optimization problems Dechter Marinescu 2005,
  de Givry et al. 2006, Kitching Bacchus 2007
• To date all these works have used AND/OR search.
  So exploiting the advantages of plain
  backtracking search remains work to be done
  Kitching in progress.
• Better exploiting micro structure also remains
  work to be done.

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Conclusions

• Caching is a technique that has great potential
  for making a material difference in the
  effectiveness of backtracking search.
• The range of practical mechanisms for exploiting
  caching remains a very fertile area for future
  research.
• Research in this direction might well change
  present day accepted practice in constraint
  solving.

81
References

• Marques-Silva and Sakallah, 1996
• J. P. Marques-Silva and K. A. Sakallah. Graspa
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