Branch and Bound Search

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Branch and Bound Search in Smodels
Gang Wu Department of Computing
Science University of Alberta
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Definition
In Answer Set Programming (ASP), a problem is
solved by writing a logic program whose answer
sets correspond to solutions to the problem.
A logic program in Answer Set Programming is a
finite collection of rules.
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Rules
A typical rule in Answer Set Programming a -
b1,b2,, bm, not c1, not c2,,not cn
a , bi s, and cj s are atoms
a is head. b1,b2,, bm, not c1, not c2,,not cn
is body
It says that if bi s are in an answer set and cj
s are not, then a must be in the same answer
set.
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Rules
- b1,b2,, bm, not c1, not c2,,not cn Rule
without head. Body must be false in any answer
set
a Rule without body. head must be true in any
answer set (fact).
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ASP under Stable Model Semantics
In ASP under the stable model semantics, an
answer set is a stable model.
A set R of atoms is a stable model of program P
if and only if for each rule a - b1,b2,, bm,
not c1, not c2,,not cn in P, if bi s are in R
and cj s are not, then a must be in R and for
each atom h in R, there exists a rule h -
b1,b2,, bm, not c1, not c2,,not cn such that
bi s are in R and none of cj s are in R.
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An Example

• b.
• c - a.
• d - c, b.
• e - f.
• g - not e.

Stable model a, b, c, d, g
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Extended Rules in Smodels
Smodels system is one of the most efficient
implementations of ASP under the stable model
semantics
Cardinality rule Lh1,h2,hmU - body If
the body is satisfied by a model then the rule is
satisfied by the model only when the cardinality
of the subset of h1,h2,hm satisfied by the
model is between integers L and U, inclusive.
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Extended Rules in Smodels
Optimization rule maximize a1w1, a2 w2, ,
an wn wis are integers representing the
weights for the atoms at the left. Generate only
those stable models where the sum of wis for the
corresponding true atoms ais is maximized
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An Example on Solving MQC Problem
1,34,5 1,23,4 1,23,5 1,23,6 1,24,5
1,24,6 1,25,6 1,63,4 1,35,6 1,45,6
2,53,4 2,63,4 2,35,6 2,45,6 3,45,6
Quartet Set Q
Phylogeny T
Smodels Program
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Search for Optimized Stable Models
To search a stable model to satisfy the
optimization rule maximize a1w1, a2 w2, ,
an wn

• Search a single stable model, count the total
  weight as a bound
• Search for another stable model, count its total
  weight, if it is greater than the bound, then
  update it.
• Output the best stable model.

In general, Smodels puts the computation of bound
at the end of a stable model generation.
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Search for Optimized Stable Models
We modify Smodels code so that it can calculate
the bound during the computation of stable
models. In this case, the objective of Smodels is
changed to be optimization-specific.
Advantage This may prune search space since most
stable models may not be optimal. Disadvantage
This may cause overhead on the computation of
bounds. Therefore, a careful implementation is
needed.
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Preliminary Results
We compare three methods the original
Smodels(Org), Smodels with branch and bound
search(BnB), and Smodels with branch and bound
search and known optimal bound(BnB-Opt) on the
same programs for MQC problem. In the third
method, we predefine the exactly number of
satisfied quartets of the optimal solution and
ask Smodels to terminate when it finds a stable
model on that number.
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Preliminary Results
Smodels program runs on the instances with 5
quartet errors