Using Maximal Independent Sets of a Constraint Graph to Improve Search (with Validation on Local Search) Joel Gompert and Berthe Y. Choueiry Constraint Systems Laboratory Department of Computer Science

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Using Maximal Independent Sets of a Constraint
Graph to Improve Search (with Validation on
Local Search) Joel Gompert and Berthe Y.
ChoueiryConstraint Systems Laboratory
Department of Computer Science Engineering
University of Nebraska-Lincoln

• Idea Using IndSet with search
• Let I be a maximal independent set of a
  constraint graph, and be the set of the remaining
  nodes
• Perform search (e.g., stochastic local search) on
  I
• Enforce arc-consistency on I
• Repeat from Step 2 until no domain is anihilated.
  
• Advantages
• Improves search performance.
• Yields families of solutions
• Validation On stochastic local search (SLS) and
  random binary CSPs

Application to Stochastic Local Search
Idea Guiding SLS When solving PI , we improve
the performance of SLS on I by exploiting
information about the connection between I and I

• SLS operates by counting the number of broken
  constraints in PI in a given instantiation of the
  variables in I.
• We include in this count some measurement of the
  number of the constraints between I and I broken
  as a result of this instantiation.

Preliminary results random binary CSPs
Basic Principle illustrative example

• Step 1 Given an independent set, split the CSP
• I Independent Set
• I Vertices not in I (vertex cover)

Step 2 Find a solution to PI , the CSP induced
by I , which is smaller than the original problem.
Step 3 Apply arc-consistency to extend partial
solution into global solution or prove
inconsistency

• Each of these experiments
• SLSIndSet typically returns a number of
  solutions greater than can be represented by a
  32-bit integer.
• SLS returns only 1 solution

• Step 4 When arc-consistency
• Succeeds it yeilds a family of solutions. Here
  
• c,d x a,c,d x a,b,d (18 solutions)
• Fails go to Step 2

Future work

• Validation on more diverse CSP instances
• Comparison against other search algorithms

Gompert J. Local search with Maximal Independent
Sets. Proceedings of CP 2004, Springer.
This research is supported by NSF CAREER Award
0133568. Experiments were conducted utilizing
the Research Computing Facility of the University
of Nebraska-Lincoln.