# Heuristic Search - PowerPoint 1

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```					Different Local Search
Algorithms in STAGE for
Solving Bin Packing Problem

Gholamreza Haffari

Sharif University of Technology
haffari@ce.sharif.edu
Overview
   Combinatorial Optimization Problems
and State Spaces
   STAGE Algorithm
   Local Search Algorithms
   Results
   Conclusion and Future works
Optimization Problems
   Objective function: F(x1, x2, …, xn)

Find vector X=(x1, x2, …, xn) which minimizes
(maximizes) F

   Constraints:
g1(X)  0
g2(X)  0
.
.
.
gm(X)  0
Combinatorial Optimization
Problems (COP)

   Special kind of Optimization Problems
which are Discrete

   Most of the COPs are NP-Hard, I.e.
there is not any polynomial time
algorithm for solving them.
Satisfiability
   SAT: Given a formula in propositional
calculus, is there an assignment to its
variables making it true?
f(x1, x2, .., xn)

   Problem is NP-Complete. (Cook 1971)
Bin Packing Problem             (BPP)

   Given a list (a1, a2, …) of items, each of
which has a size s(ai)>0, and a bin
Capacity C, what is the minimum
number of bins for packing items?

   Problem is NP-Complete (Garey and
Johnson 1979)
An Example of BPP

a1    a2    a3    a4       b1    b2   b3   b4

Objects list: a1, a2, …, an        Objective function: m

Bin’s capacity (bj) is C           ai < C, aibj, 1j m
Definition of State in BPP
   A particular permutation of items in the
object list is called state.

Greedy Algorithm

a1   a2   a3   a4                 b1   b2   b3   b4
State Space of BPP

a1, a2, a3, a4

a1, a2, a4, a3    a1, a4, a2, a3   . . .

a2, a4, a3, a1   . . .                . . .
A Local Search Algorithm
1) s0 : a random start state

2) for i = 0 to +

- generate new solutions set S from the current
solution si

- decide whether si+1 = s’S or si

- if a stopping condition is satisfied

return the best solution found
Local Optimum Solutions
   The quality of a local optimum resulted
from a local search process depends
on a starting state.
Multi-Start LSA
   Runs the base local search algorithms
from different starting states and
returns the best result found.

    Is it possible to choose a promising
new starting state?
Other Features of a State
   Other features of a state can help the
search process.

(Boyan 1998)
Previous Experiences

   There is a relationship among local optima of
a COP, so previously found local optima can
help to locate more promising start states.
Core ideas
   Using an Evaluation Function to predict
the eventual outcome of doing a local
search from a state.

   The EF is a function of some features of a
state.

   The EF is retrained gradually.
STAGE Algorithm

Execution Phase
   Uses an Evaluation Function to
locate a good start state.

   Does local search.
Learning Phase
   Retrains EF with the new generated
search trajectory
Evaluation Function
   EF can be used by another local search
algorithm for finding a good new
starting point.

   Applying EF on a state

State       Features         EF   Prediction
Diagram of STAGE

(Boyan 98)
Analysis of STAGE
   What is the effect of using different local search
algorithms?

   Local search algorithms:
   Best Improvement Hill Climbing (BIHC)
   First Improvement Hill Climbing (FIHC)
   Stochastic Hill Climbing (STHC)
Best Improvement HC
   Generates all of the neighboring states,
and then selects the best one.

1

4   7    …   2
First Improvement HC
   Generates neighboring states
systematically, and then selects the first
good one.

5

4   7
Stochastic HC
   Stochastically generates some of the
neighboring states, and then selects the
best one.

   The size of the set containing neighbors
is called PATIENCE.
Different LSAs

Different LSAs for solving U250_00 instance
http://www.ms.ic.ac.uk/info.html
Different LSAs, bounded steps
Some Results
    The higher the accuracy in choosing the next state,
the better the quality of the final solution, by
comparing STHC1 and STHC2 (PATIENCE1=350,
PATIENCE2=700)

    Deep paces result in higher quality and faster
solutions, by comparing BIHC and others.
Different LSAs, bounded moves
Some Results
• It is better to search the solution space
randomly rather than systematically, by
comparing STHC and others.
Future works
   Using other learning structures in
STAGE
   Verifying these results on another
problem (for example Graph Coloring)
   Using other LSA, such as Simulated
Annealing.
Questions

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