University of California Santa Barbara Department of Computer Science

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University of CaliforniaSanta BarbaraDepartment
of Computer Science

• Tabu Search

Local Search and Metaheuristics Teofilo F.
Gonzalez
Arturo Gonzalez-Gutierrez
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Outline

• Definitions
• Linear Ordering Problem
• Steiner Tree Problem

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Definitions

• TS is a metaheuristic that guides a local
  heuristic search procedure to explore the
  solution space beyond local optimality.
• TS emphasis on adaptive memory to exploit
  strategies of human problem-solving/to mimic
  processes in lower orders of natural phenomena
  and behavior
• TS begins as ordinary local or neighborhood
  search, proceeding iteratively from one solution
  to another until a chosen termination criterion
  is satisfied.
• Each solution x has an associated neighborhood
  N(x)?X, and each solution x?N(x) is reached from
  x by an operation called a move.
• TS permits moves that deteriorate the current
  objective function value, from moves chosen from
  a modified neighborhood N(x) Short and long
  term memory structures.

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Definitions

• TS based on short term N(x)?N(x), tabu
  clasification (soltns. forbidden to be visited)
  serves to identify elements in N(x)\N(x).
• TS based on long term N(x) may include soltns.
  found and evaluated in past search, or high
  quality neighbors of these past soltns.
• TS may be viewed as a dynamic neighborhood
  method N(x) is not static, change according to
  the history of the search.
• Structure of neighborhood in TS embraces moves
  used in constructive and destructive processes.

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Definitions

• TS uses attributive memory structures based on
  recording attributes, for guiding processes to
  compute N(x) Recency-based memory and
  frequency-based memory.
• Recency-based memory Keeps track of soltns.
  attributes that have changed during the recent
  past.
• Selected attributes occurred in soltns. recently
  visited are labeled tabu-active.
• Soltns. that contain tabu-active elements become
  tabu.
• Recent soltns. do not belong to N(x), and then
  are not revisited.

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• Frequency-based memory Select preferred moves
  and consists of a ratio p/q.
• p
• Transition Measure iterations where an
  attribute changes the soltns. visited on a
  particular trajectory
• Residence Measure iterations where an
  attribute belongs to soltns. visited on a
  particular trajectory
• q
• Total Number of occurrences
• Sum or Average (Sum of Absolute Values)
• Maximum (Absolute Value)
• Ratios produce transition frequencies (how often
  attributes change) and residence frequencies (how
  often attributes are members of soltns.
  generated)
• Indicates when phases of greater diversification
  are appropriate

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• Adaptive memory creates a balance between search
  intensification and diversification.
• Intensification strategies Modify choice rules
  to encourage move combinations
• Diversification strategies Seek to incorporate
  new attribute combinations not included within
  soltns. previously generated.
• Both of them are mutually reinforcing.

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• Path Relinking Strategy of creating trajectories
  of moves passing through high quality soltns.
• PR operates by starting from a initial solution,
  selected from a subset of high quality soltns.,
  and generating a path in the neighborhood space
  leading toward other soltns. (guided soltns) in
  the subset.
• This is accomplished by selecting moves that
  introduce attributes contained in the guiding
  soltns.

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The Linear Ordering Problem (LOP)

• Given a matrix of weight E(ei,j)mxm the LOP
  consists of finding a permutation p of columns
  (and rows) to maximize
• CE(p) ?1im ?i1jmep(i)p(j)
• NP-hard problem (Chenery Watanabe, 1958)
• Tabu Search (Laguna, et. al. 1999)

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LOP

• INSERT_MOVE(pj,i)
• iltj ? p(p1,,pi-1,pj,pi,,pj-1,pj1,..,pm)
• jlti ? p(p1,,pj-1,pj1,,pi-1,pj,pi,..,pm)
• Example p(1 2 3 4 5 6), pINSERT_MOVE(p6,2)

0 8 12 5 3 1 3 9 0
1 6 2 13 4 6 4 0 3
6 4 2 8 0 5 0 5 7
3 -2 6 7 2 0 -3 0 8
4 0 3 -1 0 1 2 1 9
4 -5 8 0
p(1 6 2 3 4 5 7) CE(p)78 CE(p)86
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LOP

• Best move pINSERT_MOVE(p6,3)

p(1 2 6 3 4 5 7) CE(p)78 CE(p)86,
MoveValue8 CE(p)89, MoveValue11
0 12 8 5 3 1 3 6 0
4 3 6 4 2 9 1 0 6
2 13 4 8 5 0 0 5 7
3 -2 7 6 2 0 -3 0 8 0
4 3 -1 0 1 2 9 1 4
-5 8 0
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LOP

• Neighborhood N
• NpINSERTION_MOVE(pj,i), for 1i,jm and i?j
• Partition of N into m neighborhoods Nj, 1jm
• Nj pINSERTION_MOVE(pj,i), for 1im and i?j
• Basic Tabu Search
• Generate p randomly
• Alternates between an intensification and a
  diversification phase
• An iteration of intensification phase begins by
  randomly selecting a sector.

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Algorithm

• p(1 2 m) the best solution found over the
  entire search
• Global_Iteration_Counter0
• WhileGlobal_Iteration_Counter lt maxGlo)
• Tabu_List
• Intensification_Iteration_Counter0
• Generate p randomly
• // Intensification Phase
• counter(1,..,n)0 //counting the number of
  iterations j is tabu-active
• While(Intensification_Iteration_CounterltMaxInt)
  //MaxInt Consecutive iterations without
  improvement
• Generate j randomly while j ? Tabu_List and
  counter(j)TabuTenure
• pargmaxCE(p)pINSERT_MOVE(pj,i)?Nj
• Tabu_ListTabu_List ?j //j is tabu-active for
  TabuTenure iterations, and j cannot be selected
  for insertions during this time (Short term
  memory)
• freq(j)freq(j)1 //for diversification
  purposes
• If CE(p)ltCE(p) then Intensification_Iteration_C
  ounterIntensification_Iteration_Counter1
• pLocalSearch(Nj,p) //without tabu
  restrictions, local optimum
• pp
• //Additional Intensification Phase by
  implementing a long term relinking phase
• // Lets save in EltSol the best solutions found
  during the entire search

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Results

• 25 instances, m75, _at_0.5 secs.
• Greedy 150 times, CK 10 times

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Steiner Tree Problem

• Given G(V,E,X,w) where X?V a connected
  undirected graph, the Steiner Problem consists of
  finding a minimum-weighted connected subtree of G
  spanning all vertices in X.

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OR-Library Collection of test data sets for a
variety of Operations Research (OR)
problems http//people.brunel.ac.uk/mastjjb/jeb/o
rlib/steininfo.html
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The End

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