Bounds for short covering codes and reactive tabu search

Given a prime power q , c q ( n , R ) denotes the minimum cardinality of a subset H in F q n such that every word in this space differs in at most R coordinates from a multiple of a vector in H . In this work, two new classes of short coverings are established. As an application, a new optimal recor...

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Published inDiscrete Applied Mathematics Vol. 158; no. 5; pp. 522 - 533
Main Authors Mendes, Carlos, Monte Carmelo, Emerson L., Poggi, Marcus
Format Journal Article
LanguageEnglish
Published Elsevier B.V 06.03.2010
Subjects
Covering code
Covering radius
Dominating set
Finite field
Tabu search
Finite field
Tabu search
Dominating set
Covering code
Covering radius
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Abstract Given a prime power q , c q ( n , R ) denotes the minimum cardinality of a subset H in F q n such that every word in this space differs in at most R coordinates from a multiple of a vector in H . In this work, two new classes of short coverings are established. As an application, a new optimal record-breaking result on the classical covering code is obtained by using short covering. We also reformulate the numbers c q ( n , R ) in terms of dominating set on graphs. Departing from this reformulation, the reactive tabu search (a variation of tabu search heuristics) is developed to obtain new upper bounds on c q ( n , R ) . The algorithm is described and conclusions on the results are drawn; they identify the advantages of using the reactive mechanism for this problem. Tables of lower and upper bounds on c q ( n , R ) , q = 3 , 4 , n ≤ 7 , and R ≤ 3 , are also presented.
AbstractList Given a prime power q , c q ( n , R ) denotes the minimum cardinality of a subset H in F q n such that every word in this space differs in at most R coordinates from a multiple of a vector in H . In this work, two new classes of short coverings are established. As an application, a new optimal record-breaking result on the classical covering code is obtained by using short covering. We also reformulate the numbers c q ( n , R ) in terms of dominating set on graphs. Departing from this reformulation, the reactive tabu search (a variation of tabu search heuristics) is developed to obtain new upper bounds on c q ( n , R ) . The algorithm is described and conclusions on the results are drawn; they identify the advantages of using the reactive mechanism for this problem. Tables of lower and upper bounds on c q ( n , R ) , q = 3 , 4 , n ≤ 7 , and R ≤ 3 , are also presented.
Author Mendes, Carlos
Monte Carmelo, Emerson L.
Poggi, Marcus
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10.1002/jos.122
10.1002/(SICI)1520-6610(1997)5:1<71::AID-JCD6>3.0.CO;2-V
10.1287/ijoc.6.2.126
10.1109/18.532916
10.1109/18.79926
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Tabu search
Dominating set
Covering code
Covering radius
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References Cohen, Honkala, Litsyn, Lobstein (b6) 1997
Monte Carmelo, Nakaoka, Gernimo (b16) 2007; 20
Östergård (b18) 1991; 37
Haas, Schlage-Puchta, Quistorff (b10) 2009; 116
Östergård (b19) 1997; 5
Battiti, Tecchiolli (b1) 1994; 6
Glover (b9) 1990; 2
Garey, Johnson (b7) 1979
acessed November (2008) and April (2009)
Bhandari, Durairajan (b2) 1996; 42
Carnielli (b4) 1985; 57
Kéri, Östergård (b13) 2003; 40
Brualdi, Litsyn, Pless (b3) 1998
Monte Carmelo, Nakaoka (b15) 2008; 29
Glover (b8) 1989; 1
Osman, Wassan (b17) 2002; 5
Honkala, Östergård (b11) 1997
W.A. Carnielli, E.L. Monte Carmelo, M. Poggi de Aragão, C.C. de Souza, Upper bounds for minimum covering codes by tabu search, II Ofic. Nac. de Problemas Combinatórios: Teoria, Algorit. e Aplicações, 1995, pp. 50–58
J. Linderoth, F. Margot, G. Thain, The tera-gridiron: A natural turf for high-throughput computing, Technical Report 07T-001, Industrial and Systems Engineering, Lehigh University, 2007
G. Kéri, Tables on
homepage
Taussky, Todd (b20) 1948; 21
Battiti (10.1016/j.dam.2009.11.006_b1) 1994; 6
Carnielli (10.1016/j.dam.2009.11.006_b4) 1985; 57
Glover (10.1016/j.dam.2009.11.006_b8) 1989; 1
Glover (10.1016/j.dam.2009.11.006_b9) 1990; 2
Monte Carmelo (10.1016/j.dam.2009.11.006_b15) 2008; 29
Haas (10.1016/j.dam.2009.11.006_b10) 2009; 116
Taussky (10.1016/j.dam.2009.11.006_b20) 1948; 21
Östergård (10.1016/j.dam.2009.11.006_b19) 1997; 5
Cohen (10.1016/j.dam.2009.11.006_b6) 1997
Bhandari (10.1016/j.dam.2009.11.006_b2) 1996; 42
Brualdi (10.1016/j.dam.2009.11.006_b3) 1998
Monte Carmelo (10.1016/j.dam.2009.11.006_b16) 2007; 20
Kéri (10.1016/j.dam.2009.11.006_b13) 2003; 40
Garey (10.1016/j.dam.2009.11.006_b7) 1979
10.1016/j.dam.2009.11.006_b14
Osman (10.1016/j.dam.2009.11.006_b17) 2002; 5
10.1016/j.dam.2009.11.006_b5
Honkala (10.1016/j.dam.2009.11.006_b11) 1997
10.1016/j.dam.2009.11.006_b12
Östergård (10.1016/j.dam.2009.11.006_b18) 1991; 37
References_xml – volume: 37
  start-page: 660
  year: 1991
  end-page: 664
  ident: b18
  article-title: Upper bounds for
  publication-title: IEEE Trans. Inform. Theory
  contributor:
    fullname: Östergård
– volume: 2
  start-page: 4
  year: 1990
  end-page: 32
  ident: b9
  article-title: Tabu search—Part II
  publication-title: ORSA J. Comput.
  contributor:
    fullname: Glover
– volume: 57
  start-page: 9
  year: 1985
  end-page: 16
  ident: b4
  article-title: On covering and coloring problems for rook domains
  publication-title: Discrete Math.
  contributor:
    fullname: Carnielli
– year: 1997
  ident: b11
  article-title: Code design
  publication-title: Local Search in Combinatorial Optimization
  contributor:
    fullname: Östergård
– volume: 5
  start-page: 71
  year: 1997
  end-page: 80
  ident: b19
  article-title: Constructing covering codes by tabu search
  publication-title: J. Combin. Des.
  contributor:
    fullname: Östergård
– volume: 29
  start-page: 227
  year: 2008
  end-page: 233
  ident: b15
  article-title: Short coverings in tridimensional spaces arising from sum-free sets
  publication-title: European J. Combin.
  contributor:
    fullname: Nakaoka
– volume: 42
  start-page: 1640
  year: 1996
  end-page: 1642
  ident: b2
  article-title: A note on bounds for
  publication-title: IEEE Trans. Inform. Theory
  contributor:
    fullname: Durairajan
– volume: 116
  start-page: 478
  year: 2009
  end-page: 484
  ident: b10
  article-title: Lower bounds on covering codes via partition matrices
  publication-title: J. Combin. Theory Ser. A
  contributor:
    fullname: Quistorff
– volume: 20
  start-page: 875
  year: 2007
  end-page: 886
  ident: b16
  article-title: A covering problem on finite spaces and rook domains
  publication-title: Int. J. Appl. Math.
  contributor:
    fullname: Gernimo
– volume: 5
  start-page: 287
  year: 2002
  end-page: 305
  ident: b17
  article-title: A reactive tabu search meta-heuristic for the vehicle routing problem with back-hauls
  publication-title: J. Sched.
  contributor:
    fullname: Wassan
– year: 1997
  ident: b6
  article-title: Covering Codes
  contributor:
    fullname: Lobstein
– volume: 21
  start-page: 303
  year: 1948
  end-page: 305
  ident: b20
  article-title: Covering theorems for groups
  publication-title: Ann. Soc. Pol. Math.
  contributor:
    fullname: Todd
– year: 1979
  ident: b7
  article-title: Computers and Intractability: A Guide to the Theory of NP-Completeness
  contributor:
    fullname: Johnson
– volume: 1
  start-page: 190
  year: 1989
  end-page: 206
  ident: b8
  article-title: Tabu search—Part I
  publication-title: ORSA J. Comput.
  contributor:
    fullname: Glover
– volume: 40
  start-page: 243
  year: 2003
  end-page: 256
  ident: b13
  article-title: On the covering radius of small codes
  publication-title: Studia Sci. Math. Hungar.
  contributor:
    fullname: Östergård
– volume: 6
  start-page: 126
  year: 1994
  end-page: 140
  ident: b1
  article-title: The reactive tabu search
  publication-title: ORSA J. Comput.
  contributor:
    fullname: Tecchiolli
– start-page: 755
  year: 1998
  end-page: 826
  ident: b3
  article-title: Covering radius
  publication-title: Handbook of Coding Theory
  contributor:
    fullname: Pless
– year: 1997
  ident: 10.1016/j.dam.2009.11.006_b6
  contributor:
    fullname: Cohen
– ident: 10.1016/j.dam.2009.11.006_b5
– volume: 2
  start-page: 4
  year: 1990
  ident: 10.1016/j.dam.2009.11.006_b9
  article-title: Tabu search—Part II
  publication-title: ORSA J. Comput.
  doi: 10.1287/ijoc.2.1.4
  contributor:
    fullname: Glover
– volume: 29
  start-page: 227
  year: 2008
  ident: 10.1016/j.dam.2009.11.006_b15
  article-title: Short coverings in tridimensional spaces arising from sum-free sets
  publication-title: European J. Combin.
  doi: 10.1016/j.ejc.2006.09.008
  contributor:
    fullname: Monte Carmelo
– volume: 116
  start-page: 478
  year: 2009
  ident: 10.1016/j.dam.2009.11.006_b10
  article-title: Lower bounds on covering codes via partition matrices
  publication-title: J. Combin. Theory Ser. A
  doi: 10.1016/j.jcta.2008.06.008
  contributor:
    fullname: Haas
– volume: 1
  start-page: 190
  year: 1989
  ident: 10.1016/j.dam.2009.11.006_b8
  article-title: Tabu search—Part I
  publication-title: ORSA J. Comput.
  doi: 10.1287/ijoc.1.3.190
  contributor:
    fullname: Glover
– volume: 57
  start-page: 9
  year: 1985
  ident: 10.1016/j.dam.2009.11.006_b4
  article-title: On covering and coloring problems for rook domains
  publication-title: Discrete Math.
  doi: 10.1016/0012-365X(85)90152-9
  contributor:
    fullname: Carnielli
– year: 1997
  ident: 10.1016/j.dam.2009.11.006_b11
  article-title: Code design
  contributor:
    fullname: Honkala
– volume: 5
  start-page: 287
  issue: 4
  year: 2002
  ident: 10.1016/j.dam.2009.11.006_b17
  article-title: A reactive tabu search meta-heuristic for the vehicle routing problem with back-hauls
  publication-title: J. Sched.
  doi: 10.1002/jos.122
  contributor:
    fullname: Osman
– volume: 20
  start-page: 875
  year: 2007
  ident: 10.1016/j.dam.2009.11.006_b16
  article-title: A covering problem on finite spaces and rook domains
  publication-title: Int. J. Appl. Math.
  contributor:
    fullname: Monte Carmelo
– start-page: 755
  year: 1998
  ident: 10.1016/j.dam.2009.11.006_b3
  article-title: Covering radius
  contributor:
    fullname: Brualdi
– volume: 5
  start-page: 71
  year: 1997
  ident: 10.1016/j.dam.2009.11.006_b19
  article-title: Constructing covering codes by tabu search
  publication-title: J. Combin. Des.
  doi: 10.1002/(SICI)1520-6610(1997)5:1<71::AID-JCD6>3.0.CO;2-V
  contributor:
    fullname: Östergård
– volume: 40
  start-page: 243
  year: 2003
  ident: 10.1016/j.dam.2009.11.006_b13
  article-title: On the covering radius of small codes
  publication-title: Studia Sci. Math. Hungar.
  contributor:
    fullname: Kéri
– volume: 6
  start-page: 126
  issue: 2
  year: 1994
  ident: 10.1016/j.dam.2009.11.006_b1
  article-title: The reactive tabu search
  publication-title: ORSA J. Comput.
  doi: 10.1287/ijoc.6.2.126
  contributor:
    fullname: Battiti
– year: 1979
  ident: 10.1016/j.dam.2009.11.006_b7
  contributor:
    fullname: Garey
– volume: 42
  start-page: 1640
  issue: 5
  year: 1996
  ident: 10.1016/j.dam.2009.11.006_b2
  article-title: A note on bounds for q-ary covering codes
  publication-title: IEEE Trans. Inform. Theory
  doi: 10.1109/18.532916
  contributor:
    fullname: Bhandari
– volume: 37
  start-page: 660
  issue: 3
  year: 1991
  ident: 10.1016/j.dam.2009.11.006_b18
  article-title: Upper bounds for q-nary covering codes
  publication-title: IEEE Trans. Inform. Theory
  doi: 10.1109/18.79926
  contributor:
    fullname: Östergård
– volume: 21
  start-page: 303
  year: 1948
  ident: 10.1016/j.dam.2009.11.006_b20
  article-title: Covering theorems for groups
  publication-title: Ann. Soc. Pol. Math.
  contributor:
    fullname: Taussky
– ident: 10.1016/j.dam.2009.11.006_b14
– ident: 10.1016/j.dam.2009.11.006_b12
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SubjectTerms Covering code
Covering radius
Dominating set
Finite field
Tabu search
Title Bounds for short covering codes and reactive tabu search
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