{"title": "A Neural-Network Solution to the Concentrator Assignment Problem", "book": "Neural Information Processing Systems", "page_first": 775, "page_last": 782, "abstract": null, "full_text": "775 \n\nA  NEURAL-NETWORK  SOLUTION TO  THE  CONCENTRATOR \n\nASSIGNNlENT  PROBLEM \n\nGene  A.  Tagliarini \nEdward  W.  Page \n\nDepartment of Computer  Science,  Clemson University,  Clemson,  SC \n\n29634-1906 \nABSTRACT \n\nNetworks  of simple analog  processors  having  neuron-like properties have \nbeen  employed  to  compute  good  solutions  to  a  variety  of optimization  prob(cid:173)\nlems.  This  paper presents  a  neural-net solution to  a  resource allocation prob(cid:173)\nlem that arises  in  providing  local  access  to  the  backbone of a  wide-area  com(cid:173)\nmunication  network.  The  problem is  described in  terms of an energy function \nthat can be  mapped onto an analog computational network.  Simulation results \ncharacterizing  the  performance  of the  neural  computation  are  also  presented. \n\nINTRODUCTION \n\nThis  paper  presents  a  neural-network  solution  to  a  resource  allocation \nproblem  that  arises  in  providing  access  to  the  backbone  of  a  communication \nnetwork. 1 In the  field  of operations  research,  this  problem was  first  known  as \nthe  warehouse  location problem and  heuristics  for finding  feasible,  suboptimal \nsolutions  have  been developed  previously.2. 3  More  recently it  has  been known \nas the multifacility location problem4  and as  the concentrator assignment prob(cid:173)\nlem.1 \n\nTHE HOPFIELD  NEURAL  NETWORK  MODEL \n\nThe  general  structure of the  Hopfield  neural  network model5 \u2022 6,7 is  illus(cid:173)\n\ntrated  in  Fig.  1.  Neurons  are  modeled  as amplifiers  that have  a  sigmoid  input! \noutput curve  as  shown  in  Fig.  2.  Synapses  are  modeled  by  permitting  the  out(cid:173)\nput  of  any  neuron  to  be  connected  to  the  input  of  any  other  neuron.  The \nstrength of the synapse is modeled by a  resistive connection between the output \nof a  neuron and the input to another.  The amplifiers  provide integrative analog \nsummation of the currents that result from the  connections to  other neurons  as \nwell  as  connection  to  external  inputs.  To  model  both  excitatory and  inhibitory \nsynaptic  links,  each amplifier provides  both a  normal output V and an inverted \noutput  V.  The  normal  outputs  range  between  0  and  1  while  the  inverting  am(cid:173)\nplifier  produces  corresponding  values  between 0 and  -1.  The  synaptic  link be(cid:173)\ntween  the  output  of  one  amplifier  and  the  input  of  another  is  defined  by  a \nconductance Tij  which connects one of the outputs of amplifier j to  the input of \namplifier i.  In the  Hopfield  model,  the  connection  between  neurons  i and  j  is \nmade with a  resistor having  a  value Rij =  1rrij  . To  provide an excitatory synap(cid:173)\ntic  connection  (positive  Tij ),  the  resistor  is  connected  to  the  normal  output of \n\nThis  research  was  supported  by  the  U.S.  Army  Strategic Defense  Command. \n\n\u00a9 American Institute of Physics 1988 \n\n\f776 \n\n13 \n\n14 \n\ninputs \n\nVI \n\nV4 \n\nV3 \n\nV2 \n\noutputs \nFig.  1.  Schematic  for  a  simplified \nHopfield  network  with  four  neurons. \n\n1 \n\nV \n\no \n\n-u \n\no \n\n+u \n\nFig.  2.  Amplifier  input/output \n\nrelationship \n\namplifier  j.  To  provide  an  inhibitory  connection  (negative  Tij),  the  resistor  is \nconnected  to  the  inverted  output  of  amplifier  j.  The  connections  among  the \nneurons  are  defined  by  a  matrix  T  consisting  of  the  conductances  Tij .  Hop-\nfield  has  shown  that a  symmetric  T  matrix  (Tij  = Tji )  whose  diagonal  entries \nare  all  zeros,  causes  convergence  to  a  stable  state  in  which  the  output of each \namplifier is  either 0 or 1.  Additionally,  when the amplifiers are operated in the \nhigh-gain mode,  the  stable  states of a  network of n  neurons correspond  to the \nlocal  minima  of the  quantity \n\nn \n\nn \nE  = (-112)  L  L \nj=l \n\ni=l \n\nT\u00b7V.V\u00b7 \nIJ  1  J \n\nn \nL  V.I\u00b7 \n\nI  1 \n\n(1) \n\nwhere  Vi  is  the  output of the  ith  neuron  and  Ii  is  the  externally  supplied  input \nto  the  ph  neuron.  Hopfield  refers  to  E  as  the  computational  energy  of the  sys(cid:173)\ntem. \n\nTHE CONCENTRATOR ASSIGNMENT  PROBLEM \n\nConsider a  collection of n  sites  that are to  be connected  to  m  concentra(cid:173)\n\ntors  as  illustrated  in  Fig.  3(a).  The  sites  are  indicated  by  the  shaded  circles \nand  the  concentrators  are  indicated  by  squares.  The  problem  is  to  find  an \nassignment of sites to concentrators that minimizes the total cost of the  assign(cid:173)\nment  and  does  not  exceed  the  capacity  of  any  concentrator.  The  constraints \nthat must be  met can  be  summarized  as  follows: \n\na)  Each  site  i  (  i = 1,  2, ... ,  n  )  is  connected to  exactly one concentrator; \n\nand \n\n\f777 \n\nb)  Each concentrator j  (j = 1,  2, ... , m  )  is  connected to  no  more than  kj \n\nsites  (where  kj  is  the  capacity  of concentrator D. \n\nFigure  3(b)  illustrates  a  possible  solution  to  the  problem  represented  in  Fig. \n3(a). \n\n0 \n\n\u2022 \n\u2022 \u2022 \u2022 \n\n\u2022 \n\u2022 \u2022 \n\u2022 \n\n0 \n\n\u2022 \u2022 \n\n0 \n\n\u2022 \n\n\u2022 \n\n0 \n\no Concentrators  \u2022  Sites \n\n(a).  Site/concentrator  map \n\n(b).  Possible  assignment \n\nFig.  3.  Example  concentrator assignment problem \n\nIf the cost of assigning  site i to  concentrator j  is  cij  , then the total cost of \n\na  particular  assignment  is \n\ntotal  cost  = \n\nn  m \nL  L \nj=l \ni=l \n\nx \u00b7\u00b7  c\u00b7\u00b7 \nIJ \n\nIJ \n\n(2) \n\nwhere  Xij = 1 only if we  actually decide to assign site i to concentrator j and is  0 \notherwise.  There  are  mn  possible  assignments  of  sites  to  concentrators  that \nsatisfy  constraint  a).  Exhaustive  search  techniques  are  therefore  impractical \nexcept for  relatively  small  values  of  m  and  n. \n\nTHE NEURAL NETWORK  SOLUTION \n\nThis  problem  is  amenable  to  solution  using  the  Hopfield  neural  network \n\nmodel.  The  Hopfield  model  is  used  to  represent  a  matrix  of  possible  assign(cid:173)\nments of sites to concentrators as illustrated in Fig.  4.  Each square corresponds \n\n\f778 \n\nS \n\nITES  ~~  , . .   \u2022 \n\nCONCENTRATORS \n1 \nm \nr , ; - - - - - - ; - ,  \n/ r  1  ,II  11- --III ---III, \n2  ,~ .---~ ---~I \n\u2022 \u2022 \u2022 \u2022  \n\u2022  I The  darkly  shaded  neu-\ni  III  11- --II ---II I ron  corresponds  to  the \nhypothesis  that site  i \n'~n 'Ii  \u2022 ---Ii ---Ii ' should  be  as~igned to \n~ \n\"  n+l  II  111---11---11 \nSLACK .... < n+2  III  II ---~ ---\u2022 \n\u2022 \n\u2022 \n,~n+k j II  11- --III ---III \n\n:.J  concentrator J. \n\n: :   : \n\n~ -\n\n-\n\n-\n\n-\n\n-\n\n-\n\n: \n\n2 \n\nj \n\n\u2022 \n\nFig.  4.  Concentrator  assignment array \n\nto  a  neuron  and  a  neuron  in  row  i  and  column  j  of the  upper  n  rows  of the \narray  represents  the  hypothesis  that site  i should  be  connected  to  concentrator \nj.  If the  neuron in  row  i and  column j  is  on,  then  site  i should  be  assigned  to \nconcentrator j;  if it is  off,  site  i should  not be  assigned  to  concentrator j. \n\nThe  neurons  in  the  lower  sub-array,  indicated  as  \"SLACK\",  are  used  to \nimplement  individual  concentrator capacity constraints.  The  number  of  slack \nneurons  in a  column should  equal  the  capacity  (expressed  as  the  number sites \nwhich  can  be  accommodated)  of  the  corresponding  concentrator.  While  it  is \nnot  necessary  to  assume  that  the  concentrators  have  equal  capacities,  it  was \nassumed here that they did and that their cumulative capacity is  greater than or \nequal  to  the  number  of sites. \n\nTo  ena~le the  neurons  in  the  network  illustrated  above  to  compute  solu(cid:173)\n\ntions  to  the  concentrator problem,  the  network must realize  an energy function \nin which the  lowest energy states correspond to the  least cost assignments.  The \nenergy  function  must therefore  favor  states which  satisfy  constraints  a)  and b) \nabove  as  well  as  states  that  correspond  to  a  minimum  cost  assignment.  The \nenergy function  is  implemented  in terms of connection strengths  between neu(cid:173)\nrons.  The  following  section  details  the  construction  of an  appropriate  energy \nfunction. \n\n\fTHE ENERGY  FUNCTION \n\nConsider the  following  energy  equation: \n\n2 \nE  =  A  L  ( L  y ..  - 1  ) \n\nm \n\nn \n\n.  1 \n1= \n\n.  1 \nJ= \n\n1J \n\nm \n\nn+k\u00b7 \n\nB  L  ( L  J y  ..  - k  .  )2 \n\n+ \n\nj=1 \n\ni=1 \n\nIJ \n\nJ \n\n779 \n\n(3~ \n\nm  n+kj \n\n+  C  L  L  y..  (  1  - Yij ) \n\nj=1 \n\ni=1 \n\n1J \n\nwhere  Yij  is  the  output  of the  amplifier  in  row  i  and  column  j  of the  neuron \nmatrix,  m  and  n  are  the  number  of  concentrators  and  the  number  of  sites \nrespectively,  and  kj  is  the  capacity of concentrator  j. \n\nThe first term will  be minimum when the sum of the outputs in each row \nof  neurons  associated  with  a  site  equals  one.  Notice  that  this  term influences \nonly those rows of neurons which correspond to sites;  no term is  used to coerce \nthe  rows  of  slack  neurons  into  a  particular  state. \n\nThe  second  term of the  equation  will  be  minimum  when  the  sum of the \n\noutputs in each column equals the  capacity  kj  of the  corresponding concentra(cid:173)\ntor.  The  presence of the  kj  slack neurons  in  each column allows  this  term  to \nenforce  the  concentrator capacity restrictions.  The effect of this  term upon the \nupper  sub-array  of  neurons  (those  which  correspond  to  site  assignments)  is \nthat  no  more  than  kj  sites  will  be  assigned  to  concentrator  j.  The  number of \nneurons  to  be  turned  on  in  column j  is  kj ;  consequently,  the  number  of  neu(cid:173)\nrons  turned  on  in  column  j  of  the  assignment  sub-array  will  be  less  than  or \nequal  to  kj \n\n. \n\nThe third term causes the  energy function to  favor the  \"zero\"  and \"one\" \nstates of the  individual neurons by being  minimum when all neurons are in one \nor the  other of these  states.  This  term  influences  all  neurons  in  the  network. \nIn summary,  the  first  term  enforces  constraint  a)  and  the  second  term \n\nenforces  constraint b)  above.  The  third  term guarantees  that a  choice  is  actu(cid:173)\nally  made;  it assures  that  each  neuron  in  the  matrix  will  assume  a  final  state \nnear  zero or one  corresponding  to  the  Xij  term of the  cost  equation  (Eq.  2). \nAfter some algebraic re-arrangement, Eq.  3 can be written in the form of \n\nEq.  1  where \n\nT  IJ  kl  = \n\n., \n, \n\nC  *  8U,I)  *  (1-8(i,k\u00bb,  if  i>n  or k>n. \n\n{A *  8(i,k)  *  (1-8U,I)  +  B  *  8U,1)  *  (1-8(i,k\u00bb,  if  i<n and  k<n \n\n(4) \n\nHere  quadruple  subscripts are  used  for  the  entries in the  matrix T.  Each entry \nindicates  the  strength of the  connection  between  the  neuron in  row  i  and  col(cid:173)\numn j  and  the  neuron  in  row  k and  column  I of the  neuron  matrix.  The  func(cid:173)\ntion  delta  is  given  by \n\n\f780 \n\n8( i  ,  j  )  =  {  1,  if i = j \n\n0,  otherwise. \n\n(5) \n\nThe  A  and B  terms  specify  inhibitions  within  a  row  or a  column of the  upper \nsub-array  and  the  C  term  provides  the  column  inhibitions  required  for  the \nneurons  in  the  sub-array of slack neurons. \n\nEquation 3 specifies the form of a  solution but it does not include a  term \nthat  will  cause  the  network  to  favor  minimum  cost  assignments.  To  complete \nthe  formulation,  the  following  term  is  added  to  each Tij,kl: \n\nD  \u2022  8( j  ,  I )  \u2022  (  1  - 8(  i  ,  k  )  ) \n\n(cost [  i  ,  j  ]  +  cost [  k  ,  I  ]) \n\nwhere  cost[  i  , j  ]  is  the  cost of assigning  site  i to  concentrator j.  The  effect of \nthis term is  to  reduce the inhibitions  among the  neurons that correspond to low \ncost assignments.  The sum of the costs of assigning both site i to concentrator j \nand  site  k  to  concentrator I  was  used  in order to  maintain the  symmetry of T. \nThe external input currents were derived from the energy equation (Eq.3) \n\nand  are  given  by \n\nI .. _ {2. k j ,  if  i  <  n \n\nIJ  -\n\n2  \u2022  k j  - 1,  otherwise. \n\n(6) \n\nThis exemplifies a  teChnique  for combining  external input currents which arise \nfrom  combinations  of certain  basic  types  of constraints. \n\nAN EXAMPLE \n\nThe  neural  network  solution for  a  concentrator assignment problem con(cid:173)\nsisting  of twelve  sites  and five  concentrators  was  simulated.  All  sites  and con(cid:173)\ncentrators  were  located  within  the  unit square on a  randomly  generated  map. \nFor this  problem,  it was  assumed  that no  more  than  three  sites  could  be \n\nassigned  to  a  concentrator.  The  assignment  cost  matrix  and  a  typical  assign(cid:173)\nment  resulting  from  the  simulation  are  shown  in  Fig.  5.  It  is  interesting  to \nnotice  that the  network proposed an assignment which made no  use of concen(cid:173)\ntrator 2. \n\nBecause  the  capacity  of  each  concentrator  kj  was  assumed  to  be  three \nsites,  the  external  input current for  each  neuron  in the  upper  sub-array  was \n\nI ij  = 6 \n\nwhile  in  the  sub-array  of slack  neurons  it was \n\nI ij  = 5. \n\nThe  other  parameter values  used  in  the  simulation  were \n\nand \n\nA  = B  = C  =-2 \n\nD  = 0.1  . \n\n\f781 \n\nCONCENTRATORS \n\n1 \n\n2 \n\n3 \n\n4 \n\n5 \n\nSITES \n\nA \n\nB \n\nC \n\nD \n\nE \n\nF \n\nG \n\n.31 \n\n.62 \n\n.25 \n\n.51 \n\n.17 \n\n.39 \n\nH  @  .81 \n\n.67 \n\n.84 \n\n.33 \n\nI \n\nJ \n\nK \n\nL \n\n.60 \n\n@ \n\n.42 \n\n@ \n\n.47 \n\n.28 \n\n.72 \n\n.75 \n\n.55  @  .46 \n@ \n\n.40 \n\n.63 \n\n.95 \n\n.88 \n\n.71  @  .39 \n.78  @  .38 \n\n.92 \n\n.82 \n\n.81 \n\n.77 \n\n.76 \n\n.54 \n\n.56  @ \n.46  G \n.41  G \n.44  G  .51 \n.55  B  .38 \n\n.76 \n\n.66 \n\n.48 \n\n.52 \n\n.56 \n\n.60  1.05 \n\n.71 \n\n.18 \n\nFig.  5.  The  concentrator assignment cost  matrix  with  choices  circled. \n\nSince  this  choice  of  parameters  results  in  a  T  matrix  that  is  symmetric \nand  whose  diagonal  entries  are  all  zeros,  the  network  will  converge  to  the \nminima of Eq.  3.  Furthermore,  inclusion of the  term which  is  weighted  by  the \nparameter D  causes  the  network  to  favor  minimum  cost assignments. \n\nTo  evaluate  the  performance  of  the  simulated  network,  an  exhaustive \n\nsearch of all  solutions to the problem was conducted using  a backtracking algo(cid:173)\nrithm.  A  frequency distribution of the  solution costs associated with the assign(cid:173)\nments generated by the  exhaustive search is  shown in Fig.  6.  For comparison, \na  histogram  of the  results  of  one  hundred  consecutive  runs  of  the  neural-net \nsimulation is  shown in Fig.  7.  Although the  neural-net simulation did  not find \na  global  minimum,  ninety-two  of the  one  hundred  assignments  which  it  did \nfind  were  among  the  best 0.01 %  of all  solutions  and  the  remaining  eight were \namong  the  best 0.3%. \n\nNeural  networks  can  be  used  to  find  good,  though  not  necessarily  opti(cid:173)\n\nmal,  solutions  to  combinatorial  optimization  problems  like  the  concentrator \n\nCONCLUSION \n\n\f782 \n\nFrequency \n\n4000000 \n3500000 \n3000000 \n250000 \n\n1500000 \n100000 \n500000 \n\nOL----\n3.2 \n4.2 \n\n5.2 \n\n6.2 \n\n7.2 \n\nCost \n8.2 \n\nFig.  6.  Distribution  of assignment \ncosts  resulting  from  an exhaustive \nsearch of all  possible  solutions. \n\nFrequency \n25 \n\n20 \n\n15 \n\n10 \n\n5 \n\no \n\nFig.  7.  Distribution  of assignment \ncosts  resulting  from  100  consecu(cid:173)\ntive  executions  of the  neural  net \nsimulation. \n\nassignment problem.  In order to  use  a  neural  network to  solve  such problems, \nit is  necessary to be able to represent a  solution to the problem as a  state of the \nnetwork.  Here  the  concentrator  assignment  problem  was  successfully  mapped \nonto a  Hopfield  network by associating  each neuron with  the  hypothesis  that a \ngiven  site  should  be assigned  to  a  particular  concentrator.  An energy  function \nwas  constructed to  determine  the  connections  that were  needed  and the  result-\ning  neural  network  was  simulated. \n\nWhile  the  neural  network  solution  to  the  concentrator  assignment  prob(cid:173)\nlem  did  not  find  a  globally  minimum  cost  assignment,  it  very  effectively  re(cid:173)\njected poor solutions.  The network was even able to suggest assignments which \nwould  allow  concentrators  to  be  removed  from  the  communication network. \n\nREFERENCES \n\n1.  A.  S.  Tanenbaum,  Computer Networks  (Prentice-Hall:  Englewood  Cliffs, \n\nNew  Jersey,  1981),  p.  83. \n\n2.  E.  Feldman,  F.  A.  Lehner  and  T.  L.  Ray,  Manag.  Sci.  V12,  670  (1966). \n\n3.  A.  Kuehn  and  M.  Hamburger,  Manag.  Sci.  V9,  643  (1966). \n\n4.  T.  Aykin  and A.  1.  G.  Babu,  1.  of the  Oper.  Res.  Soc.  V38,  N3,  241  (1987). \n\n5.  J.  1.  Hopfield,  Proc.  Natl.  Acad.  Sci.  U.  S.  A.,  V79,  2554  (1982). \n\n6.  J.  1.  Hopfield  and  D.  W.  Tank,  Bio.  Cyber.  V52,  141  (1985). \n\n7. D.  W.  Tank and  1.  1.  Hopfield,  IEEE Trans.  on Cir.  and Sys. CAS-33,  N5, \n\n533  (1986). \n\n\f", "award": [], "sourceid": 6, "authors": [{"given_name": "Gene", "family_name": "Tagliarini", "institution": null}, {"given_name": "Edward", "family_name": "Page", "institution": null}]}