IMA, Google Corp., Microsoft Corp., Yandex Corp

Gharan, Shayan Oveis

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Scholarly Materials And Research @ Georgia Tech

Online	  Stochastic	  Matching	  ProblemShayan	  Oveis	  GharanJoint	  work	  with	  Vahideh	  Manshadi,	  Amin	  SaberiMonday, April 9, 2012Given	  a	  set	  of	  girls,Boys	  arrive	  online,Match	  each	  boy,	  irrevocably,	  maximizing	  the	  size	  of	  the	  matching.	  	  	  2Online	  Bipartite	  Matching	  ProblemMonday, April 9, 2012Given	  the	  set	  of	  bins,Balls:	  adversarially/stochasticallyMeasure	  the	  performance	  against	  optimum	  (offline)	  solution	  	  	  3Online	  Bipartite	  Matching	  Problem	  (cont’)Bins	  are	  givenBalls	  arrive	  onlineMonday, April 9, 2012Demand	  is	  offline	  from	  advertisers	  	   Targeting	  a	  quantity	  (“1M	  ads	  a	  day”)Supply	  is	  online	  based	  on	  page	  views	  4Applications:	  Display	  AdsMonday, April 9, 2012Models: Adversarial:	  Balls	  arrive	  adversarially. Unknown	  Dist:	  Balls	  are	  sampled	  i.i.d.	  from	  an	  unknown	  dist. Known	  Dist:	  Balls	  are	  sampled	  i.i.d.	  from	  a	  given	  dist.	  	  	  5Models	  /	  ResultsModel Lower	  Bound Upper	  BoundAdversarial 1-­‐1/e	  (KVV’90-­‐BM) 1-­‐1/e	  (KVV’90)Known	  Dist.(integral	  rates)0.677	  (FMMM’09)0.699	  (BK’10),	  0.705	  (MOS’10)	  0.729	  (JX’11)0.86	  (MOS’10)Unknown	  Dist. 0.655	  	  (KMT’11)0.696	  (MY’11)0.823	  (MOS’10)Monday, April 9, 2012Given	  G	  with	  n	  ball	  types.For	  t=1,…,n,	  select	  a	  ball	  uniformly,	  with	  replacement	  from	  the	  set	  of	  types.Assign	  the	  ball	  to	  an	  empty	  bin,	  maximizing	  the	  expected	  number	  of	  assigned	  balls.	  	  	  Expected	  graph	  G1	  	  	  	  	  	  	  2	  	  	  	  	  	  	  	  3	  	  	  	  	  	  	  	  	  4Sample	  graph	  G(ω)	  2 1 426Online	  Stochastic	  Matching	  Problem	  (Known	  Dist.)Core	  Difficulty:	  oversampling/undersamplingMonday, April 9, 20127Outline Rounding	  by	  Sampling Applying	  Rounding	  by	  Sampling	  to	  St.	  Matching	  Problem Offline	  Statistics A	  More	  Adaptive	  Algorithm Open	  Problems/Future	  WorksMonday, April 9, 2012Given a fractional vector f in an integral polytope.Goal: Round f to an integer pointWrite	  	  f	  as	  a	  convex	  combination	  of	  integer	  points	  	  	  	  	  f	  =	  p1	  M1	  +	  p2	  M2	  +	  …	  +	  pk	  MkPick	  Mi	  	  with	  probability	  pi	   	  	   Pr	  M~D	  [	  e	  ∈	  M	  ]	  =	  f(e)For	  any	  cost	  function	  c(.)	   E	  M~D	  [	  c(M)	  ]	  =	  c(f).8Rounding	  By	  Sampling	  fMonday, April 9, 2012Distribution Properties DomainPipage	  Rounding Negative	  Correlaion Matroid	  PolytopesMaximum	  Entropy	   Strongest	  form	  of	  Negative	  Dep.Spanning	  TreesRandomized	  Swap	  RoundingNegative	  Correlation,Lower	  tail	  bounds	  for	  Monotone	  Submodular	  fMatroidPolytopes9Additional	  Features	  of	  Particular	  DistributionsMonday, April 9, 2012 Maximizing	  a	  Submodular	  Function	  w.r.t.	  Matroid	  Constraint	  [Calenscu,	  Checkuri,	  Pal,	  Vondrak	  07,	  Vondrak	  08] Pipage	  Rounding Asymmetric	  TSP	  [Asadpour,	  Goemans,	  Madry,	  O.,	  Saberi’09] Maximum	  Entropy	  dist. Symmetric	  TSP	  [O.	  Saberi,	  Singh’10] Maximum	  Entropy	  dist. TSP	  Path	  [An,	  Kleinberg,	  Shmoys’11] Any	  dist.10Applications	  in	  Approximation	  AlgorithmsMonday, April 9, 201211Outline Rounding	  by	  Sampling Applying	  Rounding	  by	  Sampling	  to	  St.	  Matching	  Problem Offline	  Statistics A	  More	  Adaptive	  Algorithm Open	  Problems/Future	  WorksMonday, April 9, 2012Let	  f:	  E	   [0,1]	  be	  the	  expected	  optimum	  offline	  solution	  	  	  	  	  f((x,y))	  =	  PrG(ω)~	  G	  [	  x	  is	  matched	  to	  y	  in	  the	  optimum]	   	   	   E[OPT]	  =	  ∑(x,y)	  f(x,y))f	  is	  a	  fractional	  matching,	  since Each	  ball	  type	  is	  sampled	  once	  in	  expectation Each	  bin	  is	  allocated	  at	  most	  once	  in	  each	  G(ω)	  Note:	  f	  can	  be	  estimated	  in	  polynomial	  time	  …12Offline	  StatisticsMonday, April 9, 2012Using	  Offline	  StatisticsFractional matchingIntegral matchingsRounding by SamplingMonday, April 9, 2012The	  algorithm	  has	  a	  competitive	  ratio	  of	  1-­‐1/e:E[ALG]	  ≥	  (1-­‐1/e)	  |M|	  	  	  	  For	  all,	  (x,y)	  ∈	  M:	  	  	  	  P[y	  is	  allocated]	  =	  P[x	  is	  sampled]	   	   	  	  	  	  	  	  	  	  =	  1-­‐(1-­‐1/n)n	  =	  1-­‐1/eE[OPT]	  =	  ∑e	  f(e)	  =	  E[	  |M|	  ]	  14Rounding	  by	  Sampling	  1	  MatchingOffline:	  	  	  	  Estimate	  f(.),	  Compute	  D	  	  	  	  Sample	  a	  matching	  M~D.	  Online:	  	  	  	  Allocate	  first	  ball	  of	  each	  type	  to	  its	  match	  in	  M,	  and	  	   drop	  the	  restGyxMonday, April 9, 2012	  	  	  15Rounding	  by	  Sampling	  Two	  MatchingsOffline:	  	  Sample	  M1	  and	  M2	  independently	  from	  DOnline:	  	  Allocate	  the	  first	  ball	  of	  each	  type	  according	  to	  	  M1	  	  Allocate	  the	  second	  ball	  of	  each	  type	  according	  to	  M2GMonday, April 9, 2012Algorithm,	  in	  expectation	  matches	   (1-­‐1/e)|M1|	  +	  1/e	  (1-­‐2/e)	  |M2	  \	  M1|P[y	  is	  allocated]=P[x1	  sampled]	  +	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  +P[x1	  not	  sampled].P[x2	  sampled	  twice	  |	  x1	  not	  sampled]	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  =	  (1-­‐1/e)	  +	  1/e	  (1-­‐2/e)	  	  	  16Lower	  Bounding	  ALGOffline:	  	  Sample	  M1	  and	  M2	  independently	  from	  DOnline:	  	  Allocate	  the	  first	  ball	  according	  to	  	  M1	  	  Allocate	  the	  second	  ball	  according	  to	  M2M1 M2yx1 x2Monday, April 9, 2012Theorem	  [Manshadi,O,Saberi’10]:	  the	  algorithm	  has	  a	  competitive	  ratio	  of	  0.668Proof.	  	  E[	  |M1|	  ]	  =	  Σe	  	  f(e)	  =	  E[OPT]	   E[	  |M2\M1|	  ]	  =	  Σ(x,y)	  Pr[(x,y)	  ∈	  M2,	  (x,y)	  ∉	  M1]	  	   	   	  	  	  	  	  	  	  	  	  	  	  =	  Σ(x,y)	  	  f(x,y)	  (1-­‐f(x,y))	   	   	  	  	  	  	  	  	  	  	  	  	  ≥	  E[OPT]/e	  Lower	  Bounding	  E[|M1|],E[|M2|]f(x,y)≤	  Pr[x	  sampled	  once]=1-­‐1/eBy	  independenceMonday, April 9, 201218Outline Rounding	  by	  Sampling Applying	  Rounding	  by	  Sampling	  to	  St.	  Matching	  Problem Offline	  Statistics A	  More	  Adaptive	  Algorithm Open	  Problems/Future	  WorksMonday, April 9, 2012We	  can	  make	  the	  algorithm	  more	  adaptive	  by	  revising	  the	  decisions	  if	  the	  chosen	  bin	  is	  allocated.	  19A	  More	  Adaptive	  Algorithmf(x,y1)xf(x,ym)f(x,y2)Offline:	  Estimate	  optimum	  offline	  solution	  f(.)Online:	  When	  a	  ball	  x	  arrives,	  	  	  	  For	  i=1	   T,	   Sample	  y~f(x,.)	   If	  y	  is	  empty,	  assign	  x	  to	  y.y1 y2 ymMonday, April 9, 2012Theorem	  [MOS’10]:	  A	  variant	  of	  the	  above	  algorithm	  for	  T=2	  has	  a	  0.705	  competitive	  ratio.Theorem	  [JX’11]:	  A	  variant	  of	  the	  above	  algorithm	  for	  T=3	  has	  a	  0.729	  competitive	  ratio.20A	  More	  Adaptive	  AlgorithmOffline:	  Estimate	  optimum	  offline	  solution	  f(.),	  and	  dist.	  DOnline:	  When	  a	  ball	  x	  arrives,	  	  	  	  For	  i=1	   T,	   Sample	  y~f(x,.)	   If	  y	  is	  empty,	  assign	  x	  to	  y.Monday, April 9, 2012OPT	  allocates	  all	  of	  the	  n	  binsBut,	  any	  online	  algorithm	  allocates	  at	  most	  n(1-­‐1/e2).21Hard	  Examplen	  vertices n/e	  verticesA BDifficulty:	  n/e	  of	  the	  ball	  types	  in	  A	  will	  not	  be	  sampled	  at	  allBut,	  we	  do	  not	  know	  that	  until	  the	  end	  of	  the	  input.	  Competitive	  ratio	  ≤	  0.86Monday, April 9, 201222Outline Rounding	  by	  Sampling Applying	  Rounding	  by	  Sampling	  to	  St.	  Matching	  Problem Offline	  Statistics A	  More	  Adaptive	  Algorithm Open	  Problems/Future	  WorksMonday, April 9, 2012Since	  the	  arrival	  distribution	  is	  known,	  using	  Dynamic	  Programming,	  	  	  	  	  	  	  Optimum	  Online	  algorithm	  is	  computable	  in	  time	  O(n.2n).	  -­‐ How	  well	  can	  the	  optimum	  online	  be	  approximated?-­‐ Does	  it	  admit	  a	  PTAS?23BIG	  OPEN	  PROBLEM:	  Competing	  with	  Online	  OptimumMonday, April 9, 2012Stochastic	  Matching	  problem	  as	  a	  special	  case	  of	  online	  stochastic	  SWP:-­‐ 1-­‐1/e	  is	  achievable	  for	  online	  stochastic	  SWP	  [Devanur,	  Jain,Sivan,Wilkens’11]-­‐ There	  are	  submodular	  functions	  s.t.	  online	  SWP	  can	  not	  be	  approximated	  better	  than	  1-­‐1/e	  [Mirrokni,Shapira,Vondrak’08]-­‐ What	  about	  the	  functions	  that	  can	  be	  approximated	  efficiently?[Haeupler,Mirrokni,Zadimoghaddam’11]	  obtained	  0.667	  for	  weighted	  matching	  problem24Future	  Works	  1:Online	  Submodular	  Welfare	  ProblemMonday, April 9, 2012SCSP:	  Given	  a	  directed	  graph	  G,	  find	  the	  smallest	  k	  strongly	  connected	  subgraph	  of	  G.A	  union	  of	  in-­‐aborescence	  and	  out-­‐arborescence	  is	  a	  2-­‐app.[Laekhanukit,O.,Singh’12]:	  Sampling	  the	  in-­‐arborescence	  and	  out-­‐aborescence	  independently	  based	  on	  LP	  solution	  gives	  a	  1+1/k.-­‐ Obtaining	  better	  than	  2	  for	  the	  weighted	  version?25Future	  Works	  2:k-­‐Strongly	  Connected	  Subgraph	  problemMonday, April 9, 201226Conclusion	  and	  Open	  ProblemsModel Lower	  Bound Upper	  BoundKnown	  Dist.With	  integer	  rates0.729	  (JX’11) 0.86	  (MOS’10)Known	  Dist. 0.708	  (JX’11) 0.823	  (MOS’10)Known	  Dist.SWP1-­‐1/e	  (DJSV’11) 1-­‐1/e	  (MSV’08)Monday, April 9, 2012

Online Stochastic Matching Problem

https://smartech.gatech.edu/bitstream/1853/43298/1/Gharan.pdf

Online Stochastic Matching Problem

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