Finding Small Balanced Separators

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• Finding Small Balanced Separators
• Author Uriel Feige
• Mohammad
  Mahdian
• Presented by
  Yang Liu

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Separator

• Given a graph G(V,E), a separator is a vertex
  set S½V such that the deletion of vertices in S
  results in more than one component.
• A k-separator is a separator S such that S k.

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(?,k)-separator

• A (?,k)-separator is a separator S such that
  there is no component larger than ?V when S is
  removed.
• Finding a (?,k)-separator is not in FPT. But this
  paper provides a FPT algorithm to find a (?e,
  k)-separator for any fixed ? .

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VC-dimension

• A set T is shattered by a collection C of subsets
  S1 , S2 , ? of 1,2, ? ,n, if for every subset P
  ½ T, there is some Si 2 C such that T ? Si P.
• The VC-dimension of C is the cardinality of the
  largest T shattered by C.

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Lemma 1

• Given a graph G(V,E) and kltn, a collection C is
  defined as follows one vertex set P ½ V belongs
  to C if there is a k-separator S, and one
  component when S is deleted has either P or V\(S
  ? P) as its vertex set.
• Lemma 1 the VC-dimension of C is at most ck,
  where c is some constant.

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? -sample

• An ? -sample with respect to a collection C of
  subsets S1 , S2 , ? of 1,2, ? ,n, is a set W
  such that for every subset Si , we have
  (Si/n-
  ?)W Si?W (Si/n ?)W

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Lemma 2

• For some constant c, for every collection C over
  1, ? ,n of VC dimension d, a random set W½ 1,
  ? ,n of size c/?2 (dlog(1/?)log(1/?))
  has probability at least 1-? of being an ?
  -sample for C.
• What goodness can this lemma have?

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(?, k)-sample

• Given a graph G(V,E), an (? , k)-sample is a
  vertex set W such that for every k-separator S
  and for every vertex set P that forms a component
  when S is deleted from G, following is true
• (P/n- ?)W P?W (P/n ?)W

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(?, k)-sample ? -sample

• This is correct with respect to the collection C
  we defined before.
• Reminder of C given a graph G(V,E) and kltn, a
  collection C is defined as follows one vertex
  set P ½ V belongs to C if there is a k-separator
  S, and one component when S is deleted has either
  P or V\(S ? P) as its vertex set.

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Corollary

• For some constant c, for every collection C over
  1, ? ,n of VC dimension d, a random set W½ 1,
  ? ,n of size c/?2 (dlog(1/?)log(1/?)
  ) has probability at least 1-? of being an (?,
  k)-sample for C.
• What goodness can this corollary have?

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(?,k, W)-separator

• A (?, k, W)-separator is a vertex set S such that
  (1) S 6 k, and (2) the remaining graph when S
  is removed has no component with more than ?W
  vertices from W.

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Lemma 3

• Let W be an (?, k)-sample in a graph G(V,E).
  Then for every 0lt \alpha lt1
• Every (?,k)-separator is also an (??
  ,k,W)-separator.
• Every (? ,k,W)-separator is also an (??
  ,k)-separator

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Benefits of (?, k)-sample

• By lemma 3, if we can find a (? ,k,W)-separator
  where W is one (?, k)-sample with an FPT
  algorithm, then finding a (?? ,k)-separator
  is in FPT.
• Since finding a (?,k)-separator is not in FPT,
  this method provides a viable way at the cost of
  small increase at the maximum ration of the
  largest component.

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Theorem

• For ? 2/3 and arbitrary k, if G(V,E) has an
  (?,k)-separator , then for every \epsilon gt0,
  there is a randomized algorithm with running time
  nO(1)2O(k?-2log(1/?)) that with probability at
  least half finds an (? ?,k)-separator in G.

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Algorithm

• Pick a random set W of O(k?-2log(1/?)) .
• For each partition of W into A and B, find the
  minimum cut that separate A and B.
• The minimum cut found is one (?2?,k)-separator
  for G.

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Thanks

• Question ?