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

Separator Given a graph G=(V,E), a separator is a vertex set S ½ V 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.

( ,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 ( , k)-separator for any fixed 

VC-dimension A set T is shattered by a collection C of subsets S 1, S 2,  of {1,2, ,n}, if for every subset P ½ T, there is some S i 2 C such that T  S i =P. The VC-dimension of C is the cardinality of the largest T shattered by C.

Lemma 1 Given a graph G=(V,E) and k<n, 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.

 -sample An  -sample with respect to a collection C of subsets S 1, S 2,  of {1,2, ,n}, is a set W such that for every subset S i, we have (|S i |/n-  )|W| · |S i  W| · (|S i |/n+  )|W|

Lemma 2 For some constant c, for every collection C over {1, ,n} of VC dimension d, a random set W ½ {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?

( , 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|

( , 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 k<n, 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.

Corollary For some constant c, for every collection C over {1, ,n} of VC dimension d, a random set W ½ {1, ,n} of size c/  2 (d*log(1/  )+log(1/  )) has probability at least 1-  of being an (  k)-sample for C. What goodness can this corollary have?

( ,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.

Lemma 3 Let W be an ( , k)-sample in a graph G=(V,E). Then for every 0< \alpha <1 1. Every ( ,k)-separator is also an (  + ,k,W)-separator. 2. Every ( ,k,W)-separator is also an (  + ,k)-separator

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.

Theorem For  ¸ 2/3 and arbitrary k, if G=(V,E) has an ( ,k)-separator, then for every \epsilon >0, there is a randomized algorithm with running time n O(1) 2 O(k  -2 log(1/  )) that with probability at least half finds an (  + ,k)-separator in G.

Algorithm Pick a random set W of O(k  -2 log(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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