Finding Small Balanced Separators Author: Uriel Feige

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

2. 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.

3. (,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  .

4. 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.

5. 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.

6. -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|

7. 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?

8. (, 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|

9. (, 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.

10. 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 (d*log(1/)+log(1/)) has probability at least 1- of being an (,k)-sample for C. • What goodness can this corollary have?

11. (,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.

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

13. 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.

14. 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 nO(1)2O(k-2log(1/)) that with probability at least half finds an ( +,k)-separator in G.

15. 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.

16. Thanks • Question ?