1. Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj, University of Warwick)

2. Christian Sohler 2 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Property Testing[Rubinfeld, Sudan]: Formal framework to analyze „Sampling“-algorithms for decision problems Decide with help of a random sample whether a given object has a property or is far away from it Property Far away from property Close to property

3. Christian Sohler 3 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Property Testing[Rubinfeld, Sudan]: Formal framework to analyze „Sampling“-algorithms for decision problems Decide with help of a random sample whether a given object has a property or is far away from it Definition: An object is  -far from a property , if it differs in more than an  -fraction of ist formal description from any object with property .

4. Christian Sohler 4 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Bounded degree graphs Graph (V,E) with degree bound d V={1,…,n} Edges as adjacency lists through function f: V  {1,…,d}  V f(v,i) is i-th neighbor of v or ■, if i-th neighbor does not exist Query f(v,i) in O(1) time 1 2 4 3 1234 2442 41 ■ 3 ■■■ 1 d n

5. Christian Sohler 5 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Definition: A graph (V,E) with degree bound d and n vertices is  -far from a property , if more than  dn entries in the adjacency lists have to be modified to obtain a graph with property . Example (Bipartiteness): 1 2 4 3 1234 2442 41 ■ 3 ■■■ 1 d n 1/7-far from bipartite

6. Christian Sohler 6 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Goal: Accept graphs that have property  with probability at least 2/3 Reject graphs that are  -far from  with probability at least 2/3 Complexity Measure: Query (sample) complexity Running time

7. Christian Sohler 7 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Definition [Neighborhood] N(U) denotes the neighborhood of U, i.e. N(U) = {v  V-U:  u  U such that (v,u)  E} Definition [Expander]: A Graph is an  -Expander, if N(U)   |U| for each U  V with |U|  |V|/2.

8. Christian Sohler 8 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Testing Expansion: Accept every graph that is an  -expander Reject every graph that is  -far from an  *-expander If not an  -expander and not  -far then we can accept or reject Look at as few entries in the graph representation as possible

9. Christian Sohler 9 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Related results: Definition of bounded degree graph model; connectivity, k-connectivity, circle freeness [Goldreich, Ron; Algorithmica] Conjecture: Expansion can be tested O(  n polylog(n)) time [Goldreich, Ron; ECCC, 2000] Rapidly mixing property of Markov chains [Batu, Fortnow, Rubinfeld, Smith, White; FOCS‘00] Parallel / follow-up work: An expansion tester for bounded degree graphs [Kale, Seshadhri, ICALP’08] Testing the Expansion of a Graph [Nachmias, Shapira, ECCC’07]

10. Christian Sohler 10 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Difficulty: Expansion is a rather global property Expander with n/2 vertices Expander with n/2 vertices Case 1: A good expander

11. Christian Sohler 11 University of Dortmund Testing Expansion in Bounded Degree Graphs Introduction Difficulty: Expansion is a rather global property Expander with n/2 vertices Expander with n/2 vertices Case 2:  -far from expander

12. Christian Sohler 12 University of Dortmund Testing Expansion in Bounded Degree Graphs The algorithm of Goldreich and Ron How to distinguish these two cases? Perform a random walk for L= poly(log n, 1  ) steps Case 1: Distribution of end points is essentially uniform Case 2: Random walk will typically not cross cut -> distribution differs significantly from uniform Expander with n/2 vertices Expander with n/2 vertices Case 1: A good expander Expander with n/2 vertices Expander with n/2 vertices Case 2:  -far from expander

13. Christian Sohler 13 University of Dortmund Testing Expansion in Bounded Degree Graphs The algorithm of Goldreich and Ron How to distinguish these two cases? Perform a random walk for L= poly(log n, 1  ) steps Case 1: Distribution of end points is essentially uniform Case 2: Random walk will typically not cross cut -> distribution differs significantly from uniform Expander with n/2 vertices Expander with n/2 vertices Case 1: A good expander Expander with n/2 vertices Expander with n/2 vertices Case 2:  -far from expander Idea: Count the number of collisions among end points of random walks

14. Christian Sohler 14 University of Dortmund Testing Expansion in Bounded Degree Graphs The algorithm of Goldreich and Ron ExpansionTester(G, ,l,m,s) 1. repeat s times 2. choose vertex v uniformly at random from V 3. do m random walks of length L starting from v 4. count the number of collisions among endpoints 5.if #collisions> (1+  E[#collisions in uniform distr.] then reject 6. accept

15. Christian Sohler 15 University of Dortmund ExpansionTester(G, ,l,m,s) 1. repeat s times 2. choose vertex v uniformly at random from V 3. do m random walks of length L starting from v 4. count the number of collisions among endpoints 5.if #collisions> (1+  E[#collisions in uniform distr.] then reject 6. accept Theorem:[This work] Algorithm ExpansionTester with s=  (1/ , m=  (  n/poly(  ) and L= poly(log n, d, 1/ , 1/  ) accepts every  -expander with probability at least 2/3 and rejects every graph, that is  -far from every  *-expander with probability 2/3, where  * =  (  ²/(d² log (n/  )). Testing Expansion in Bounded Degree Graphs Main result

16. Christian Sohler 16 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Overview of the proof: Algorithm ExpansionTester accepts every  -expander with probability at least 2/3

17. Christian Sohler 17 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Overview of the proof: Algorithm ExpansionTester accepts every  -expander with probability at least 2/3 (Chebyshev inequality)

18. Christian Sohler 18 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Overview of the proof: Algorithm ExpansionTester accepts every  -expander with probability at least 2/3 (Chebyshev inequality) If G is  -far from an  *-expander, then it contains a set U of  n vertices such that N(U) is small U G

19. Christian Sohler 19 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Overview of the proof: Algorithm ExpansionTester accepts every  -expander with probability at least 2/3 (Chebyshev inequality) If G is  -far from an  *-expander, then it contains a set U of  n vertices such that N(U) is small If G has a set U of  n vertices such that N(U) is small, then ExpansionTester rejects U G

20. Christian Sohler 20 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Overview of the proof: Algorithm ExpansionTester accepts every  -expander with probability at least 2/3 (Chebyshev inequality) If G is  -far from an  *-expander, then it contains a set U of  n vertices such that N(U) is small If G has a set U of  n vertices such that N(U) is small, then ExpansionTester rejects Random walk is unlikely to cross cut -> more collisions U G

21. Christian Sohler 21 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm If G is  -far from an  *-expander, then it contains a set U of  n vertices such that N(U) is small U G

22. Christian Sohler 22 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm If G is  -far from an  *-expander, then it contains a set U of  n vertices such that N(U) is small Lemma: If G is  -far from an  *-expander, then for every A  V of size at most  n/4 we have that G[V-A] is not a (c  *)-expander U G

23. Christian Sohler 23 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm If G is  -far from an  *-expander, then it contains a set U of  n vertices such that N(U) is small Lemma: If G is  -far from an  *-expander, then for every A  V of size at most  n/4 we have that G[V-A] is not a (c  *)-expander Procedure to construct U: As long as U is too small apply lemma with A=U Since G[V-A] is not an expander, we have a set B of vertices that is badly connected to the rest of G[V-A] Add B to U U G

24. Christian Sohler 24 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Lemma: If G is  -far from an  *-expander, then for every A  V of size at most  n/4 we have that G[V-A] is not a (c  *)-expander Proof (by contradiction): Assume A as in lemma exists with G[V-A] is (c  *)-expander Construct from G an  *-expander by changing at most  dn edges Contradiction: G is not  -far from  *-expander A G (c  *)-Expander

25. Christian Sohler 25 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Lemma: If G is  -far from an  *-expander, then for every A  V of size at most  n/4 we have that G[V-A] is not a (c  *)-expander Proof (by contradiction): A G (c  *)-Expander Construction of  *-expander: 1. Remove edges incident to A 2. Add (d-1)-regular c‘-expander to A 3. Remove arbitrary matching M of size |A|/2 from G[V-A] 4. Match endpoints of M with points from A

26. Christian Sohler 26 University of Dortmund Testing Expansion in Bounded Degree Graphs Analysis of the algorithm Lemma: If G is  -far from an  *-expander, then for every A  V of size at most  n/4 we have that G[V-A] is not a (c  *)-expander Proof (by contradiction): A G (c  *)-Expander Construction of  *-expander: 1. Remove edges incident to A 2. Add (d-1)-regular c‘-expander to A 3. Remove arbitrary matching M of size |A|/2 from G[V-A] 4. Match endpoints of M with points from A X Show that every set X has large neighborhood by case distinction

27. Christian Sohler 27 University of Dortmund Testing Expansion in Bounded Degree Graphs Main result ExpansionTester(G, ,l,m,s) 1. repeat s times 2. choose vertex v uniformly at random from V 3. do m random walks of length L starting from v 4. count the number of collisions among endpoints 5.if #collisions> (1+  E[#collisions in unif. Distr.] then reject 6. accept Theorem:[This work] Algorithm ExpansionTester with s=  (1/ , m=  (  n/poly(  ) and L= poly(log n, d, 1/ , 1/  ) accepts every  -expander with probability at least 2/3 and rejects every graph, that is  -far from every  *-expander with probability 2/3, where  * = poly(1/log n, 1/d, ,  ).

28. Christian Sohler 28 University of Dortmund Thank you!