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)

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

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 .

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 ■ 3 ■■■ 1 d n

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): ■ 3 ■■■ 1 d n 1/7-far from bipartite

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

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.

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

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]

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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, ,  ).

Christian Sohler 28 University of Dortmund Thank you!