1. Artur Czumaj DIMAP DIMAP (Centre for Discrete Maths and it Applications) Computer Science & Department of Computer Science University of Warwick Testing Expansion in Bounded Degree Graphs

2. Topic of this talk How to distinguish good expanders from weak expanders n 1/2For graphs of bounded degree, we can distinguish expanders from graphs that are “far” even from poor expanders in O(n 1/2 ) time [in the framework of property testing] ~ Main technical challenge: Analysis of random walks on “non-expanders”

3. Expanders Informally: a graph is an expander if it expands well every set of vertices has many neighbors

4. Testing expanders in “property testing framework” To distinguish expanders from graphs that are far from being even poor expanders: PROPERTY TESTING we’ll use the framework of PROPERTY TESTING

5. Property Testing definition Given input x If x has the property  tester passes If x is  -far from any string that has the property  tester fails error probability < 1/3 Notion of  -far depends on the problem; Typically: one needs to change  fraction of the input to obtain object satisfying the property Typically we think about  as on a small constant, say,  = 0.1

6. Graph properties Measure of being far/close from a property farIs graph connected or is far from being connected? These two graphs are close to be connected

7. Graph properties Measure of being far/close from a property farIs graph connected or is far from being connected? far from being connected

8. 1 st definition Graph G is  -far from satisfying property P adjacency matrix If one needs to modify more than  -fraction of entries in adjacency matrix to obtain a graph satisfying P 01001 10111 01001 01000 11100

9. 1 st definition Graph G is  -far from satisfying property P adjacency matrix If one needs to modify more than  -fraction of entries in adjacency matrix to obtain a graph satisfying P  ¢ n 2 edges have to be added/deleted Suitable for dense graphs Usually “boring” for sparse graphs

10. 2 nd definition Graph G is  -far from satisfying property P adjacency lists If one needs to modify more than  -fraction of entries in adjacency lists to obtain a graph satisfying P 1 5 2 3 4 52 1 1453 5 2 2 2 3

11. 2 nd definition Graph G is  -far from satisfying property P adjacency lists If one needs to modify more than  -fraction of entries in adjacency lists to obtain a graph satisfying P Suitable for sparse graphs Main model: graphs of bounded degree

12. Adjacency matrix model There are very fast property testers They’re very simple –Typical algorithm: The analysis is (often) very hard We understand this model very well –mostly because of very close relation to combinatorics Select a random set of vertices U Select a random set of vertices U Test the property on the subgraph induced by U Test the property on the subgraph induced by U

13. General result constant-timeEvery hereditary property can be tested in constant-time! hereditaryProperty is hereditary if –It holds if we remove vertices [Alon & Shapira, 2003-2005]

14. Adjacency matrix model There are very fast property testers They’re very simple –Typical algorithm: The analysis is (often) very hard We understand this model very well –mostly because of very close relation to combinatorics –Typical running time: Select a random set of vertices U Select a random set of vertices U Test the property on the subgraph induced by U Test the property on the subgraph induced by U

15. What’s about adjacency lists model ? We consider bounded-degree model d –graph has maximum degree d [constant] Much less is known Less connection to combinatorics random walksConnection to random walks!

16. Bounded-degree adjacency list model Testing bipartitness (2-colorability) O(n 1/2 /  O(1) ) –Can be done in O(n 1/2 /  O(1) ) time (Goldreich & Ron) ~ Algorithm: O(1/  )Select O(1/  ) starting vertices poly(log n/  ) n 1/2 poly(log n/  )For each vertex run poly(log n/  ) n 1/2 random walks of length poly(log n/  ) rejectIf any of the starting vertices lies on an odd-length cycle then reject acceptOtherwise accept

17. Bounded-degree adjacency list model Testing bipartitness (2-colorability) O(n 1/2 /  O(1) ) –Can be done in O(n 1/2 /  O(1) ) time (Goldreich & Ron) –Cannot be done faster (Goldreich & Ron) So: no constant-time algorithms ~ But we had O(1/  O(1) )-time tester in the adjacency matrix model For general bounded degree graphs, testing most of natural properties require superconstant-time (typically,  (n 1/2 ) )

18. This talk: testing expansion Can we quickly test if a (bounded degree) graph has good expansion?  (n 1/2 ) lower bound [Goldreich & Ron] –even to distinguish between a very good expander and disconnected graph with several huge components Most property testing results in the bounded degree model use expansion

19. This talk: testing expansion Can we test if a (bounded degree) graph has good expansion in O(n 1/2 ) time? Combinatorial expansion: –Expander = graphs without small cuts Every vertex set U (of size at most n/2) has neighborhood of size  |U| (for certain positive constant  ) Algebraic expansion: –Expander = graph with large second largest eigenvalue

20. Algorithm of Goldreich and Ron Choose s = O(1/  ) vertices at random For each chosen vertex v –run m = O(n 1/2 ) random walks of length O(log n) –count the number of collisions at the end-vertices –If the number of collisions is too large then STOP & Reject If no STOP then –accept Random walks are on regular graphs: for each node v: choose a random neighbor with prob. 1 / 2d otherwise stay

21. Algorithm of Goldreich and Ron Key use of the well-known fact: –If a graph is expander then random walk of length O(log n) will reach a random vertex –If we run c n 1/2 random walks (for an appropriate constant c) then we expect the number of collisions to be close to expected: ~ c 2 /2 this is testing of uniform distribution Key task – prove the following: If graph is  -far from expander then for many starting vertices random walk won’t mix cn 1/2 2 1/n ()

22. Can graphs far from expanders rapidly mix? We don’t understand well non-expanders We understand even less graphs that are far from expanders Goldreich and Ron suggested algorithm Couldn’t analyze it Gave a conjecture – which if true – would yield property tester

23. Testing vertex expansion  -expanderGraph G = (V,E) is an  -expander if For every X 4 V, |X|  |V|/2 holds: |N(X)|   |X| Our goal: –Distinguish graphs with vertex expansion  from those  -far from having vertex expansion  *,  * ¿   In our case  * = O(   /log n) Goldreich & Ron analyzed algebraic notion of expansion Czumaj & Sohler, FOCS’2007 Perhaps main conceptual contribution: moving from algebraic notion of expansion to the combinatorial one

24. Algorithm of Goldreich and Ron Choose s = O(1/  ) vertices at random For each chosen vertex v –run m = O(n 1/2 ) random walks of length O(log n) –count the number of collisions at the end-vertices –If the number of collisions is too large then STOP & Reject If no STOP then –accept m  12 s n 1/2 /  2 l  16 d 2 ln(n/  )/  2 s  16/   (1+7  ) ( ) /n m2m2 Easy to see:  -expander will be accepted (with prob.  0.99) Task: prove that a poor expander will be rejected

25. Testing vertex expansion Key Property: If G is  -far from  * -expander then there is a set of vertices X 4 V such that –  |V|/4  |X|  (1+  )|V|/2 – |N(X)|  c *  * |X| Think: –G is  -far from  *-expander  c  2 |X| / log n there is a large set X with  c  2 |X| / log n neighbors

26. Small ratio cut  bad mixing Think  =  (1) What if we have set X with |N(X)|  c|X|/log n ? Run a random walk of length < c log n/2 that starts at a random vertex from X With a constant probability it won’t leave X !

27. Small ratio cut  bad mixing Start random walk at a random node at V Suppose it starts at a node at X Until it’s in X, in each step it has “probability” |N(X)|/|X| of “leaving” X If random walk is shorter than |X|/|N(X)|  we don’t expect to leave X Collision probability will be large We’ll reject! X has small neigborhood X V - X

28. Small ratio cut  bad mixing We have a large (of size   |V|/4) set X with small neighborhood With a constant probability a node from X will be a starting node for random walks With a constant probability, we will have too many collisions for such a node With a constant probability we will REJECT

29. It suffices to prove “Key Property” Key Property: If G is  -far from  * -expander then there is a set of vertices X 4 V such that –  |V|/4  |X|  (1+  )|V|/2 – |N(X)|  c *  * |X|

30. Auxiliary lemma If G=(V,E) has A 4 V with |A|   n /4 such that G[V – A] is an c  * - expander then G is not  -far from  * -expander If G is  -far from  * -expander: every “small” set can be removed so that the remaining graph is still not an expander

31. Auxiliary lemma If G=(V,E) has A 4 V with |A|   n /4 such that G[V – A] is an c  * - expander then G is not  -far from  * -expander We can modify  dn/2 edges in G to obtain an  * -expander A V – A c  *-expander

32. Auxiliary lemma If G=(V,E) has A 4 V with |A|   n /4 such that G[V – A] is an c  * - expander then G is not  -far from  * -expander We can modify  dn/2 edges in G to obtain an  * -expander 1.Remove all edges incident to A 2.Add (d-1)-regular good expander in A 3.Remove a matching M of size |A|/2 in G[V-A] 4.Add arbitrary matching between A and M

33. Proving “Key Property” If G=(V,E) has A 4 V with |A|   n /4 such that G[V – A] is an c  * - expander then G is not  -far from  * -expander If G is  -far from  * -expander: every “small” set can be removed so that the remaining graph is still not an expander 1.Start with X = ; 2.G[V-A] is not an expander  9 A V-X with small neighborhood 9 A 4 V-X with small neighborhood 3.X = A [ X 4.Repeat step 2 with new A until |X| |V| /4 4.Repeat step 2 with new A until |X|   |V| /4 Proves “Key Property”

34. Summarizing We can distinguish between graphs (of maximum degree d) that have  -vertex expansion and are  -far from graph with (c  2 /log n)-vertex expansion in time O(d 2 ln(n/  ) n 1/2 /(  2  3 ))

35. Further developments: Can we distinguish (in O(n 1/2 ) time) between graphs that have  -vertex expansion and are  -far from graph with  /c-vertex expansion? Partial answer (Kale & Seshadhri’2007): O(n 1/2 )-time to distinguish between graphs of max-degree d that have  -vertex expansion and those with max-degree 2d and  -far from graphs with   /c-vertex expansions ~ ~

36. Further developments: Can we distinguish (in O(n 1/2 ) time) between graphs that have  -vertex expansion and are  -far from graph with  /c-vertex expansion? Similar result for algebraic definition of expansion Full answer (Kale & Seshadhri’2007 AND Nachmias & Shapira’2007): O(n 1/2 )-time to distinguish between graphs of max-degree d that have  -vertex expansion and those with max-degree d and  -far from graphs with    c-vertex expansions ~ ~

37. Key improvement More direct/tighter analysis of the random walks (via conductance) leads to the following: C & Sohler: If G is  -far from  * -expander then there is a set of vertices X 4 V such that –  |V|/4  |X|  (1+  )|V|/2 – |N(X)|  c *  * |X| In the set X with small cut as defined above, for a constant fraction of vertices in X O(log n) random walks won’t mix well (intuition: random walk of O(log n) length will typically stay in X)

38. Further developments: Can we distinguish (in O(n 1/2 ) time) between graphs that have  -vertex expansion and are  -far from graph with  /c-vertex expansion? Similar result for algebraic definition of expansion Full answer (Kale & Seshadhri’2007 AND Nachmias & Shapira’2007): O(n 1/2 )-time to distinguish between graphs of max-degree d that have  -vertex expansion and those with max-degree d and  -far from graphs with    c-vertex expansions ~ ~

39. Open questions: Understand “non-expanding” graphs Understand random walks on “non-???” graphs –That is, on graphs that don’t satisfy certain property