Artur Czumaj Dept of Computer Science & DIMAP University of Warwick Testing Expansion in Bounded Degree Graphs Joint work with Christian Sohler

Topic of this talk How to distinguish between good expanders and weak expanders I will show that for graphs of bounded degree, we can distinguish between expanders and graphs that are “far” even from poor expanders in O(n 1/2 ) time [in the framework of property testing] ~

Property testing Classical decision problem: –Given a property P and input instance I –Does I has property P? What we want to study [relaxation]: –Is I close to satisfy property P? Often it’s hard (NP-complete or even undecidable) Can work fast even for NP-hard or undecidable properties

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

Examples Early motivation: –Program checking –Program verification –Learning theory Testing properties of functions: –Linearity test –Low total degree polynomial tests –Low complexity functions –… Properties of distributions: –Is given distribution uniform? –… Useful in –Program checking –PCP constructions

Study of combinatorial properties [Goldreich Goldwasser Ron] Graph properties Hypergraph properties Monotonicity Set properties Geometric properties String properties Membership in low complexity languages (regular languages, constant width branching programs, context-free languages …)

Properties of graphs [Goldwasser, Goldreich, Ron] Graph properties: –Colorability –Not containing a forbidden subgraph –Connectivity –Acyclicity –Rapid mixing –Max-Cut …

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

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

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

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

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

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

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

What’s about adjacency lists model ? We consider bounded-degree model d –graph has maximum degree d [constant] Much less is known

Constant time testing? Very few properties known (for general graphs) –connectivity –k-connectivity –H-freeness –… –very few more

Bounded-degree adjacency list model Recent result (C & Sohler, 2007): –Any hereditary property is testable in constant-time if the input graph belongs to a hereditary and non-expanding family of graphs –Corollary: Testing hereditary properties in planar graphs can be done in constant time. But this doesn’t deal with general graphs

Constant time testing? Very few properties known (for general graphs) –connectivity –k-connectivity –H-freeness –… –very few more

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 ) )

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

This talk: testing expansion Can we test if a (bounded degree) graph has good expansion in O(n 1/2 ) time? Algebraic expansion: –Expander = graph with large second largest eigenvalue Combinatorial expansion: –Expander = graphs without small cuts

This talk: testing expansion Can we test if a (bounded degree) graph has good expansion in O(n 1/2 ) time?

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 / 2 deg(v) otherwise stay

Algorithm of Goldreich and Ron Key use of the well-known fact: –If a graph is expander (regular) 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 this is testing of uniform distribution Idea/hope: –If graph is not expander then for many starting vertices random walk won’t “mix” Key task – prove the following: If graph is  -far from expander then for many starting vertices random walk won’t mix In general: obviously wrong

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 –Conjecture: quite deep property of graphs that are far from being expander

Testing vertex expansion  -expanderGraph G = (V,E) is an  -expander if For every X 4 V, |X| b |V|/2 holds: |N(X)| r  |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

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 r 12 s n 1/2 /  2 l r 16 d 2 ln(n/  )/  2 s r 16/  r (1+7  ) ( ) /n m2m2

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 b |X| b (1+  )|V|/2 – |N(X)| b c *  * |X| Think: c  |X| / log n –G is  -far from  *-expander  X has b c  |X| / log n neighbors

Small ratio cut  bad mixing Think  =  (1) What if we have set X with |N(X)| b 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 !

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

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

Small ratio cut  bad mixing We have a large (of size r  |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

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 b |X| b (1+  )|V|/2 – |N(X)| b c *  * |X|

Auxiliary lemma If G=(V,E) has A 4 V with |A| b  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

Auxiliary lemma If G=(V,E) has A 4 V with |A| b  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

Auxiliary lemma If G=(V,E) has A 4 V with |A| b  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

Auxiliary lemma If G=(V,E) has A 4 V with |A| b  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

Proving “Key Property” If G=(V,E) has A 4 V with |A| b  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| r  |V| /4 Proves “Key Property”

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

Open questions: Can we distinguish (in O(n 1/2 ) time) between graphs that have  -vertex expansion and are  -far from graph with  /c-vertex expansion? Same question for algebraic definition of 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