Testing Expansion in Bounded Degree Graphs Artur Czumaj

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

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

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

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

Property Testing definition • Given input x • If x has the property e tester passes • If x is -far from any string that has the property e 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

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

1 st definition Graph G is -far from satisfying property P If one needs to modify more than -fraction of entries in adjacency matrix to obtain a graph satisfying P 0 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0

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

2 nd definition Graph G is -far from satisfying property P 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 • Every hereditary property can be tested in constant-time! constant-time [Alon & Shapira, 2003 -2005] • Property is hereditary if – It holds if we remove vertices

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 – Typical running time:

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

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

Bounded-degree adjacency list model • Testing bipartitness (2 -colorability) ~ – Can be done in O(n 1/2 / O(1)) time (Goldreich & Ron) – Cannot be done faster (Goldreich & Ron) But we had O(1/ O(1))-time tester in the adjacency matrix model • So: no constant-time algorithms 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? • Most property testing results in the bounded degree model use expansion • (n 1/2) lower bound [Goldreich & Ron] – even to distinguish between a very good expander and disconnected graph with several huge components

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

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 d otherwise stay

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 cn to be close to expected: 1/n( 2 ) ~ c 2/2 1/2 • 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

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

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

Algorithm of Goldreich and Ron s r 16/ • Choose s = O(1/ ) vertices at random 2 ln(n/ )/ 2 1/2 2 l r 16 d • For each chosenmvertex v r 12 s n / – 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 r(1+7 ) ( m 2) /n – accept Easy to see: -expander will be accepted (with prob. r 0. 99) Task: prove that a poor expander will be rejected

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: – G is -far from *-expander there is a large set X with b c 2|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 X has small neigborhood X V-X 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!

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 A V–A c *-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 1. 2. 3. 4. Remove all edges incident to A Add (d-1)-regular good expander in A Remove a matching M of size |A|/2 in G[V-A] Add arbitrary matching between A and M We can modify dn/2 edges in G to obtain an *-expander

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 1. Start with X = ; If G is isnot -far from -expander: 2. G[V-A] an expander 4 V-X with every 9 A “small” setsmall canneighborhood be removed so that X = A [ X graph is still not an expander the 3. remaining * 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 2/log n)-vertex expansion in time O(d 2 ln(n/ ) n 1/2/( 2 3))

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 2 d and -far from graphs with 2/c-vertex expansions

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? 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 2/c-vertex expansions Similar result for algebraic definition of expansion

Key improvement C & Sohler: 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| More direct/tighter analysis of the random walks (via conductance) leads to the following: 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)

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? 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 2/c-vertex expansions Similar result for algebraic definition of expansion

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