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Property Testing in Sparse and General Graphs

Michael Krivelevich

Tel Aviv University

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Graph Property Testing

Very general setting:

P = graph property to test(k-colorability, planarity, non-existence of a copy of H, etc.)

Input: graph G on n vertices, n→∞

Promise: GP (positive)or: G is ε-far from P (negative)

(ε-percentage of description of P should be changed to get HP)

Algorithm A (typically randomized): Queries description of PGP Pr[ A accepts G] ≥ 2/3G is ε-far from P Pr[ A rejects G] ≥ 2/3

GP, Pr[ A accepts G] =1 – one-sided error algorithm

Should be specified!

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Property Testing in Dense Graphs- Formally defined in GGR’98(appeared implicitly in combinatorial papers in 70’s, 80’s)

Input graph description: adjacency matrixG=(V,E), V=[n]

Algorithm: queries the adjacency matrix of GQuery: whether (i,j) E(G)?(vertex pair query)Distance: G is is ε-far from P if ≥εn2 entries in A(G) need to

be changed to get HP

1, ( , ) ( )

0,ij

i j E Ga

otherwise

n nA

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Property Testing in Dense Graphs – Brief Summary

“… It’s all about REGULARITY.” (AFNS’06)

• Very strong (and fruitful) connection between property testing in dense graphs and the Szemerédi Regularity Lemma and its versions

(started in AFKS’99 and culminated in AFNS’06)

• Have reached very good understanding of this setting

(though of course quite a few challenging problems remain)

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Dense Graph Model - limitations

• Suitable/tailored for dense graphs only

• Degenerate for many graph properties Ex. : P = “ G is connected”

- Always answer “YES” ( dist(G,P)≤ n-1 << εn2 )

• A typical algorithm: - sample S [n], |S|=O(1) - look inside to check whether G[S]P

- returns a.s. empty set S for |E(G)|=o(n2) useless/irrelevant

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Property Testing in Bounded Degree Graphs

Introduced by GR’97

• Assumption: Δ(input graph G) ≤ d=const; ε<< 1/d• Graph representation: by incidence lists

L(vi)=(vi,1,…,vi,d) – list of neighbors of vi

• Query: who is the j-th neighbor of vi?(neighbor query)

• Distance: G is ε-far from P if need ≥ εdn modifications in incidence lists to get HP

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Bounded Degree Graphs – an Example

Th. (GR’97): Connectivity in bounded degree model can be tested in O(1/ε2) queries

Proof: Assume: G is ε-far from being connected

G has ≥ εn connected components

G has ≥ εn/2 con. components of size ≤ 2/ε (= small components)

≥ ε/2 percentage of all vertices in small components

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Property Testing in Bounded Degree Graphs (cont.)

Algorithm: Repeat O(1/ε) times:

1. Sample a random vertex vRV

2. Explore the connected component C(v) of v till accumulate 2/ε vertices

3. If |C(v)| ≤ 2/ε – reject

If never reject – accept

One-sided error algorithm with complexity O(1/ε2)

More careful analysis (1/ε) queries O~

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Testing bounded degree graphs – basic tools

• Random sampling

• Local search

(exploring the neighborhood/ball of a vertex)

• Random walks

(a random neighbor of a random neighbor of a random neighbor…)

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Bounded degree – first results

Results from GR’97:

Can test:-connectivity:

connectivity in (1/ε) queries2-edge connectivity: (1/ε2) 3-edge connectivity: (1/ε3) k-vertex connectivity, k=2,3: (1/εk)

- one-sided error algorithms

- cycle-freeness in O(1/ε3) queries- two-sided error algorithmProof idea: G is ε-far from a forest many small components with a cycle, or large components Ci with large surplus e(Ci)-v(Ci)

O~

O~O

~

O~

Uses structural connectivity results (block, cactus, etc.)

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Testing bipartiteness in bounded degree graphsP = “G is bipartite”

Lower bound (GR’97): Ω(√n) queries- in very sharp contrast to the dense case

Proof idea: Negative distribution DN= Hamilton cycle + random perfect matching

(O(1)-far from being bipartite a.s.)Positive distribution DP=Hamilton cycle + random perfect matching between

vertices of different parity

= DN = DP

Any tester: can’t distinguish between DP, DN before having seen a cycle

Takes Ω(√n) queries by birthday paradox

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Testing bipartiteness in bounded degree graphs (cont.)

Th. (GR’99): There is a one-sided error algorithm for testing bipartiteness in the bounded degree model in (√n) queries.

Algorithm: Repeat T= O(1/ε) times:

1. Choose a random vertex sRV2. Perform K:= (√n) random walks of length L:=polylog(n) starting from s3. If get to the same endvertex by an odd and an even path – reject

If no rejection - accept

O~

O~

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Testing bipartiteness in bounded degree graphs (cont.)

Analysis: very elaborate

- relatively easy for rapidly mixing case

[ s Pr[a random walk of length L starting from s] = Θ(1/n) )]

- for general case:

no rapid mixing small cut (M’89)

use them to decompose the graph and the problem

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Testing k-colorability

P = “G is k-colorable”; k≥3 – fixed

Obviously can be done in O(n) queries(just get all O(dn) edges of G)

Th. (BOT’02): For every fixed k ≥3, testing k-colorability in the bounded degree model requires Ω(n) queries

No room for sophisticated testing algorithms

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Testing k-colorability (cont.)

Proof Idea: For one-sided error: Can use classical result of Erdős’62:

Th.: There exists G=(V,E), |V|=n, Δ(G)=O(1), G is ε-far from 3-colorable, but: every δn edges form a 3-colorable graph

tester has to obtain ≥ δn edges to catch G0 G with χ(G0)>3

For two-sided error algorithm:- Two distributions (positive, negative) over instances of systems of

linear equations; Any algorithm can’t distinguish between them in o(n) time- Then: gap preserving reductions from linear equations to 3-

colorability

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Testing in non-expanding bounded degree graphs

Czumaj, Shapira, Sohler’07

Notion of hereditary non-expanding graphs:Def: G is λ-expanding if for every V0 V(G), |V0| ≤n/2,

|N(V0)|≥ λ |V_0|

Def: Graph family F is non-expanding if there exists n0=n0(F) s.t. for all GF , |V(G)|≥ n0, G is not (1/log2n)-expanding

Ex.: F =planar graphs – non-expanding(exists separator of size O(√|V(G)|)

Use: G non-expanding family F , bounded degree can repeatedly cut G to decompose it into constant sized pieces H1,H2,

…, number of edges between pieces ≤ ε n/2

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Testing in non-expanding graphs (cont.)

Th. (CSS): P= hereditary property(closed under taking induced subgraphs, say, 3-colorability)

Assume: Input G non-expanding family F of bounded degree subgraphs

P can be tested over F in constant time f(ε)

Proof idea: Decompose G=(H1,H2,…) as aboveG=negative instance

many of Hi’s are witnesses can be found by random sampling + local search

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Testing planarity

Th. (BSS’08) P = “G is planar”

P can be tested in time Oε(1) in bounded degree graphs by a 2-sided error algorithm(proved more: every minor-closed property P is testable in constant time)

Proof idea: Local statistics in planar graphs differ substantially from those in graphs ε-far from planar

(related to hyper-finite graphs, converging sequences of sparse graphs, etc.)

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Testing planarity (cont.)

Remarks: 1. Get two-sided error algorithm, query complexity exp(exp(exp(1/ε))). Better query complexity?

2. Two-sided vs one-sidedEx: G= bounded degree expander of high girth (Θ(log n))

(say, LPS graph)- Θ(1)-far from planar- every c logn edges form a forest planar subgraph LB=Ω(log n)

can strengthen to Ω(√ n) of GR’97

Conj: P= “G is H-minor free”P can be tested with a one-sided error algorithm in O(√n) queries

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Bounded degree graphs –open questions

• Characterization of testable properties? (testable := testable in Oε(1) queries)

or at least: wide classes of testable properties

• One-sided vs two-sided? Comparative study for various properties

• Testing in restricted graph classes?(á la CSS)

• Tolerant testing? Estimating distance to a given property?

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Bounded degree model - limitations

Opposite/similar to the dense model

• Suitable/tailored only for bounded degree graphs

• Distance notion is “hardwired” – measured always w.r.t. to dn

• Degenerates for certain properties

(e.g. √ n-colorability – always answer “YES”)

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Testing in graphs of general density- Introduced in KKR’03

Main principles:1. Distance in measured w.r.t. to the actual size of the input graph

(latter can be approximated first if necessary)G=(V,E) is ε-far from P if ≥ ε|E| edges need to be changed to get HP(appeared already in PR’02)

2. Queries allowed:a) vertex pair queries: whether (i,j) E(G)?(like in the dense model)b) neighbor queries: j-th neighbor of i V(G)?(like in the sparse model)c) degree queries: what is dG(i)?

No inherent limitation on input graph density

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Testing bipartiteness in general graphs

Th. (KKR’03):

1. Testing bipartiteness can be done in (min(√n, n/d)) queries,where d=2|E|/|V| is the average degree of G;

2. Lower bound of Ω(min(√n, n/d))

- continuous interpolation between the sparse and the dense cases

queries

n√

dn√ n

O~

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Testing bipartiteness for general graphs - proofs

Upper bound:Case d≤√n – same as in the bounded degree model

K:= Oε(√n), L:=polylogε(n)

Repeat T= O(1/ε) times:1. Choose a random vertex sRV2. Perform K random walks of length L starting from s3. A0 = endpoints of walks corresponding to paths of even length A1 = endpoints of walks corresponding to paths of odd length4. If A0∩ A1 ≠Ø – reject, found an odd cycle

Never rejected - accept

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Testing bipartiteness for general graphs – proofs (cont.)

Upper bound:Case d≥√n

Now: K:= Oε(√(n/d)), L:=polylogε(n)

A0 , A1 – as before

Check whether A0 or A1 spans an edge(here use vertex pair queries)

If happens – reject

Never happens - accept

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Testing bipartiteness for general graphs – proofs (cont.)

Lower bound:

Negative distribution DN= Gn,d – random d-regular graphPositive distribution DP=Gn/2,n/2,d – random bipartite d-regular graph

- choose an equipartition V=(V1,V2) u.a.r.- construct a random d-regular bipartite graph between V1, V2

Proof idea: ALG = arbitrary algorithm• o(n/d) vertex pair queries a.s. do not produce an edge• have seen o(√n) vertices a.s. no neighbor query closes a cycle

(birthday paradox)

o(min(n/d, √n)) queries – both items apply, can’t distinguish between DP, DN

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Testing triangle-freeness in general graphs

Result of AKKR’06

Property P to test = “G is K3-free”

Most interesting part – Lower Bound

d:=average degree of the input graph• d≤ n1-δ(n), δ(n)→ 0 Ω(n1/3) queries are needed• d=Θ(n) Oε(1) queries are enough (AFKS’99) Threshold-like behavior for query complexity, abrupt change around

d=Θ(n)

Proof Idea: Cayley graphs, set of generators – random subset of a dense 3AP-free set (c.f. A’02 for the dense case)

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Comparative study of strength of different query types

- BKKR’08Test case: k-colorability, k≥3 fixedModels to compare: vertex pair queries neighbor queries combined model (pair+neighbor queries) new query type – group query

Group query: vV - vertex, S – vertex subset

? Whether there is an edge between v and S in G ?

YES/NO(and then can find a random edge between v and S in O(log n) queries if needed)

- motivated by Group Testing

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Comparative study of strength of different query types -results

On the qualitative level:• vertex pair, neighbor < combined model < group query

• vertex pair queries are better for dense graphs, neighbor queries are better for sparse graphs

• for group queries: UB=O(n/d)

LB= Ω(n/d)

(d := average degree of the input graph)

Say, in testing bipartiteness

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Testing general graphs – open problems

Results for (other) concrete problems?(testing H-freeness, k-colorability, etc.)

Develop technology for proving lower bounds

One-sided vs two-sided error algorithms?

What if given ability to sample a random edge?(to eliminate hiding small dense hard instances)

Further query types, their comparison? Query types driven by practical applications?