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Focus: sub-linear time algorithms – performing the task by inspecting the object at few locations.

Property Testing: informal definitionA relaxation of a decision problem:

For a fixed property Pand any object O,

determine whether O has property P,

or whether O is far from having property P(i.e., far from any other object having P).

Property Testing: the standard (one-sided error) def’n

- A property P = nPn , where Pn is a set of functions with domain Dn.
- The tester gets explicit input n and ,
- and oracle access to a function with domain Dn.
- If f Pn then Prob[Tf(n,) accepts] = 1.
- If f is -far from Pn then Prob[Tf(n,) rejects] > 2/3.

Focus: query complexity q(n,)=q() ( « | Dn |)

Terminology:is called the proximity parameter.

How does a tester use the proximity parameter

Some testers use the proximity parameter merely in order to determine the number of times that a basic test is performed, where the basic test is oblivious of the proximity parameter.

We call such basic tests proximity oblivious testers.

- Example:the BLR (linearity) tester.
- On input (prox.par.) and oracle f,
- repeat the following test O(1/ ) times:
- Select uniformly x,y in Dn
- Accept iff f(x)+f(y)=f(x+y).

Proximity Oblivious Testing: the basic definition

- A property P = nPn ’where Pn is a set of functions with domain Dn.
- A P.O. Tester (POT) gets explicit input n (but not),
- and oracle access to a function with domain Dn.
- If fPn then Prob[Tf(n) accepts] = 1.
- If fPn then Prob[Tf(n) rejects] > (P(f)),
- where :(0,1] (0,1] (is the “detection rate”)
- andP(f) denotes the distance of f fromP.

N.B.: A standard tester is obtained by repeating the POT (i.e., on prox. par. , repeat O(1/()) times).

Focus: constant query complexity q(n)=q ( « | Dn |)

Questions addressed in this work

- Which “testable” properties have POTs?
- How does the complexity of the standard tester obtained by repeating the POT compare to the complexity of the best possible standard tester .
- These questions are studied mainly in two standard models
- of testing graph properties:
- (i) the adjacency matrix model and (ii) the bounded-degree model.

Example:the BLR (linearity) tester.

The complexity of the (std.) tester obtained

by repeating the POT equals (up to a constant)

the complexity of the best possible standard tester.

PART 1: In the adjacency matrix model

A graph G=(V,E) is represented

by a function g:[N][N]{0,1}

(i.e., g(u,v)=1 iff (u,v) is an edge in G).

The adjacency matrix model: two simple examples

A graph G=(V,E) is represented

by a function g:[N][N]{0,1}.

Example 1: Clique.The property of being a clique has a “trivial” two-query POT with ()=.

Example 2: BiClique. The property of being a biclique has a three-query POT with ()=.

Select s[N] arbitrarily, and random u,v[N], and accept iff the induced subgraph is a biclique (i.e., has an even number of edges).

Example 2: analysis of the 3-query POT

Select s[N] arbitrarily, and random u,v[N], and accept iff the induced subgraph is a biclique (i.e., has an even number of edges).

Analysis technique: consider an induced partition.

s

(s)

[N] \ (s)

Suppose that the graph is -far from Biclique. Then

#edges in same side + #non-edges between sides > N2

induced subgraph induced subgraph has 1 or 3 edges has a single edge

Example 3: triangle-freeness [AFKS, Alon]

THM:-freenesshas a 3-query POT with ()=1/Tower(1/), but no O(1)-query POT with ()=poly().

The point is that being-far from-freenessmeans thatN2edges must be omitted to obtain a-free graph,but this does not mean that the graph hasN3 (norpoly()N3 ) triangles.

Conclusion: easy testability and POT-ness are “far from straightforward”.

Example 4: testing bipartiteness

Recall that Bipartitness is efficiently testable with poly(1/) queries.

THM:Bipartitnesshas no O(1)-query POT.

PF:A graph can be-far fromBipartitenessstill all its O(1)-vertex induced subgraphs may be bipartite. E.g., a super-cycle of (1/)(equal-sized) independent sets such that each adjacent pairs of sets is connected by a complete bipartite graph.

Conclusion: easily testable properties may not have POTs.

Characterization of graph properties having a POT

THM (oversimplified):PropertyPhas an O(1)-query POT iff P equals the set of F-free graphs, where F is a fixed set of O(1)-size graphs.

PF idea:Given a POT , we derive a canonical POT (a la [GT]), which yields a characterization of P in terms of forbidden subgraphs (equiv., allowed induced subgraphs). In the other direction, use [AFKS].

Clarification:For a set of graphsFand a graphG, we say thatGisF-free if no induced subgraph of G belongs to F.

THM (actual):PropertyP = NPNhas a O(1)-query POT iff for some constant c and every N, it holds that PN equals the set of FN-free graphs, where FN is a set of c-size graphs.

Example 5: testing Clique Collection (CC)

Recall that CC is efficiently testable with Õ(1/) queries [GR], and even Õ(-4/3) non-adaptive queries suffice.

THM:CChas a 3-query POT with ()=O(2), and no O(1)-query POT can do better.

PF (of the lower bound): Consider a collection of 1/4 balanced bicliques, each of size 4N. This graph is -far fromCCwhile rejecting it requires hitting some biclique at least three times.

Conclusion:The (std.) tester obtained by repeating

the best POT may have significantly higher complexity than the standard tester.

Example 6: testing c-Clique Collection (c-CC)

Recall that c-CC is testable with Õ(1/) queries [GR],even non-adaptively!

THM:For every c2, the property c-CChas a (c+1)-query POT with ()=O(c/2), and no O(1)-query POT can do better.

PF (of the lower bound): Consider a graph consisting of c small cliques, each of size sqrt()N and a large clique of size (1-sqrt())N. This graph is -far fromc-CCwhile rejecting it requires hitting each of the c small cliques.

Conclusion:The (std.) tester obtained by repeating

the best POT may have tremendously higher complexity than the standard tester.

PART 2: In the bounded-degree model

A graph G=(V,E) of degree bound d,

is represented by a function g:[N][d][N]{0}

(i.e., g(u,i)=v iff v is the ithneighbor of u in G

and g(u,i)=0 iff v has less than i neighbors).

The bounded-degree model: preliminaries to the characterization

- DEF (generalized subgraph freeness):graphs with vertices marked full, semi-full, and partial such that a disallowed mapping of F=([n],EF) to G=([N],E) satisfies
- for full vertex v, map(neigh(v)) = neigh(map(v))
- for semi-full vertex v, map(neigh(v)) = neigh(map(v)) map([n])
- for partial vertex v, map(neigh(v)) neigh(map(v))
- E.g., induced (resp., non-induced) graph-freeness corresponds to the special case of using only semi-full (resp., partial) markings.

Generalized subgraph freeness: non-propagation

- DEF (abbrev.):a disallowed mapping of F=([n], EF) to G=([N],E) satisfies
- for full vertex v, map(neigh(v)) = neigh(map(v))
- for semi-full vertex v, map(neigh(v)) = neigh(map(v)) map([n])
- for partial vertex v, map(neigh(v)) neigh(map(v)).

Def:Fisnon-propagating if there exists :(0,1](0,1] such that if every mapping of every marked graph in Fto the graph G uses a vertex in B, then G is (|B|/N)-close to being F-free.

- Not all setsFare non-propagating.
- For any Fwith no full vertices, F is non-propagating.
- Degree-regularity is captured by a non-propagating F.Note that this is a non-hereditary property.

The bounded-degree model: characterization

Def:Fisnon-propagatingif there exists :(0,1](0,1] such that if every mapping of every marked graph in Fto the graph G uses a vertex in B, then G is (|B|/N)-close to being F-free.

- Not all setsFarenon-propagating.
- For any F with no full vertices, F is non-propagating.
- Degree-regularity is captured by a non-propagating F.Note that this is a non-hereditary property.

THM (ov. sim.):A property P has an O(1)-query POT iff for some non-propagating F it holds that P equals F-freeness.

OPEN:Can every generalized subgraph freeness property be captured by F-freeness for some non-propagating F?

Other Models (of property testing)

THM: If property P is testable by a non-adaptive tester that (i) makes a number of queries that only depends on the proximity parameter and (ii) rejects based on a constant-sized “witness”, then P has a POT.

Note: strong codeword tests (cf. [GS]) correspond to POTs.

OPEN:Do codes of 1/polylog rate have O(1)-query codeword POT?

The End

The slides of this talk are available at http://www.wisdom.weizmann.ac.il/~oded/T/pot.ppt

The paper itself is available at http://www.wisdom.weizmann.ac.il/~oded/p_testPOT.html

A companion paper is available at http://www.wisdom.weizmann.ac.il/~oded/p_testAA.html

On the companion paper “Algorithmic Aspects of Property Testing in the Dense Graphs Model”

THM [GT]: If a graph property is testable by q(N,) queries then it is testable by a canonical tester of query complexity O(q(N,)2).

A canonical tester inspects a random induced subgraph and accepted iff the inspected graph has a predetermined property.

Me (since 2001): “In this model, there is no room for algorithms -- property testing reduces to sheer combinatorics.”

Me (now): A finer examination (which cares for the quadratic blow-up) reveals the role of algorithms; as shown in the paper, adaptive algorithms outperform non-adaptive ones, which in turn outperform canonical testers.

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