Property Testing on Combinatorial
Objects
狄彥吾 (Yen-Wu Ti)
華夏技術學院資訊工程系
Email: d91010@csie.ntu.edu.tw
Outline
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Preliminaries.
Testing of digraph property.
Testing of group property.
Our plan to construct a tester.
Publications.
The basic idea of property testing
• The general notion of property testing is first
formulated by Rubinfeld and Sudan (1996).
Property testing and PCP
• Property testing emerges naturally in the context of
program checking and probabilistically checkable
proofs (PCP).
• This paradigm is followed in the theory of program
checking by Blum, Luby and Rubinfeld (1993).
Property testing and PCP—related works
• An improved lower bound for the efficient
approximability of many optimization problems
studied previously (see Håstad (1997)).
• An approach takes an atomic verification procedure
and then iterates it several times, saving queries by
recycling them between different iterations of the
atomic test (see Trevisan (1998)).
Probabilistically Checkable Proof (PCP) 1
• The class of decision problems such that a "yes"
answer can be verified by a probabilistically
checkable proof, as follows.
• The verifier is a polynomial-time Turing machine
with access to O(r(n)) uniformly random bits. It has
random access to a proof (which might be
exponentially long), but can query only O(q(n)) bits
of the proof.
Probabilistically Checkable Proof (PCP) 2
• Then we require the following:
1. If the answer is “yes,” there exists a proof such that
the verifier accepts with certainty.
2. If the answer is "no," then for all proofs the verifier
rejects with probability at least 1/2 (over the choice
of the O(r(n)) random bits).
Question of property testing (I)
• Let Π be a fixed property of functions, and f be an
unknown function.
• Our goal is to determine (possibly probabilistically) if
f has property Π or if it is far from any function that
has property Π.
• We are given examples of the form (x; f(x)), where x
is distributed according to the uniform probability
distribution.
Question of property testing (II)
• We are allowed to query f on instances of our choice.
• The complexity measures we focus on are the query
complexity (the number of function queries made)
and the running time of the tester.
Property testing is an interesting topic
• We believe that property testing is a natural notion
whose relevance to applications goes beyond program
checking.
Property testing on combinatorial objects
• We study property testing as applied to group
properties and graph properties.
• We hope to derive extremely fast algorithms for
testing natural properties.
• We only consider the uniform probability distribution
on these combinatorial objects, as well as the
consideration of algorithms that only obtain samples
randomly.
Efficient Testing of Large Graphs
• Goldreich, Goldwasser and Ron showed that certain
individual graph properties, like k-colorability, admit
an ε-test.
• Alon, Fischer, Krivelevich and Szegedy use this
theorem to prove that first order graph properties
not containing a quantifier alternation of type ``  ''
are always testable.
The number of generators of an abelian finite
group (I)
• A generator set for a finite group is a subset of the
group's elements such that repeated multiplications of
the generators alone can produce all the group
elements.
• A generator set with size k is called a k-generator set.
The number of generators of an abelian
finite group (II)
• We can transform a complete graph K into a
quasigroup Q.
• Babai and Erdös prove that the number of the
generators of Q is the lower bound of the size of a
specific subset of a one- factorization of K. The union
of the elements of this subset is a connected
graph.(see Babai and Erdös (1982)).
The number of generators of an abelian
finite group (III)
• Sometimes it is easy to find the number of generators
of a non-abelian group (see Menegazzo (2003)).
• We plan to investigate testing whether an abelian
group has a k-generator set.
Property testing and learning theory (I)
• In both cases one is given access to an unknown
target function.
• The goal of a learning algorithm is to find a good
approximation to the target function.
• A testing algorithm should only determine whether
the target function is in a predetermined function
class or is far away from it.
Property testing and learning theory (II)
• Goldreich, Goldwasser and Ron show that the
relation between learning and testing is nontrivial (see
Goldreich, Goldwasser and Ron (1998)).
• There are function classes for which testing is harder
than learning, provided that NP is not a subset of BPP.
Property testing on strong connectivity of
digraphs (I)
• Let Π be a property of digraphs, that is, a family of
digraphs closed under isomorphism.
• Let 0< ε <1, a digraph G = (V, E) is ε-close to having
property Π if there exists a digraph G’ = (V, E’)
having property Π such that the symmetric difference
between E and E’ is at most ε(|V| choose 2).
• We say that a digraph G is ε-far from having property
Π if it is not ε-close to having property Π.
Property testing on strong connectivity of
digraphs (II)
• T is an ε-tester for Π if for every G = (V, E) and every
ε, the following two conditions hold:
• (1) if G has property Π, then Pr[ T accepts G] ≧2/3;
• (2) if G is ε-far from having property Π, then Pr[ T
accepts G] ≦1/3.
Property testing on strong connectivity of
digraphs (III)
• The probability 2/3 can be replaced by any constant
smaller than 1 because the algorithm can be repeated
if necessary.
• A graph property is testable if the property has a
tester and the total number of queries is o(|V | ).
2
Research work related to graph property testing
• Holt and Reingold show that the graph properties of
connectivity and the existence of cycles are both
Ω( |V | )-evasive (see Holt and Reingold (1972)).
• For any digraph H, Alon and Shapira discuss the exist
of the tester for testing H-freeness in digraphs (see
Alon and Shapira (2004)).
2
Reduction between group property and digraph
property (I)
• Let G be a group,。 be the group multiplication and S
be a subset of the group's elements not containing the
identity element.
• The Cayley graph associated with S is defined as the
digraph having one vertex associated with each group
element G and directed edges (g, h) whenever
g  h 1  S .
Reduction between group property and digraph
property (II)
• The Cayley graph associated with a subset of a
group's elements (but not containing the identity
element) is strongly connected iff the subset generates
the group.
Reduction between group property and digraph
property (III)
• Our plan relies on the strong connectivity of Cayley
graphs to test if a finite group-like structure s has a kgenerator set.
• For an input group-like structure s, if we can test
whether there exists a k-subset of the groundset of s
with a corresponding strongly connected Cayley
graph, then we can test whether s has a k-generator
set.
Finite group-like structure
• A finite group-like structure s is a four-tuple (Γ,。, i,
1), where Γ is the groundset of s,。 is a binary
operator, i is the inverse operator, and 1 is the identity
element.
ε-tester for a group property (I)
• An ε-tester for a property Π is a randomized
algorithm that is given a finite group-like structure s
and a distance parameter ε.
• The tester can make queries as to the results of
operations on elements of s.
ε-tester for a group property (II)
• Let the property Π be {si}.
• Given an upper bound M on the size of the groundset,
the tester needs to distinguish with probability at least
2/3 between the case of s having Π and the case of the
minimal cost to transform s to any si being at
2
least M .
• In the latter case, s is said to be ε-far from having
property Π.
The difficulties in constructing a group property
tester (I)
• We need to make sure that every instance being ε-far
from any instance with property Π={si} will be
rejected with high property
• We would like to have a method to find the lower
bound of the distance between any group-like
structure and {si}.
• But for any group-like structure s, it is very difficult
to find a group which is close to s.
The difficulties in constructing a group property
tester (II)
1
a  b  c  ( a  b )c  1
If we replace the value a  b to be d
then (a  b)c 1  ?
Research work related to group property testing
• Friedl, Ivanyos and Santha construct a tester which,
given a finite group-like structure, tests if it is an
abelian group (see Friedl, Ivanyos and Santha (2005)).
• Blum, Luby and Rubinfeld construct the first
homomorphism tester for abelian groups (see Blum,
Luby and Rubinfeld (1993)).
Thank You for your attention.
Happy weekend to you!