狄彥吾 (Yen-Wu Ti)

華夏技術學院資訊工程系

Email: d91010@csie.ntu.edu.tw

Property Testing on CombinatorialObjects

• Preliminaries.• Testing of digraph property.• Testing of group property.• Our plan to construct a tester.• Publications.

Outline

• The general notion of property testing is first formulated by Rubinfeld and Sudan (1996).

The basic idea of property testing

• 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

• 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)).

Property testing and PCP—related works

• 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) 1

• 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).

Probabilistically Checkable Proof (PCP) 2

• 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 (I)

• 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.

Question of property testing (II)

• We believe that property testing is a natural notion whose relevance to applications goes beyond program checking.

Property testing is an interesting topic

• 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.

Property testing on combinatorial objects

• 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.

Efficient Testing of Large Graphs

• 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 (I)

• 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 (II)

• 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.

The number of generators of an abelian finite group (III)

• 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 (I)

• 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 and learning theory (II)

• 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 (I)

• 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 (II)

• 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( ).

Property testing on strong connectivity of digraphs (III)

2||V

• Holt and Reingold show that the graph properties of connectivity and the existence of cycles are both Ω( )-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)).

Research work related to graph property testing

2||V

• 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

.

Reduction between group property and digraph property (I)

Shg 1

• 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 (II)

• Our plan relies on the strong connectivity of Cayley graphs to test if a finite group-like structure s has a k-generator 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.

Reduction between group property and digraph property (III)

• 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.

Finite group-like structure

• 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 (I)

• 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 least .

• In the latter case, s is said to be ε-far from having property Π.

ε-tester for a group property (II)

2M

• 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 (I)

The difficulties in constructing a group property tester (II)

?)(then

be to value thereplace weIf

1)(

1

1

cba

dba

cbacba

• 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)).

Research work related to group property testing

Thank You for your attention.Happy weekend to you!