In computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, a property testing algorithm for a decision problem

Decision problem

In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

 is an algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

 whose query complexity to its input is much smaller than the instance size of the problem. Typically property testing algorithms are used to decide if some mathematical object (such as a graph

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

 or a boolean function) has a "global" property, or is "far" from having this property, using only a small number of "local" queries to the object.

For example, the following promise problem

Promise problem

In computational complexity theory, a promise problem is a generalization of a decision problem where the input is promised to belong to a subset of all possible inputs. Unlike decision problems, the yes instances and no instances do not exhaust the set of all inputs...

 admits an algorithm whose query complexity is independent of the instance size (for an arbitrary constant ε > 0):

"Given a graph G on n vertices, decide if G is bipartite

Bipartite graph

In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets...

, or G cannot be made bipartite even after removing an arbitrary subset of at most edges of G."

Property testing algorithms are important in the theory of probabilistically checkable proofs.

Definition and variants

Formally, a property testing algorithm with query complexity q(n) and proximity parameter ε for a decision problem L is a randomized algorithm

Randomized algorithm

A randomized algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits...

 that, on input x (an instance of L) makes at most q(|x|) queries to x and behaves as follows:

• If x is in L, the algorithm accepts x with probability at least ⅔.
• If x is ε-far from L, the algorithm rejects x with probability at least ⅔.

Here, "x is ε-far from L" means that the Hamming distance between x and any string in L is at least ε|x|.

A property testing algorithm is said to have one-sided error if it satisfies the stronger condition that the accepting probability for instances x ∈ L is 1 instead of ⅔.

A property testing algorithm is said be non-adaptive if it performs all its queries before it "observes" any answers to previous queries. Such an algorithm can be viewed as operating in the following manner. First the algorithm receives its input. Before looking at the input, using its internal randomness, the algorithm decides which symbols of the input are to be queried. Next, the algorithm observes these symbols. Finally, without making any additional queries (but possibly using its randomness), the algorithm decides whether to accept or reject the input.

Features and limitations

The main efficiency parameter of a property testing algorithm is its query complexity, which is the maximum number of input symbols inspected over all inputs of a given length (and all random choices made by the algorithm). One is interested in designing algorithms whose query complexity is as small as possible. In many cases the running time of property testing algorithms is sublinear in the instance length. Typically, the goal is first to make the query complexity as small as possible as a function of the instance size n, and then study the dependency on the proximity parameter ε.

Unlike other complexity-theoretic settings, the asymptotic query complexity of property testing algorithms is affected dramatically by the representation of instances. For example, when ε = 0.01, the problem of testing bipartiteness of dense graphs (which are represented by their adjacency matrix) admits an algorithm of constant query complexity. In contrast, sparse graphs on n vertices (which are represented by their adjacency list) require property testing algorithms of query complexity .

The query complexity of property testing algorithms grows as the proximity parameter ε becomes smaller for all non-trivial properties. This dependence on ε is necessary as a change of fewer than ε symbols in the input cannot be detected with constant probability using fewer than O(1/ε) queries. Many interesting properties of dense graphs can be tested using query complexity that depends only on ε and not on the graph size n. However, the query complexity can grow enormously fast as a function of ε. For example, for a long time the best known algorithm for testing if a graph does not contain any triangle

Triangle-free graph

In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally...

 had a query complexity which is a tower function

Tetration

In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

of poly(1/ε), and only in 2010 this has been improved to a tower function of log(1/ε).