Property testing

In computer science, a property testing algorithm for a decision problem is an algorithm 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 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. In computer science, a property testing algorithm for a decision problem is an algorithm 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 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 admits an algorithm whose query complexity is independent of the instance size (for an arbitrary constant ε > 0): Property testing algorithms are central to the definition of probabilistically checkable proofs, as a probabilistically checkable proof is essentially a proof that can be verified by a property testing algorithm. Formally, a property testing algorithm with query complexity q(n) and proximity parameter ε for a decision problem L is a randomized algorithm that, on input x (an instance of L) makes at most q(|x|) queries to x and behaves as follows: 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. 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 Ω ( n ) {displaystyle Omega ({sqrt {n}})} .