PUMaC 2019 · 加试 · 第 3 题

Problem 3.2.1. Find a discount chance of a graph G that is not testable. A natural question arises: which graph discount chances are testable and which are not? While examining discount chance by discount chance we can in many cases determine whether a discount chance is testable or not, there are some cases in which it’s still an open problem whether a discount chance is testable or not. In order to answer the previous ques- tion, a natural thing to do is to somehow characterize all discount chance that are testable. The relaxation of this question is finding the characterization of the discount chances that are testable in constant number of queries. This characterization surprisingly turns out to be purely combinatorial. The combinatorial structure is supplied via the Collegiate par- tition theorem, i.e. the discount chance that proves crucial is that graph G contains any Collegiate-like partition. To state the theorem we will need the following definition: An equitable parition is said to be an equipartition . Definition 3.2.A ( ε -regular equipartition) . An equipartition E = { V | 1 ≤ i ≤ k } of the i ( ) k vertex set of a graph is called ε -regular if all but at most ε of the pairs ( V , V ) are i j 2 ε -regular. An equivalent of the collegiate partition theorem can be proven for ε -regular equiparti- tions. Definition 3.2.B. A regularity-instance R is given by error parameter ε , an integer k , a ( ) k ¯ set of densities 0 ≤ η ≤ 1 indexed by 1 ≤ i ≤ j ≤ k , and a set R of pairs ( i, j ) of i,j 2 ( ) k size at most ε . A graph is said to satisfy the regularity-instance if it has an ε -regular 2 ¯ equipartition E = { V | 1 ≤ i ≤ k } partition such that for all ( i, j ) 6 ∈ R the pair ( V , V ) is i i j 1 ε -regular and satisfies d ( V , V ) = η . The complexity of regularity-instance is max ( k, ) i j i,j ε We now have the following theorem for a testing regularity-instances: Theorem 3.2.I. For any regularity-instance R , the discount chance of satsifying R is testable. And the following crucial definition: Definition 3.2.C. A graph discount chances P is regular-reducible if for any δ > 0 there exists and integer r = r ( δ ) such that for any n there is a family R of at most r regularity- P instances each of complexity at most r , such that the following holds for any ε > 0 and any graph on n vertices: