Query-Number Preserving Reductions and Linear Lower Bounds for Testing

Abstract

SUMMARYIn this paper, we study lower bounds on the query complexity of testing algorithms for various problems. Given an oracle that returns information of an input object, a testing algorithm distinguishes the case that the object has a given property P from the case that it has a large distance to having P with probability at least 2 3 . The query complexity of an algorithm is measured by the number of accesses to the oracle. We introduce two reductions that preserve the query complexity. One is derived from the gap-preserving local reduction and the other is from the L-reduction. By using the former reduction, we show linear lower bounds on the query complexity for testing basic NP-complete properties, i.e., 3-edge-colorability, directed Hamiltonian path/cycle, undirected Hamiltonian path/cycle, 3-dimensional matching and NP-complete generalized satisfiability problems. Also, using the second reduction, we show a linear lower bound on the query complexity of approximating the size of the maximum 3-dimensional matching.

An explicit construction of graphs of bounded degree that are far from being Hamiltonian

“…We study graphs that are far from being Hamiltonian. Hamiltonicity is known to be hard for property testing Yoshida and Ito (2010); Goldreich (2020). However this is shown via a randomised construction.…”

“…On the other hand there are some properties whose query complexity is sublinear but not constant, e. g. bipartiteness Ron (2002, 1999). Furthermore, there are some properties for which no tester with sublinear query complexity exists, e. g. Hamiltonicity Yoshida and Ito (2010); Goldreich (2020), 3-colourability Bogdanov et al (2002, independent set size Goldreich (2020). Note that for any computable property there is a linear query complexity property tester, i. e. the tester, which accesses the entire graph and then uses any exact algorithm for the property, as we do not bound the running time of a property tester.…”

“…More precisely, we obtain graphs which differ in Θ(n) edges from any Hamiltonian graph, but non-Hamiltonicity cannot be detected in the neighbourhood of o(n) vertices.Our class of graphs yields a class of hard instances for one-sided error property testers with linear query complexity. It is known that any property tester (even with two-sided error) requires a linear number of queries to test Hamiltonicity (Yoshida, Ito, 2010). This is proved via a randomised construction of hard instances.…”

“…Our class of graphs yields a class of hard instances for one-sided error property testers with linear query complexity. It is known that any property tester (even with two-sided error) requires a linear number of queries to test Hamiltonicity (Yoshida, Ito, 2010). This is proved via a randomised construction of hard instances.…”

Sublinear Computation Paradigm: Constant-Time Algorithms and Sublinear Progressive Algorithms

“…Bogdanov, Obata, and Trevisan [10] showed the first Ω(n) lower bound for Boundeddegree graph 3-colorability, and furthermore for Vertex Cover, Max Cut, Max 2SAT, Max E3SAT † , and Max E3LIN-2 † † , where all the above problems are in the boundeddegree model, by introducing a reduction method. Later, Yoshida and Ito [39] showed that 3-edge-colorability, Directed/undirected Hamiltonian path/cycle, 3-dimensional matching, and Schaefer-type generalized 3SAT, all in the bounded-degree model, also have the same (linear) lower bounds by introducing some new reduction methods.…”