Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation

Abstract

We study two simple yet general complexity classes, which provide a unifying framework for efficient query evaluation in areas like graph databases and information extraction, among others. We investigate the complexity of three fundamental algorithmic problems for these classes: enumeration, counting and uniform generation of solutions, and show that they have several desirable properties in this respect. Both complexity classes are defined in terms of non deterministic logarithmic-space transducers (NL transducers). For the first class, we consider the case of unambiguous NL transducers, and we prove constant delay enumeration, and both counting and uniform generation of solutions in polynomial time. For the second class, we consider unrestricted NL transducers, and we obtain polynomial delay enumeration, approximate counting in polynomial time, and polynomialtime randomized algorithms for uniform generation. More specifically, we show that each problem in this second class admits a fully polynomial-time randomized approximation scheme (FPRAS) and a polynomial-time Las Vegas algorithm (with preprocessing) for uniform generation. Remarkably, the key idea to prove these results is to show that the fundamental problem #NFA admits an FPRAS, where #NFA is the problem of counting the number of strings of length n (given in unary) accepted by a non-deterministic finite automaton (NFA). While this problem is known to be #P-complete and, more precisely, SpanL-complete, it was open whether this problem admits an FPRAS. In this work, we solve this open problem, and obtain as a welcome corollary that every function in SpanL admits an FPRAS.

Counting Problems over Incomplete Databases

“…Besides, it is known that #P = SpanP if and only if NP = UP [35]. 10 Therefore, it is widely believed that #P is properly included in SpanP. The following easy observation can be seen as a first hint that SpanP is a good alternative to describe the complexity of counting completions.…”

“…For every variable , 1 ⩽ ⩽ , we have a null ⊥ , and the (uniform) domain is {0, 1}. For ( , , ) ∈ {0, 1} 3 , we have a relation of arity 3, and we fill it with every tuple of the form ( ′ , ′ , ′ ) with ( ′ , ′ , ′ ) ∈ {0, 1} 3 such that = ′ ∨ = ′ ∨ = ′ holds; 10 Recall that UP is the class Unambiguous Polynomial-Time introduced in [49], and that ∈ UP if and only if there exists a polynomial-time nondeterministic Turing Machine such that if ∈ , then accept ( ) = 1, and if ∉ , then accept ( ) = 0.…”

Effectively Counting s-t Simple Paths in Directed Graphs

“…The subgraph of G induced by A is the graph whose vertices are A and edges are all the edges in E(G) that have both endpoints in A. 1 It is important to not confuse our studied problem with the problem wherein the constraint "simple" is removed, i.e., we count the number of all paths (simple or non-simple) between two vertices s and t. In this problem, a node may appear for several times in a path. Unlike our studied problem (which is #P-complete), this problem can be efficiently solved in polynomial time, using e.g., dynamic programming or matrix multiplication.…”

“…For example and as described in e.g., [3], the number of all paths of size k (simple or non-simple) from s to t is equal to the st-th entry of A k , where A is the adjacency matrix of the graph and A k is the k-th power of A. Arenas et.al. [1] study the problem of counting the number of all paths between two vertices s and t of a length at most k, and show that it admits a fully polynomial-time randomized approximation scheme (FPRAS).…”

Chile's new interdisciplinary institute for foundational research on data

“…Moreover, we have explored new paradigms for answering queries over large volumes of data; in particular, we have studied the problems of enumerating, uniformly generating, and counting the answers to a query, proposing a simple yet general unifying framework to investigate these fundamental algorithmic problems, in particular for the case of graph databases. 5 Query answering methods for emerging requirements. As the requirements for data management systems continue to evolve at an accelerating rate, the diversity of proposed solutions addressing these requirements likewise continues to grow.…”