If the Current Clique Algorithms are Optimal, So is Valiant's Parser

Abstract

The CFG recognition problem is: given a context-free grammar G and a string w of length n, decide if w can be obtained from G. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in O(n ω ) time, where ω < 2.373 is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic O(n 3 / log 3 n) complexity. Lee (JACM'01) provided evidence that fast matrix multiplication is needed for CFG parsing, and that very efficient and practical algorithms might be hard or even impossible to obtain. Lee showed that any algorithm for a more general parsing problem with running time O(|G| · n 3−ε ) can be converted into a surprising subcubic algorithm for Boolean Matrix Multiplication. Unfortunately, Lee's hardness result required that the grammar size be |G| = Ω(n 6 ). Nothing was known for the more relevant case of constant size grammars.In this work, we prove that any improvement on Valiant's algorithm, even for constant size grammars, either in terms of runtime or by avoiding the inefficiencies of fast matrix multiplication, would imply a breakthrough algorithm for the k-Clique problem: given a graph on n nodes, decide if there are k that form a clique.Besides classifying the complexity of a fundamental problem, our reduction has led us to similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding, a central problem in computational biology, and Dyck Language Edit Distance, answering an open question of Saha (FOCS'14).

Homomorphisms are a good basis for counting small subgraphs

“…• For k-edge paths or generally any pattern of bounded pathwidth, a "meet in the middle" approach yields n k/2+O (1) time algorithms [39,4]. For a while, this approach appeared to be a barrier for faster algorithms, until Björklund et al [5] gave an algorithm for counting k-paths, matchings on k vertices, and other k-vertex patterns of bounded pathwidth in time n 0.455k+O(1) .…”

“…We now turn to hardness results for counting subgraphs. Here, the vertex-cover number of the pattern H plays a special role: When it is bounded by a fixed constant b ∈ N, we have an n b+O (1) time algorithm even when the size of the pattern is otherwise unbounded. However when it is unbounded, e.g., for k-paths, the best known running times are of the form n k for some ∈ (0, 1).…”

“…It is known that ω < 2.373 holds [56,26]. This approach can be generalized from triangles to k-cliques with k ∈ N [47], for a running time of n ωk/3+O (1) . Fast matrix multiplication is also used to improve on exhaustive search for counting cycles of length at most seven [3] and various other problems [38,23].…”

“…A problem is #W [1]-complete if it is equivalent under parameterized reductions to the problem of counting k-cliques, where the allowed reductions are Turing reductions that run in time f (k) · poly(n). As mentioned before, it is known that #ETH implies FPT = #W [1]. In this setting, Theorem 1.4 takes on the following form: The original proof of Theorem 1.5 relied on the #W[1]-completeness of counting k-matchings, which was nontrivial on its own [15,6].…”

“…In this framework, the problem is parameterized by |V (H)|. A problem is #W [1]-complete if it is equivalent under parameterized reductions to the problem of counting k-cliques, where the allowed reductions are Turing reductions that run in time f (k) · poly(n). As mentioned before, it is known that #ETH implies FPT = #W [1].…”