Quantum Algorithms for Finding Constant-sized Sub-hypergraphs

We develop a general framework to construct quantum algorithms that detect if a $3$-uniform hypergraph given as input contains a sub-hypergraph isomorphic to a prespecified constant-sized hypergraph. This framework is based on the concept of nested quantum walks recently proposed by Jeffery, Kothari and Magniez [SODA'13], and extends the methodology designed by Lee, Magniez and Santha [SODA'13] for similar problems over graphs. As applications, we obtain a quantum algorithm for finding a $4$-clique in a $3$-uniform hypergraph on $n$ vertices with query complexity $O(n^{1.883})$, and a quantum algorithm for determining if a ternary operator over a set of size $n$ is associative with query complexity $O(n^{2.113})$.Comment: 18 pages; v2: changed title, added more backgrounds to the introduction, added another applicatio

sub-cubic

© 2019 Society for Industrial and Applied Mathematics It is a major open problem whether the (min, +)-product of two n × n matrices has a truly subcubic (i.e., O(n3−ε) for ε > 0) time algorithm; in particular, since it is equivalent to the famous all-pairs-shortest-paths problem (APSP) in n-vertex graphs. Some restrictions of the (min, +)-product to special types of matrices are known to admit truly subcubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different, and powerful restriction in which all matrix entries are integers and one matrix can be arbitrary, as long as the other matrix has “bounded differences” in either its columns or rows, i.e., any two consecutive entries differ by only a small amount. We obtain the first truly subcubic algorithm for this bounded-difference (min, +)-product (answering an open problem of Chan and Lewenstein). Our new algorithm, combined with a strengthening of an approach of Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly subcubic algorithms for the following problems: language edit distance (a major problem in the parsing community), RNA folding (a major problem in bioinformatics), and optimum stack generation (answering an open problem of Tarjan)

Fast 2-approximate All-Pairs Shortest Paths

In this paper, we revisit the classic approximate All-Pairs Shortest Paths (APSP) problem in undirected graphs. For unweighted graphs, we provide an algorithm for 2-approximate APSP in O~(n2.5−r+nω(r)) time, for any r∈[0,1]. This is O(n2.032) time, using known bounds for rectangular matrix multiplication nω(r) [Le Gall, Urrutia, SODA 2018]. Our result improves on the O~(n2.25) bound of [Roditty, STOC 2023], and on the O~(mn−−√+n2) bound of [Baswana, Kavitha, SICOMP 2010] for graphs with m≥n1.532 edges. For weighted graphs, we obtain (2+ϵ)-approximate APSP in O~(n3−r+nω(r)) time, for any r∈[0,1]. This is O(n2.214) time using known bounds for ω(r). It improves on the state of the art bound of O(n2.25) by [Kavitha, Algorithmica 2012]. Our techniques further lead to improved bounds in a wide range of density for weighted graphs. In particular, for the sparse regime we construct a distance oracle in O~(mn2/3) time that supports 2-approximate queries in constant time. For sparse graphs, the preprocessing time of the algorithm matches conditional lower bounds [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer, STOC 2023]. To the best of our knowledge, this is the first 2-approximate distance oracle that has subquadratic preprocessing time in sparse graphs. We also obtain new bounds in the near additive regime for unweighted graphs. We give faster algorithms for (1+ϵ,k)-approximate APSP, for k=2,4,6,8. We obtain these results by incorporating fast rectangular matrix multiplications into various combinatorial algorithms that carefully balance out distance computation on layers of sparse graphs preserving certain distance information