Uri Zwick

We present two new algorithms for solving the {\em All

Pairs Shortest Paths\/} (APSP) problem for weighted directed

graphs. Both algorithms use fast matrix multiplication algorithms.

The first algorithm

solves the APSP problem for weighted directed graphs in which the edge

weights are integers of small absolute value in ...
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Ryan Williams

We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm; more precisely, it is a constant factor improvement in the base of ... more >>>

Artur Czumaj, Andrzej Lingas

We show that for any $\epsilon > 0$, a maximum-weight triangle in an

undirected graph with $n$ vertices and real weights assigned to

vertices can be found in time $\O(n^{\omega} + n^{2 + \epsilon})$,

where $\omega $ is the exponent of fastest matrix multiplication

algorithm. By the currently best bound ...
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Noga Alon, Amir Shpilka, Chris Umans

We present several variants of the sunflower conjecture of Erd\H{o}s and Rado and discuss the relations among them.

We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd and Cohn et al. regarding possible approaches for obtaining fast matrix multiplication algorithms. ... more >>>

Marcos Villagra, Masaki Nakanishi, Shigeru Yamashita, Yasuhiko Nakashima

In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input ... more >>>

Shiva Manne, Manjish Pal

In this paper, we present novel deterministic algorithms for multiplying two $n \times n$ matrices approximately. Given two matrices $A,B$ we return a matrix $C'$ which is an \emph{approximation} to $C = AB$. We consider the notion of approximate matrix multiplication in which the objective is to make the Frobenius ... more >>>

Andris Ambainis, Yuval Filmus, Francois Le Gall

Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time $O(n^{2.3755})$. Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an improved algorithm running in time $O(n^{2.3729})$. These algorithms are obtained by analyzing higher ... more >>>

Stasys Jukna, Georg Schnitger

We prove a general lower bound on the size of branching programs over any semiring of zero characteristic, including the (min,+) semiring. Using it, we show that the classical dynamic programming algorithm of Bellman, Ford and Moore for the shortest s-t path problem is optimal, if only Min and Sum ... more >>>

Joshua Grochow, Cris Moore

In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x 2 matrix multiplication could be performed with ... more >>>

Swastik Kopparty, Guy Moshkovitz, Jeroen Zuiddam

Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the ... more >>>

Matthias Christandl, François Le Gall, Vladimir Lysikov, Jeroen Zuiddam

We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give optimal algorithms. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously ... more >>>

Vahid Reza Asadi, Alexander Golovnev, Tom Gur, Igor Shinkar

We present a new framework for designing worst-case to average-case reductions. For a large class of problems, it provides an explicit transformation of algorithms running in time $T$ that are only correct on a small (subconstant) fraction of their inputs into algorithms running in time $\widetilde{O}(T)$ that are correct on ... more >>>

Robert Andrews

We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting $\underline{R}(n)$ denote the border rank of $n \times n \times n$ matrix multiplication, we construct a hitting set generator with seed length $O(\sqrt{n} \cdot ... more >>>

Paul Beame, Niels Kornerup

We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems---including matrix-vector product, matrix inversion, matrix multiplication and powering---existing ... more >>>