An optimal algorithm for computing Frieze–Kannan regular partitions

A Deterministic Algorithm for the Frieze-Kannan Regularity Lemma

The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. Williams [25] recently as...

A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem

We describe a 2O(r √ n(logn)5/2) ǫ −2 time randomized algorithm which estimates the number of feasible solutions of a multidimensional knapsack problem within 1± ǫ of the exact number. (Here r is the number of constraints and n is the number of integer variables.) The algorithm

A fast new algorithm for weak graph regularity

We provide a deterministic algorithm that finds, in $\epsilon^{-O(1)} n^2$ time, an $\epsilon$-regular Frieze-Kannan partition of a graph on $n$ vertices. The algorithm outputs an approximation of a given graph as a weighted sum of $\epsilon^{-O(1)}$ many complete bipartite graphs. As a corollary, we give a deterministic algorithm for estimating the number of copies of $H$ in an $n$-vertex grap...

An Elementary Method for Computing the Kostka Coefficients

Efficient Learning of Simplices

We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan Frieze et al. (1996). Our result can also be interpreted as efficiently learning the interse...