Linear time Constructions of some dd-Restriction Problems

We give new linear time globally explicit constructions for perfect hash families, cover-free families and separating hash functions

A Simple Algorithm for Hamiltonicity

We develop a new algebraic technique that solves the following problem: Given a black box that contains an arithmetic circuit $f$ over a field of characteristic $2$ of degree~$d$. Decide whether $f$, expressed as an equivalent multivariate polynomial, contains a multilinear monomial of degree $d$. This problem was solved by Williams \cite{W} and Bj\"orklund et. al. \cite{BHKK} for a white box (the circuit is given as an input) that contains arithmetic circuit. We show a simple black box algorithm that solves the problem with the same time complexity. This gives a simple randomized algorithm for the simple $k$-path problem for directed graphs of the same time complexity\footnote{$O^*(f(k))$ is $O(poly(n)\cdot f(k))$} $O^*(2^k)$ as in \cite{W} and with reusing the same ideas from \cite{BHKK} with the above gives another algorithm (probably not simpler) for undirected graphs of the same time complexity $O^*(1.657^k)$ as in \cite{B10,BHKK}

Optimal Query Complexity for Reconstructing Hypergraphs

In this paper we consider the problem of reconstructing a hidden weighted hypergraph of constant rank using additive queries. We prove the following: Let $G$ be a weighted hidden hypergraph of constant rank with n vertices and $m$ hyperedges. For any $m$ there exists a non-adaptive algorithm that finds the edges of the graph and their weights using $$O(\frac{m\log n}{\log m})$$ additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal Query Complexity Bounds for Finding Graphs. {\em STOC}, 749--758,~2008]. When the weights of the hypergraph are integers that are less than $O(poly(n^d/m))$ where $d$ is the rank of the hypergraph (and therefore for unweighted hypergraphs) there exists a non-adaptive algorithm that finds the edges of the graph and their weights using $$O(\frac{m\log \frac{n^d}{m}}{\log m}).$$ additive queries. Using the information theoretic bound the above query complexities are tight