Related papers: On the Peterson hit problem

We consider the problem of minimal correction of the training set to make it consistent with monotonic constraints. This problem arises during analysis of data sets via techniques that require monotone data. We show that this problem is…

The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict…

We consider the algorithmic problem of generating each subset of $[n]:=\{1,2,\ldots,n\}$ whose size is in some interval $[k,l]$, $0\leq k\leq l\leq n$, exactly once (cyclically) by repeatedly adding or removing a single element, or by…

In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in $\mathbf{SL}_2(\mathbb{R})$ from which…

Forman's Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoeldberg we show that this theory can be extended to chain complexes of free modules over a ring. We provide three applications…

Dealing with large numbers of symmetries is often problematic. One solution is to focus on just symmetries that generate the symmetry group. Whilst there are special cases where breaking just the symmetries in a generating set is complete,…

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a…

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G cong H. For several decades, the n^(log_p n + O(1)) generator-enumeration bound (where p is the smallest prime…

In the classical linear degeneracy testing problem, we are given $n$ real numbers and a $k$-variate linear polynomial $F$, for some constant $k$, and have to determine whether there exist $k$ numbers $a_1,\ldots,a_k$ from the set such that…

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to…

We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests,…

In the recent paper arXiv:0710.4085 was shown that any solution of "the polynomial moment problem", which asks to describe polynomials Q orthogonal to all powers of a given polynomial P on a segment, may be obtained as a sum of some…

Let $G$ be a complete edge-weighted graph on $n$ vertices. To each subset of vertices of $G$ assign the cost of the minimum spanning tree of the subset as its weight. Suppose that $n$ is a multiple of some fixed positive integer $k$. The…

The partition problem is a well-known basic NP-complete problem. We mainly consider the optimization version of it in this paper. The problem has been investigated from various perspectives for a long time and can be solved efficiently in…

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about…

We investigate the geometric hitting set problem in the online setup for the range space $\Sigma=({\cal P},{\cal S})$, where the set $\P\subset\mathbb{R}^2$ is a collection of $n$ points and the set $\cal S$ is a family of geometric objects…

{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a…

We analyze the stable isomorphism type of polynomial rings on degree 1 generators as modules over the sub-algebra A(1) = <Sq^1, Sq^2> of the mod 2 Steenrod algebra. Since their augmentation ideals are Q_1-local, we do this by studying the…

A theorem of O. Forster says that if $R$ is a noetherian ring of Krull dimension $d$, then any projective $R$-module of rank $n$ can be generated by $d+n$ elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there…

We consider the operation of division in Pimenov algebras. We obtain necessary and sufficient conditions for prime elements in Pimenov algebras with a number of generators less than five. We adduce examples of the factorization of elements…