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Lov\'asz gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a different short proof by reducing to the cubic case. Then we show how to extend the result to (online) list coloring via the Kernel Lemma.

We study the settings where we are given a function of n variables defined in a given box of integers. We show that in many cases we can replace the given objective function by a new function with a much smaller domain. Our approach allows…

We present three new complexity results for classes of planning problems with simple causal graphs. First, we describe a polynomial-time algorithm that uses macros to generate plans for the class 3S of planning problems with binary state…

We extend the methods and results of [arXiv 1603.04896] to the setting of multinomial distributions satisfying certain properties. These include all the multinomial distributions arising from the direct proof of the Central Limit Theorem…

It is shown that the length of the algorithmic minimal sufficient statistic of a binary string x, either in a representation of a finite set, computable semimeasure, or a computable function, has a length larger than the computational depth…

We give a triplet of short proofs, each of which answers a question raised by Erd\H{o}s. The first concerns the small prime factors of $\binom{n}{k}$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$…

We give a proof of a result of Bonet, Engli\v{s} and Taskinen filling in several details and correcting some flaws.

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential…

A criterion is established for the transitivity of connectedness in a transfinite graph. Its proof is much shorter than a prior argument published previously for that criterion.

This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity…

Modern software for propositional satisfiability problems gives a powerful automated reasoning toolkit, capable of outputting not only a satisfiable/unsatisfiable signal but also a justification of unsatisfiability in the form of resolution…

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with…

A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in…

We provide new families of minimal codes in any characteristic. Also, an inductive construction of minimal codes is presented.

A short survey about combinatorics on words and algorithmic methods in a ring. Special attention is given to Shirshov's results. Adopted for undegraduate students.

Given a convex set $Q \subseteq R^m$ and an integer matrix $W \in Z^{m \times n}$, we consider statements of the form $ \forall b \in Q \cap Z^m$ $\exists x \in Z^n$ s.t. $Wx \leq b$. Such statements can be verified in polynomial time with…

We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given where each…

We show that the nearring $(\mathbb{Z}[x],+,\circ)$ of integer polynomials, where the nearring multiplication is the composition of polynomials, has uncountably many subnearrings, and we give an explicit description of those nearrings that…

Kolmogorov complexity theory is used to tell what the algorithmic informational content of a string is. It is defined as the length of the shortest program that describes the string. We present a programming language that can be used to…

We study the problems of finding a shortest synchronizing word and its length for a given prefix code. This is done in two different settings: when the code is defined by an arbitrary decoder recognizing its star and when the code is…