Related papers: Chaitin \Omega numbers and halting problems
In a recent breakthrough result, Balliu et al. [FOCS'19] proved a deterministic $\Omega(\min(\Delta,\log n /\log \log n))$-round and a randomized $\Omega(\min(\Delta,\log \log n/\log \log \log n))$-round lower bound for the complexity of…
The observation that optimum circuit size changes by at most $O(n)$ under a one-point truth table perturbation is implicit in prior work on the Minimum Circuit Size Problem. This note states the bound explicitly for arbitrary fixed finite…
Let $\Omega(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $\omega(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $\omega(N)$ and $\Omega(N)$…
In several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph…
In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is…
The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this…
The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala etal (2012) show that there exist graphs with n vertices having obstacle number in Omega(n/\log n). In this note, we up this lower bound…
Oscillator Ising machines (OIMs) are networks of coupled oscillators that seek the minimum energy state of an Ising model. Since many NP-hard problems are equivalent to the minimization of an Ising Hamiltonian, OIMs have emerged as a…
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…
Linear programming is a powerful method in combinatorial optimization with many applications in theory and practice. For solving a linear program quickly it is desirable to have a formulation of small size for the given problem. A useful…
Complex networks are everywhere. They appear for example in the form of biological networks, social networks, or computer networks and have been studied extensively. Efficient algorithms to solve problems on complex networks play a central…
Two novel computing models based on an infinite tessellation of space-time are introduced. They consist of recursively coupled primitive building blocks. The first model is a scale-invariant generalization of cellular automata, whereas the…
Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\em $\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\leq \alpha |u|$. Our main result is the $O(\alpha n)$ bound on the…
Robbins' problem of optimal stopping asks one to minimise the expected {\it rank} of observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the {\it value} of the stopped variable under the rule optimal…
Given a boolean predicate $\Pi$ on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for $\Pi$ is a distributed algorithm that can start from any initial configuration of the network (i.e., every…
We prove a lower bound of $\Omega(n^2/\log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, \ldots, x_n)$. Our approach expands and improves upon a result of Raz,…
We consider a classical scheduling problem on $m$ identical machines. For an arbitrary constant $q>1$, the aim is to assign jobs to machines such that $\sum_{i=1}^m C_i^q$ is minimized, where $C_i$ is the total processing time of jobs…
We prove that, assuming $\mathrm{ZF}$, and restricted to any pointed set, Chaitin's $\Omega_U:x\mapsto \Omega_U^x=\sum_{U^x(\sigma)\downarrow}2^{-|\sigma|}$ is not injective for any universal prefix-free Turing machine $U$, and that…
The "turbo codes", recently proposed by Berrou et. al. are written as a disordered spin Hamiltonian. It is shown that there is a threshold Theta such that for signal to noise ratios v^2 / w^2 > Theta, the error probability per bit vanishes…
The lower and the upper irredundance numbers of a graph $G$, denoted $ir(G)$ and $IR(G)$ respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a…