arXiv subjects

Elena Zamaraeva

Publications and source records attributed to Elena Zamaraeva.

7 recordsLinked to original sources

arXiv2025

MACS: Multi-Agent Reinforcement Learning for Optimization of Crystal Structures

Geometry optimization of atomic structures is a common and crucial task in computational chemistry and materials design. Following the learning to optimize paradigm, we propose a new multi-agent reinforcement learning method called Multi-Agent Crystal Structure optimization (MACS) to address periodic crystal structure optimization. MACS treats geometry optimization as a partially observable Markov game in which atoms are agents that adjust their positions to collectively discover a stable configuration. We train MACS across various compositions of reported crystalline materials to obtain a policy that successfully optimizes structures from the training compositions as well as structures of larger sizes and unseen compositions, confirming its excellent scalability and zero-shot transferability. We benchmark our approach against a broad range of state-of-the-art optimization methods and demonstrate that MACS optimizes periodic crystal structures significantly faster, with fewer energy calculations, and the lowest failure rate.

cs.LG↗

arXiv2025

Fast and Near-Optimal Collision-Free Robot Scheduling On Paths

In this paper, we address the problem of scheduling a set of robots to complete tasks in a laboratory environment, modelled as a graph, while avoiding collisions. We analyze the dynamic programming algorithm (PA) introduced in arXiv:2402.12019 and present three baselines for comparison: an integer programming approach (IP) that always yields an optimal solution, a greedy algorithm (GA), and a simple randomized algorithm (RA). We show that for a path graph, PA, GA, and RA find solutions several orders of magnitude faster than IP (the optimal baseline), with PA returning optimal results in the vast majority of cases. Our scaled experiments comparing non-optimal algorithms show that the average schedule timespan produced by PA is less than half that of RA and GA. This outperformance is consistent across varying path lengths, task durations and distributions, number and allocations of tasks and robots, and task-to-robot ratios. This work serves two purposes. First, we present three algorithms for scheduling on line graphs, including a novel integer programming formulation for finding optimal solutions. Second, we demonstrate that PA produces near-optimal schedules that outperform all non-optimal baselines while maintaining a comparable runtime. Code is available at https://github.com/sea26-robots/code.

cs.RO↗

arXiv2021

Asymptotics of the number of 2-threshold functions

A $k$-threshold function on a rectangular grid of size $m \times n$ is the conjunction of $k$ threshold functions on the same domain. In this paper, we focus on the case $k=2$ and show that the number of two-dimensional 2-threshold functions is~$\dfrac{25}{12π^4} m^4 n^4 + o(m^4n^4)$.

math.CO↗

arXiv2020

A characterization of 2-threshold functions via pairs of prime segments

A $\{0,1\}$-valued function on a two-dimensional rectangular grid is called threshold if its sets of zeros and ones are separable by a straight line. In this paper we study 2-threshold functions, i.e. functions representable as the conjunction of two threshold functions. We provide a characterization of 2-threshold functions by pairs of oriented prime segments, where each such segment is defined by an ordered pair of adjacent integer points.

math.CO↗

arXiv2018

Linear read-once and related Boolean functions

It is known that a positive Boolean function f depending on n variables has at least n + 1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n + 1 extremal points if and only if it is linear read-once. The class of linear read-once functions is known to be the intersection of the classes of read-once and threshold functions. Generalizing this result we show that the class of linear read-once functions is the intersection of read-once and Chow functions. We also find the set of minimal read-once functions which are not linear read-once and the set of minimal threshold functions which are not linear read-once. In other words, we characterize the class of linear read-once functions by means of minimal forbidden subfunctions within the universe of read-once and the universe of threshold functions. Within the universe of threshold functions the importance of linear read-once func- tions is due to the fact that they attain the minimum value of the specification number, which is n + 1 for functions depending on n variables. In 1995 Anthony et al. conjec- tured that for all other threshold functions the specification number is strictly greater than n + 1. We disprove this conjecture by exhibiting a threshold non-linear read-once function depending on n variables whose specification number is n + 1.

math.CO↗

arXiv2017

Specifying a positive threshold function via extremal points

An extremal point of a positive threshold Boolean function $f$ is either a maximal zero or a minimal one. It is known that if $f$ depends on all its variables, then the set of its extremal points completely specifies $f$ within the universe of threshold functions. However, in some cases, $f$ can be specified by a smaller set. The minimum number of points in such a set is the specification number of $f$. It was shown in [S.-T. Hu. Threshold Logic, 1965] that the specification number of a threshold function of $n$ variables is at least $n+1$. In [M. Anthony, G. Brightwell, and J. Shawe-Taylor. On specifying Boolean functions by labelled examples. Discrete Applied Mathematics, 1995] it was proved that this bound is attained for nested functions and conjectured that for all other threshold functions the specification number is strictly greater than $n+1$. In the present paper, we resolve this conjecture negatively by exhibiting threshold Boolean functions of $n$ variables, which are non-nested and for which the specification number is $n+1$. On the other hand, we show that the set of extremal points satisfies the statement of the conjecture, i.e., a positive threshold Boolean function depending on all its $n$ variables has $n+1$ extremal points if and only if it is nested. To prove this, we reveal an underlying structure of the set of extremal points.

math.CO↗

arXiv2016

On teaching sets of k-threshold functions

Let $f$ be a $\{0,1\}$-valued function over an integer $d$-dimensional cube $\{0,1,\dots,n-1\}^d$, for $n \geq 2$ and $d \geq 1$. The function $f$ is called threshold if there exists a hyperplane which separates $0$-valued points from $1$-valued points. Let $C$ be a class of functions and $f \in C$. A point $x$ is essential for the function $f$ with respect to $C$ if there exists a function $g \in C$ such that $x$ is a unique point on which $f$ differs from $g$. A set of points $X$ is called teaching for the function $f$ with respect to $C$ if no function in $C \setminus \{f\}$ agrees with $f$ on $X$. It is known that any threshold function has a unique minimal teaching set, which coincides with the set of its essential points. In this paper we study teaching sets of $k$-threshold functions, i.e. functions that can be represented as a conjunction of $k$ threshold functions. We reveal a connection between essential points of $k$ threshold functions and essential points of the corresponding $k$-threshold function. We note that, in general, a $k$-threshold function is not specified by its essential points and can have more than one minimal teaching set. We show that for $d=2$ the number of minimal teaching sets for a 2-threshold function can grow as $Ω(n^2)$. We also consider the class of polytopes with vertices in the $d$-dimensional cube. Each polytope from this class can be defined by a $k$-threshold function for some $k$. In terms of $k$-threshold functions we prove that a polytope with vertices in the $d$-dimensional cube has a unique minimal teaching set which is equal to the set of its essential points. For $d=2$ we describe structure of the minimal teaching set of a polytope and show that cardinality of this set is either $Θ(n^2)$ or $O(n)$ and depends on the perimeter and the minimum angle of the polytope.

math.CO↗