62 research outputs found
Critical branching processes in digital memcomputing machines
Memcomputing is a novel computing paradigm that employs time non-locality
(memory) to solve combinatorial optimization problems. It can be realized in
practice by means of non-linear dynamical systems whose point attractors
represent the solutions of the original problem. It has been previously shown
that during the solution search digital memcomputing machines go through a
transient phase of avalanches (instantons) that promote dynamical long-range
order. By employing mean-field arguments we predict that the distribution of
the avalanche sizes follows a Borel distribution typical of critical branching
processes with exponent . We corroborate this analysis by solving
various random 3-SAT instances of the Boolean satisfiability problem. The
numerical results indicate a power-law distribution with exponent , in very good agreement with the mean-field analysis. This indicates
that memcomputing machines self-tune to a critical state in which avalanches
are characterized by a branching process, and that this state persists across
the majority of their evolution.Comment: 5 pages, 3 figure
MemComputing Integer Linear Programming
Integer linear programming (ILP) encompasses a very important class of
optimization problems that are of great interest to both academia and industry.
Several algorithms are available that attempt to explore the solution space of
this class efficiently, while requiring a reasonable compute time. However,
although these algorithms have reached various degrees of success over the
years, they still face considerable challenges when confronted with
particularly hard problem instances, such as those of the MIPLIB 2010 library.
In this work we propose a radically different non-algorithmic approach to ILP
based on a novel physics-inspired computing paradigm: Memcomputing. This
paradigm is based on digital (hence scalable) machines represented by
appropriate electrical circuits with memory. These machines can be either built
in hardware or, as we do here, their equations of motion can be efficiently
simulated on our traditional computers. We first describe a new circuit
architecture of memcomputing machines specifically designed to solve for the
linear inequalities representing a general ILP problem. We call these
self-organizing algebraic circuits, since they self-organize dynamically to
satisfy the correct (algebraic) linear inequalities. We then show simulations
of these machines using MATLAB running on a single core of a Xeon processor for
several ILP benchmark problems taken from the MIPLIB 2010 library, and compare
our results against a renowned commercial solver. We show that our approach is
very efficient when dealing with these hard problems. In particular, we find
within minutes feasible solutions for one of these hard problems (f2000 from
MIPLIB 2010) whose feasibility, to the best of our knowledge, has remained
unknown for the past eight years
Self-Averaging of Digital MemComputing Machines
Digital MemComputing machines (DMMs) are a new class of computing machines
that employ non-quantum dynamical systems with memory to solve combinatorial
optimization problems. Here, we show that the time to solution (TTS) of DMMs
follows an inverse Gaussian distribution, with the TTS self-averaging with
increasing problem size, irrespective of the problem they solve. We provide
both an analytical understanding of this phenomenon and numerical evidence by
solving instances of the 3-SAT (satisfiability) problem. The self-averaging
property of DMMs with problem size implies that they are increasingly
insensitive to the detailed features of the instances they solve. This is in
sharp contrast to traditional algorithms applied to the same problems,
illustrating another advantage of this physics-based approach to computation.Comment: 9 pages, 13 figure
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