34 research outputs found
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
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
Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks
Recent work on quantum annealing has emphasized the role of collective
behavior in solving optimization problems. By enabling transitions of clusters
of variables, such solvers are able to navigate their state space and locate
solutions more efficiently despite having only local connections between
elements. However, collective behavior is not exclusive to quantum annealers,
and classical solvers that display collective dynamics should also possess an
advantage in navigating a non-convex landscape. Here, we give evidence that a
benchmark derived from quantum annealing studies is solvable in polynomial time
using digital memcomputing machines, which utilize a collection of dynamical
components with memory to represent the structure of the underlying
optimization problem. To illustrate the role of memory and clarify the
structure of these solvers we propose a simple model of these machines that
demonstrates the emergence of long-range order. This model, when applied to
finding the ground state of the Ising frustrated-loop benchmarks, undergoes a
transient phase of avalanches which can span the entire lattice and
demonstrates a connection between long-range behavior and their probability of
success. These results establish the advantages of computational approaches
based on collective dynamics of continuous dynamical systems
Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states
Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information on the same physical platform. It was recently proven mathematically that memcomputing machines have the same computational power of nondeterministic Turing machines. Therefore, they can solve NP-complete problems in polynomial time and, using the appropriate architecture, with resources that only grow polynomially with the input size. The reason for this computational power stems from properties inspired by the brain and shared by any universal memcomputing machine, in particular intrinsic parallelism and information overhead, namely, the capability of compressing information in the collective state of the memprocessor network. We show an experimental demonstration of an actual memcomputing architecture that solves the NP-complete version of the subset sum problem in only one step and is composed of a number of memprocessors that scales linearly with the size of the problem. We have fabricated this architecture using standard microelectronic technology so that it can be easily realized in any laboratory setting. Although the particular machine presented here is eventually limited by noise—and will thus require error-correcting codes to scale to an arbitrary number of memprocessors—it represents the first proof of concept of a machine capable of working with the collective state of interacting memory cells, unlike the present-day single-state machines built using the von Neumann architecture