3,206 research outputs found
Algorithms for Highly Symmetric Linear and Integer Programs
This paper deals with exploiting symmetry for solving linear and integer
programming problems. Basic properties of linear representations of finite
groups can be used to reduce symmetric linear programming to solving linear
programs of lower dimension. Combining this approach with knowledge of the
geometry of feasible integer solutions yields an algorithm for solving highly
symmetric integer linear programs which only takes time which is linear in the
number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title
slightly change
The Mathematical Universe
I explore physics implications of the External Reality Hypothesis (ERH) that
there exists an external physical reality completely independent of us humans.
I argue that with a sufficiently broad definition of mathematics, it implies
the Mathematical Universe Hypothesis (MUH) that our physical world is an
abstract mathematical structure. I discuss various implications of the ERH and
MUH, ranging from standard physics topics like symmetries, irreducible
representations, units, free parameters, randomness and initial conditions to
broader issues like consciousness, parallel universes and Godel incompleteness.
I hypothesize that only computable and decidable (in Godel's sense) structures
exist, which alleviates the cosmological measure problem and help explain why
our physical laws appear so simple. I also comment on the intimate relation
between mathematical structures, computations, simulations and physical
systems.Comment: Replaced to match accepted Found. Phys. version, 31 pages, 5 figs;
more details at http://space.mit.edu/home/tegmark/toe.htm
Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations
In this self-contained paper, we present a theory of the piecewise linear
minimal valid functions for the 1-row Gomory-Johnson infinite group problem.
The non-extreme minimal valid functions are those that admit effective
perturbations. We give a precise description of the space of these
perturbations as a direct sum of certain finite- and infinite-dimensional
subspaces. The infinite-dimensional subspaces have partial symmetries; to
describe them, we develop a theory of inverse semigroups of partial bijections,
interacting with the functional equations satisfied by the perturbations. Our
paper provides the foundation for grid-free algorithms for the Gomory-Johnson
model, in particular for testing extremality of piecewise linear functions
whose breakpoints are rational numbers with huge denominators.Comment: 67 pages, 21 figures; v2: changes to sections 10.2-10.3, improved
figures; v3: additional figures and minor updates, add reference to IPCO
abstract. CC-BY-S
Taming Numbers and Durations in the Model Checking Integrated Planning System
The Model Checking Integrated Planning System (MIPS) is a temporal least
commitment heuristic search planner based on a flexible object-oriented
workbench architecture. Its design clearly separates explicit and symbolic
directed exploration algorithms from the set of on-line and off-line computed
estimates and associated data structures. MIPS has shown distinguished
performance in the last two international planning competitions. In the last
event the description language was extended from pure propositional planning to
include numerical state variables, action durations, and plan quality objective
functions. Plans were no longer sequences of actions but time-stamped
schedules. As a participant of the fully automated track of the competition,
MIPS has proven to be a general system; in each track and every benchmark
domain it efficiently computed plans of remarkable quality. This article
introduces and analyzes the most important algorithmic novelties that were
necessary to tackle the new layers of expressiveness in the benchmark problems
and to achieve a high level of performance. The extensions include critical
path analysis of sequentially generated plans to generate corresponding optimal
parallel plans. The linear time algorithm to compute the parallel plan bypasses
known NP hardness results for partial ordering by scheduling plans with respect
to the set of actions and the imposed precedence relations. The efficiency of
this algorithm also allows us to improve the exploration guidance: for each
encountered planning state the corresponding approximate sequential plan is
scheduled. One major strength of MIPS is its static analysis phase that grounds
and simplifies parameterized predicates, functions and operators, that infers
knowledge to minimize the state description length, and that detects domain
object symmetries. The latter aspect is analyzed in detail. MIPS has been
developed to serve as a complete and optimal state space planner, with
admissible estimates, exploration engines and branching cuts. In the
competition version, however, certain performance compromises had to be made,
including floating point arithmetic, weighted heuristic search exploration
according to an inadmissible estimate and parameterized optimization
Generalizing Boolean Satisfiability I: Background and Survey of Existing Work
This is the first of three planned papers describing ZAP, a satisfiability
engine that substantially generalizes existing tools while retaining the
performance characteristics of modern high-performance solvers. The fundamental
idea underlying ZAP is that many problems passed to such engines contain rich
internal structure that is obscured by the Boolean representation used; our
goal is to define a representation in which this structure is apparent and can
easily be exploited to improve computational performance. This paper is a
survey of the work underlying ZAP, and discusses previous attempts to improve
the performance of the Davis-Putnam-Logemann-Loveland algorithm by exploiting
the structure of the problem being solved. We examine existing ideas including
extensions of the Boolean language to allow cardinality constraints,
pseudo-Boolean representations, symmetry, and a limited form of quantification.
While this paper is intended as a survey, our research results are contained in
the two subsequent articles, with the theoretical structure of ZAP described in
the second paper in this series, and ZAP's implementation described in the
third
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