49 research outputs found
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Hybrid VCSPs with crisp and conservative valued templates
A constraint satisfaction problem (CSP) is a problem of computing a
homomorphism between two relational
structures. Analyzing its complexity has been a very fruitful research
direction, especially for fixed template CSPs, denoted , in
which the right side structure is fixed and the left side
structure is unconstrained.
Recently, the hybrid setting, written ,
where both sides are restricted simultaneously, attracted some attention. It
assumes that is taken from a class of relational structures
that additionally is closed under inverse homomorphisms. The last
property allows to exploit algebraic tools that have been developed for fixed
template CSPs. The key concept that connects hybrid CSPs with fixed-template
CSPs is the so called "lifted language". Namely, this is a constraint language
that can be constructed from an input . The
tractability of that language for any input is a
necessary condition for the tractability of the hybrid problem.
In the first part we investigate templates for which the
latter condition is not only necessary, but also is sufficient. We call such
templates widely tractable. For this purpose, we construct from
a new finite relational structure and define
as a class of structures homomorphic to . We
prove that wide tractability is equivalent to the tractability of
. Our proof is based on the key observation
that is homomorphic to if and only if the core of
is preserved by a Siggers polymorphism. Analogous
result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706
Statistical Physics of Hard Optimization Problems
Optimization is fundamental in many areas of science, from computer science
and information theory to engineering and statistical physics, as well as to
biology or social sciences. It typically involves a large number of variables
and a cost function depending on these variables. Optimization problems in the
NP-complete class are particularly difficult, it is believed that the number of
operations required to minimize the cost function is in the most difficult
cases exponential in the system size. However, even in an NP-complete problem
the practically arising instances might, in fact, be easy to solve. The
principal question we address in this thesis is: How to recognize if an
NP-complete constraint satisfaction problem is typically hard and what are the
main reasons for this? We adopt approaches from the statistical physics of
disordered systems, in particular the cavity method developed originally to
describe glassy systems. We describe new properties of the space of solutions
in two of the most studied constraint satisfaction problems - random
satisfiability and random graph coloring. We suggest a relation between the
existence of the so-called frozen variables and the algorithmic hardness of a
problem. Based on these insights, we introduce a new class of problems which we
named "locked" constraint satisfaction, where the statistical description is
easily solvable, but from the algorithmic point of view they are even more
challenging than the canonical satisfiability.Comment: PhD thesi
Logic Column 17: A Rendezvous of Logic, Complexity, and Algebra
This article surveys recent advances in applying algebraic techniques to
constraint satisfaction problems.Comment: 30 page